Merge pull request #5425

87bddb7 Squashed 'src/secp256k1/' changes from ad2028f..b0210a9 (Pieter Wuille)
ffe3291 Remove gmp dependency and doc mentions (Pieter Wuille)

30 files changed, 1241 insertions, 773 deletions

diff --git a/src/secp256k1/.travis.yml b/src/secp256k1/.travis.yml
index 24a86b561b..3a85e8cba0 100644
--- a/src/secp256k1/.travis.yml
+++ b/src/secp256k1/.travis.yml
@@ -18,6 +18,8 @@ env:
     - FIELD=64bit     ENDOMORPHISM=yes
     - FIELD=32bit
     - FIELD=32bit     ENDOMORPHISM=yes
+    - BIGNUM=none
+    - BIGNUM=none     ENDOMORPHISM=yes
     - BUILD=distcheck
     - EXTRAFLAGS=CFLAGS=-DDETERMINISTIC
 before_script: ./autogen.sh
diff --git a/src/secp256k1/Makefile.am b/src/secp256k1/Makefile.am
index d527da6b77..dbf1790f34 100644
--- a/src/secp256k1/Makefile.am
+++ b/src/secp256k1/Makefile.am
@@ -68,12 +68,13 @@ bench_sign_LDFLAGS = -static
 bench_inv_SOURCES = src/bench_inv.c
 bench_inv_LDADD = $(COMMON_LIB) $(SECP_LIBS)
 bench_inv_LDFLAGS = -static
+bench_inv_CPPFLAGS = $(SECP_INCLUDES)
 endif
 
 if USE_TESTS
 noinst_PROGRAMS += tests
 tests_SOURCES = src/tests.c
-tests_CPPFLAGS = -DVERIFY $(SECP_TEST_INCLUDES)
+tests_CPPFLAGS = -DVERIFY $(SECP_INCLUDES) $(SECP_TEST_INCLUDES)
 tests_LDADD = $(COMMON_LIB) $(SECP_LIBS) $(SECP_TEST_LIBS)
 tests_LDFLAGS = -static
 TESTS = tests
diff --git a/src/secp256k1/build-aux/m4/bitcoin_secp.m4 b/src/secp256k1/build-aux/m4/bitcoin_secp.m4
index e6f3470ed7..4ca28f99cf 100644
--- a/src/secp256k1/build-aux/m4/bitcoin_secp.m4
+++ b/src/secp256k1/build-aux/m4/bitcoin_secp.m4
@@ -78,7 +78,13 @@ fi
 dnl
 AC_DEFUN([SECP_GMP_CHECK],[
 if test x"$has_gmp" != x"yes"; then
-  AC_CHECK_HEADER(gmp.h,[AC_CHECK_LIB(gmp, __gmpz_init,[has_gmp=yes; GMP_LIBS=-lgmp; AC_DEFINE(HAVE_LIBGMP,1,[Define this symbol if libgmp is installed])])])
+  CPPFLAGS_TEMP="$CPPFLAGS"
+  CPPFLAGS="$GMP_CPPFLAGS $CPPFLAGS"
+  LIBS_TEMP="$LIBS"
+  LIBS="$GMP_LIBS $LIBS"
+  AC_CHECK_HEADER(gmp.h,[AC_CHECK_LIB(gmp, __gmpz_init,[has_gmp=yes; GMP_LIBS="$GMP_LIBS -lgmp"; AC_DEFINE(HAVE_LIBGMP,1,[Define this symbol if libgmp is installed])])])
+  CPPFLAGS="$CPPFLAGS_TEMP"
+  LIBS="$LIBS_TEMP"
 fi
 if test x"$set_field" = x"gmp" && test x"$has_gmp" != x"yes"; then
     AC_MSG_ERROR([$set_field field support explicitly requested but libgmp was not found])
diff --git a/src/secp256k1/configure.ac b/src/secp256k1/configure.ac
index 2da5709834..6e6fccd7fd 100644
--- a/src/secp256k1/configure.ac
+++ b/src/secp256k1/configure.ac
@@ -33,10 +33,35 @@ case $host in
 esac
 
 case $host_os in
-  darwin*)
-     CPPFLAGS="$CPPFLAGS -I/opt/local/include"
-     LDFLAGS="$LDFLAGS -L/opt/local/lib"
-     ;;
+  *darwin*)
+     if  test x$cross_compiling != xyes; then
+       AC_PATH_PROG([BREW],brew,)
+       if test x$BREW != x; then
+         dnl These Homebrew packages may be keg-only, meaning that they won't be found
+         dnl in expected paths because they may conflict with system files. Ask
+         dnl Homebrew where each one is located, then adjust paths accordingly.
+
+         openssl_prefix=`$BREW --prefix openssl 2>/dev/null`
+         gmp_prefix=`$BREW --prefix gmp 2>/dev/null`
+         if test x$openssl_prefix != x; then
+           PKG_CONFIG_PATH="$openssl_prefix/lib/pkgconfig:$PKG_CONFIG_PATH"
+           export PKG_CONFIG_PATH
+         fi
+         if test x$gmp_prefix != x; then
+           GMP_CPPFLAGS="-I$gmp_prefix/include"
+           GMP_LIBS="-L$gmp_prefix/lib"
+         fi
+       else
+         AC_PATH_PROG([PORT],port,)
+         dnl if homebrew isn't installed and macports is, add the macports default paths
+         dnl as a last resort.
+         if test x$PORT != x; then
+           CPPFLAGS="$CPPFLAGS -isystem /opt/local/include"
+           LDFLAGS="$LDFLAGS -L/opt/local/lib"
+         fi
+       fi
+     fi
+   ;;
 esac
 
 CFLAGS="$CFLAGS -W"
@@ -70,7 +95,7 @@ AC_ARG_ENABLE(endomorphism,
 AC_ARG_WITH([field], [AS_HELP_STRING([--with-field=gmp|64bit|64bit_asm|32bit|auto],
 [Specify Field Implementation. Default is auto])],[req_field=$withval], [req_field=auto])
 
-AC_ARG_WITH([bignum], [AS_HELP_STRING([--with-bignum=gmp|auto],
+AC_ARG_WITH([bignum], [AS_HELP_STRING([--with-bignum=gmp|none|auto],
 [Specify Bignum Implementation. Default is auto])],[req_bignum=$withval], [req_bignum=auto])
 
 AC_ARG_WITH([scalar], [AS_HELP_STRING([--with-scalar=64bit|32bit|auto],
@@ -154,7 +179,7 @@ if test x"$req_bignum" = x"auto"; then
   fi
 
   if test x"$set_bignum" = x; then
-    AC_MSG_ERROR([no working bignum implementation found])
+    set_bignum=none
   fi
 else
   set_bignum=$req_bignum
@@ -162,8 +187,7 @@ else
   gmp)
     SECP_GMP_CHECK
     ;;
-  openssl)
-    SECP_OPENSSL_CHECK
+  none)
     ;;
   *)
     AC_MSG_ERROR([invalid bignum implementation selection])
@@ -196,9 +220,15 @@ esac
 # select bignum implementation
 case $set_bignum in
 gmp)
-  AC_DEFINE(HAVE_LIBGMP,1,[Define this symbol if libgmp is installed])
-  AC_DEFINE(USE_NUM_GMP, 1, [Define this symbol to use the gmp implementation])
-  AC_DEFINE(USE_FIELD_INV_NUM, 1, [Define this symbol to use the USE_FIELD_INV_NUM implementation])
+  AC_DEFINE(HAVE_LIBGMP, 1, [Define this symbol if libgmp is installed])
+  AC_DEFINE(USE_NUM_GMP, 1, [Define this symbol to use the gmp implementation for num])
+  AC_DEFINE(USE_FIELD_INV_NUM, 1, [Define this symbol to use the num-based field inverse implementation])
+  AC_DEFINE(USE_SCALAR_INV_NUM, 1, [Define this symbol to use the num-based scalar inverse implementation])
+  ;;
+none)
+  AC_DEFINE(USE_NUM_NONE, 1, [Define this symbol to use no num implementation])
+  AC_DEFINE(USE_FIELD_INV_BUILTIN, 1, [Define this symbol to use the native field inverse implementation])
+  AC_DEFINE(USE_SCALAR_INV_BUILTIN, 1, [Define this symbol to use the native scalar inverse implementation])
   ;;
 *)
   AC_MSG_ERROR([invalid bignum implementation])
@@ -236,10 +266,11 @@ fi
 
 if test x"$set_field" = x"gmp" || test x"$set_bignum" = x"gmp"; then
   SECP_LIBS="$SECP_LIBS $GMP_LIBS"
+  SECP_INCLUDES="$SECP_INCLUDES $GMP_CPPFLAGS"
 fi
 
 if test x"$use_endomorphism" = x"yes"; then
-  AC_DEFINE(USE_ENDOMORPHISM, 1, [Define this symbol to use endomorphism])
+  AC_DEFINE(USE_ENDOMORPHISM, 1, [Define this symbol to use endomorphism optimization])
 fi
 
 AC_MSG_NOTICE([Using field implementation: $set_field])
@@ -256,4 +287,10 @@ AC_SUBST(YASM_BINFMT)
 AM_CONDITIONAL([USE_ASM], [test x"$set_field" == x"64bit_asm"])
 AM_CONDITIONAL([USE_TESTS], [test x"$use_tests" != x"no"])
 AM_CONDITIONAL([USE_BENCHMARK], [test x"$use_benchmark" != x"no"])
+
+dnl make sure nothing new is exported so that we don't break the cache
+PKGCONFIG_PATH_TEMP="$PKG_CONFIG_PATH"
+unset PKG_CONFIG_PATH
+PKG_CONFIG_PATH="$PKGCONFIG_PATH_TEMP"
+
 AC_OUTPUT
diff --git a/src/secp256k1/include/secp256k1.h b/src/secp256k1/include/secp256k1.h
index 932bf0279f..94a6ef483f 100644
--- a/src/secp256k1/include/secp256k1.h
+++ b/src/secp256k1/include/secp256k1.h
@@ -15,18 +15,6 @@ extern "C" {
 # endif
 
 # if (!defined(__STDC_VERSION__) || (__STDC_VERSION__ < 199901L) )
-#  if SECP256K1_GNUC_PREREQ(3,0)
-#   define SECP256K1_RESTRICT __restrict__
-#  elif (defined(_MSC_VER) && _MSC_VER >= 1400)
-#   define SECP256K1_RESTRICT __restrict
-#  else
-#   define SECP256K1_RESTRICT
-#  endif
-# else
-#  define SECP256K1_RESTRICT restrict
-# endif
-
-# if (!defined(__STDC_VERSION__) || (__STDC_VERSION__ < 199901L) )
 #  if SECP256K1_GNUC_PREREQ(2,7)
 #   define SECP256K1_INLINE __inline__
 #  elif (defined(_MSC_VER))
diff --git a/src/secp256k1/src/ecdsa.h b/src/secp256k1/src/ecdsa.h
index 3b1e0484ea..5fc5230c36 100644
--- a/src/secp256k1/src/ecdsa.h
+++ b/src/secp256k1/src/ecdsa.h
@@ -7,17 +7,21 @@
 #ifndef _SECP256K1_ECDSA_
 #define _SECP256K1_ECDSA_
 
-#include "num.h"
+#include "scalar.h"
+#include "group.h"
+
+static void secp256k1_ecsda_start(void);
+static void secp256k1_ecdsa_stop(void);
 
 typedef struct {
-    secp256k1_num_t r, s;
+    secp256k1_scalar_t r, s;
 } secp256k1_ecdsa_sig_t;
 
 static int secp256k1_ecdsa_sig_parse(secp256k1_ecdsa_sig_t *r, const unsigned char *sig, int size);
 static int secp256k1_ecdsa_sig_serialize(unsigned char *sig, int *size, const secp256k1_ecdsa_sig_t *a);
-static int secp256k1_ecdsa_sig_verify(const secp256k1_ecdsa_sig_t *sig, const secp256k1_ge_t *pubkey, const secp256k1_num_t *message);
+static int secp256k1_ecdsa_sig_verify(const secp256k1_ecdsa_sig_t *sig, const secp256k1_ge_t *pubkey, const secp256k1_scalar_t *message);
 static int secp256k1_ecdsa_sig_sign(secp256k1_ecdsa_sig_t *sig, const secp256k1_scalar_t *seckey, const secp256k1_scalar_t *message, const secp256k1_scalar_t *nonce, int *recid);
-static int secp256k1_ecdsa_sig_recover(const secp256k1_ecdsa_sig_t *sig, secp256k1_ge_t *pubkey, const secp256k1_num_t *message, int recid);
-static void secp256k1_ecdsa_sig_set_rs(secp256k1_ecdsa_sig_t *sig, const secp256k1_num_t *r, const secp256k1_num_t *s);
+static int secp256k1_ecdsa_sig_recover(const secp256k1_ecdsa_sig_t *sig, secp256k1_ge_t *pubkey, const secp256k1_scalar_t *message, int recid);
+static void secp256k1_ecdsa_sig_set_rs(secp256k1_ecdsa_sig_t *sig, const secp256k1_scalar_t *r, const secp256k1_scalar_t *s);
 
 #endif
diff --git a/src/secp256k1/src/ecdsa_impl.h b/src/secp256k1/src/ecdsa_impl.h
index 4c05ec39f8..a951d0b4ad 100644
--- a/src/secp256k1/src/ecdsa_impl.h
+++ b/src/secp256k1/src/ecdsa_impl.h
@@ -8,13 +8,51 @@
 #ifndef _SECP256K1_ECDSA_IMPL_H_
 #define _SECP256K1_ECDSA_IMPL_H_
 
-#include "num.h"
+#include "scalar.h"
 #include "field.h"
 #include "group.h"
 #include "ecmult.h"
 #include "ecmult_gen.h"
 #include "ecdsa.h"
 
+typedef struct {
+    secp256k1_fe_t order_as_fe;
+    secp256k1_fe_t p_minus_order;
+} secp256k1_ecdsa_consts_t;
+
+static const secp256k1_ecdsa_consts_t *secp256k1_ecdsa_consts = NULL;
+
+static void secp256k1_ecdsa_start(void) {
+    if (secp256k1_ecdsa_consts != NULL)
+        return;
+
+    /* Allocate. */
+    secp256k1_ecdsa_consts_t *ret = (secp256k1_ecdsa_consts_t*)malloc(sizeof(secp256k1_ecdsa_consts_t));
+
+    static const unsigned char order[] = {
+        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
+        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,
+        0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B,
+        0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41
+    };
+
+    secp256k1_fe_set_b32(&ret->order_as_fe, order);
+    secp256k1_fe_negate(&ret->p_minus_order, &ret->order_as_fe, 1);
+    secp256k1_fe_normalize(&ret->p_minus_order);
+
+    /* Set the global pointer. */
+    secp256k1_ecdsa_consts = ret;
+}
+
+static void secp256k1_ecdsa_stop(void) {
+    if (secp256k1_ecdsa_consts == NULL)
+        return;
+
+    secp256k1_ecdsa_consts_t *c = (secp256k1_ecdsa_consts_t*)secp256k1_ecdsa_consts;
+    secp256k1_ecdsa_consts = NULL;
+    free(c);
+}
+
 static int secp256k1_ecdsa_sig_parse(secp256k1_ecdsa_sig_t *r, const unsigned char *sig, int size) {
     if (sig[0] != 0x30) return 0;
     int lenr = sig[3];
@@ -26,18 +64,37 @@ static int secp256k1_ecdsa_sig_parse(secp256k1_ecdsa_sig_t *r, const unsigned ch
     if (lenr == 0) return 0;
     if (sig[lenr+4] != 0x02) return 0;
     if (lens == 0) return 0;
-    secp256k1_num_set_bin(&r->r, sig+4, lenr);
-    secp256k1_num_set_bin(&r->s, sig+6+lenr, lens);
+    const unsigned char *sp = sig + 6 + lenr;
+    while (lens > 0 && sp[0] == 0) {
+        lens--;
+        sp++;
+    }
+    if (lens > 32) return 0;
+    const unsigned char *rp = sig + 4;
+    while (lenr > 0 && rp[0] == 0) {
+        lenr--;
+        rp++;
+    }
+    if (lenr > 32) return 0;
+    unsigned char ra[32] = {0}, sa[32] = {0};
+    memcpy(ra + 32 - lenr, rp, lenr);
+    memcpy(sa + 32 - lens, sp, lens);
+    int overflow = 0;
+    secp256k1_scalar_set_b32(&r->r, ra, &overflow);
+    if (overflow) return 0;
+    secp256k1_scalar_set_b32(&r->s, sa, &overflow);
+    if (overflow) return 0;
     return 1;
 }
 
 static int secp256k1_ecdsa_sig_serialize(unsigned char *sig, int *size, const secp256k1_ecdsa_sig_t *a) {
-    int lenR = (secp256k1_num_bits(&a->r) + 7)/8;
-    if (lenR == 0 || secp256k1_num_get_bit(&a->r, lenR*8-1))
-        lenR++;
-    int lenS = (secp256k1_num_bits(&a->s) + 7)/8;
-    if (lenS == 0 || secp256k1_num_get_bit(&a->s, lenS*8-1))
-        lenS++;
+    unsigned char r[33] = {0}, s[33] = {0};
+    secp256k1_scalar_get_b32(&r[1], &a->r);
+    secp256k1_scalar_get_b32(&s[1], &a->s);
+    unsigned char *rp = r, *sp = s;
+    int lenR = 33, lenS = 33;
+    while (lenR > 1 && rp[0] == 0 && rp[1] < 0x80) { lenR--; rp++; }
+    while (lenS > 1 && sp[0] == 0 && sp[1] < 0x80) { lenS--; sp++; }
     if (*size < 6+lenS+lenR)
         return 0;
     *size = 6 + lenS + lenR;
@@ -45,98 +102,67 @@ static int secp256k1_ecdsa_sig_serialize(unsigned char *sig, int *size, const se
     sig[1] = 4 + lenS + lenR;
     sig[2] = 0x02;
     sig[3] = lenR;
-    secp256k1_num_get_bin(sig+4, lenR, &a->r);
+    memcpy(sig+4, rp, lenR);
     sig[4+lenR] = 0x02;
     sig[5+lenR] = lenS;
-    secp256k1_num_get_bin(sig+lenR+6, lenS, &a->s);
+    memcpy(sig+lenR+6, sp, lenS);
     return 1;
 }
 
-static int secp256k1_ecdsa_sig_recompute(secp256k1_num_t *r2, const secp256k1_ecdsa_sig_t *sig, const secp256k1_ge_t *pubkey, const secp256k1_num_t *message) {
-    const secp256k1_ge_consts_t *c = secp256k1_ge_consts;
-
-    if (secp256k1_num_is_neg(&sig->r) || secp256k1_num_is_neg(&sig->s))
-        return 0;
-    if (secp256k1_num_is_zero(&sig->r) || secp256k1_num_is_zero(&sig->s))
-        return 0;
-    if (secp256k1_num_cmp(&sig->r, &c->order) >= 0 || secp256k1_num_cmp(&sig->s, &c->order) >= 0)
+static int secp256k1_ecdsa_sig_recompute(secp256k1_scalar_t *r2, const secp256k1_ecdsa_sig_t *sig, const secp256k1_ge_t *pubkey, const secp256k1_scalar_t *message) {
+    if (secp256k1_scalar_is_zero(&sig->r) || secp256k1_scalar_is_zero(&sig->s))
         return 0;
 
     int ret = 0;
-    secp256k1_num_t sn, u1, u2;
-    secp256k1_num_init(&sn);
-    secp256k1_num_init(&u1);
-    secp256k1_num_init(&u2);
-    secp256k1_num_mod_inverse(&sn, &sig->s, &c->order);
-    secp256k1_num_mod_mul(&u1, &sn, message, &c->order);
-    secp256k1_num_mod_mul(&u2, &sn, &sig->r, &c->order);
+    secp256k1_scalar_t sn, u1, u2;
+    secp256k1_scalar_inverse_var(&sn, &sig->s);
+    secp256k1_scalar_mul(&u1, &sn, message);
+    secp256k1_scalar_mul(&u2, &sn, &sig->r);
     secp256k1_gej_t pubkeyj; secp256k1_gej_set_ge(&pubkeyj, pubkey);
     secp256k1_gej_t pr; secp256k1_ecmult(&pr, &pubkeyj, &u2, &u1);
     if (!secp256k1_gej_is_infinity(&pr)) {
         secp256k1_fe_t xr; secp256k1_gej_get_x_var(&xr, &pr);
         secp256k1_fe_normalize(&xr);
         unsigned char xrb[32]; secp256k1_fe_get_b32(xrb, &xr);
-        secp256k1_num_set_bin(r2, xrb, 32);
-        secp256k1_num_mod(r2, &c->order);
+        secp256k1_scalar_set_b32(r2, xrb, NULL);
         ret = 1;
     }
-    secp256k1_num_free(&sn);
-    secp256k1_num_free(&u1);
-    secp256k1_num_free(&u2);
     return ret;
 }
 
-static int secp256k1_ecdsa_sig_recover(const secp256k1_ecdsa_sig_t *sig, secp256k1_ge_t *pubkey, const secp256k1_num_t *message, int recid) {
-    const secp256k1_ge_consts_t *c = secp256k1_ge_consts;
-
-    if (secp256k1_num_is_neg(&sig->r) || secp256k1_num_is_neg(&sig->s))
-        return 0;
-    if (secp256k1_num_is_zero(&sig->r) || secp256k1_num_is_zero(&sig->s))
-        return 0;
-    if (secp256k1_num_cmp(&sig->r, &c->order) >= 0 || secp256k1_num_cmp(&sig->s, &c->order) >= 0)
+static int secp256k1_ecdsa_sig_recover(const secp256k1_ecdsa_sig_t *sig, secp256k1_ge_t *pubkey, const secp256k1_scalar_t *message, int recid) {
+    if (secp256k1_scalar_is_zero(&sig->r) || secp256k1_scalar_is_zero(&sig->s))
         return 0;
 
-    secp256k1_num_t rx;
-    secp256k1_num_init(&rx);
-    secp256k1_num_copy(&rx, &sig->r);
+    unsigned char brx[32];
+    secp256k1_scalar_get_b32(brx, &sig->r);
+    secp256k1_fe_t fx;
+    VERIFY_CHECK(secp256k1_fe_set_b32(&fx, brx)); /* brx comes from a scalar, so is less than the order; certainly less than p */
     if (recid & 2) {
-        secp256k1_num_add(&rx, &rx, &c->order);
-        if (secp256k1_num_cmp(&rx, &secp256k1_fe_consts->p) >= 0)
+        if (secp256k1_fe_cmp_var(&fx, &secp256k1_ecdsa_consts->p_minus_order) >= 0)
             return 0;
+        secp256k1_fe_add(&fx, &secp256k1_ecdsa_consts->order_as_fe);
     }
-    unsigned char brx[32];
-    secp256k1_num_get_bin(brx, 32, &rx);
-    secp256k1_num_free(&rx);
-    secp256k1_fe_t fx;
-    secp256k1_fe_set_b32(&fx, brx);
     secp256k1_ge_t x;
     if (!secp256k1_ge_set_xo(&x, &fx, recid & 1))
         return 0;
     secp256k1_gej_t xj;
     secp256k1_gej_set_ge(&xj, &x);
-    secp256k1_num_t rn, u1, u2;
-    secp256k1_num_init(&rn);
-    secp256k1_num_init(&u1);
-    secp256k1_num_init(&u2);
-    secp256k1_num_mod_inverse(&rn, &sig->r, &c->order);
-    secp256k1_num_mod_mul(&u1, &rn, message, &c->order);
-    secp256k1_num_sub(&u1, &c->order, &u1);
-    secp256k1_num_mod_mul(&u2, &rn, &sig->s, &c->order);
+    secp256k1_scalar_t rn, u1, u2;
+    secp256k1_scalar_inverse_var(&rn, &sig->r);
+    secp256k1_scalar_mul(&u1, &rn, message);
+    secp256k1_scalar_negate(&u1, &u1);
+    secp256k1_scalar_mul(&u2, &rn, &sig->s);
     secp256k1_gej_t qj;
     secp256k1_ecmult(&qj, &xj, &u2, &u1);
     secp256k1_ge_set_gej_var(pubkey, &qj);
-    secp256k1_num_free(&rn);
-    secp256k1_num_free(&u1);
-    secp256k1_num_free(&u2);
     return !secp256k1_gej_is_infinity(&qj);
 }
 
-static int secp256k1_ecdsa_sig_verify(const secp256k1_ecdsa_sig_t *sig, const secp256k1_ge_t *pubkey, const secp256k1_num_t *message) {
-    secp256k1_num_t r2;
-    secp256k1_num_init(&r2);
+static int secp256k1_ecdsa_sig_verify(const secp256k1_ecdsa_sig_t *sig, const secp256k1_ge_t *pubkey, const secp256k1_scalar_t *message) {
+    secp256k1_scalar_t r2;
     int ret = 0;
-    ret = secp256k1_ecdsa_sig_recompute(&r2, sig, pubkey, message) && secp256k1_num_cmp(&sig->r, &r2) == 0;
-    secp256k1_num_free(&r2);
+    ret = secp256k1_ecdsa_sig_recompute(&r2, sig, pubkey, message) && secp256k1_scalar_eq(&sig->r, &r2);
     return ret;
 }
 
@@ -150,34 +176,30 @@ static int secp256k1_ecdsa_sig_sign(secp256k1_ecdsa_sig_t *sig, const secp256k1_
     secp256k1_fe_normalize(&r.y);
     secp256k1_fe_get_b32(b, &r.x);
     int overflow = 0;
-    secp256k1_scalar_t sigr;
-    secp256k1_scalar_set_b32(&sigr, b, &overflow);
+    secp256k1_scalar_set_b32(&sig->r, b, &overflow);
     if (recid)
         *recid = (overflow ? 2 : 0) | (secp256k1_fe_is_odd(&r.y) ? 1 : 0);
     secp256k1_scalar_t n;
-    secp256k1_scalar_mul(&n, &sigr, seckey);
+    secp256k1_scalar_mul(&n, &sig->r, seckey);
     secp256k1_scalar_add(&n, &n, message);
-    secp256k1_scalar_t sigs;
-    secp256k1_scalar_inverse(&sigs, nonce);
-    secp256k1_scalar_mul(&sigs, &sigs, &n);
+    secp256k1_scalar_inverse(&sig->s, nonce);
+    secp256k1_scalar_mul(&sig->s, &sig->s, &n);
     secp256k1_scalar_clear(&n);
     secp256k1_gej_clear(&rp);
     secp256k1_ge_clear(&r);
-    if (secp256k1_scalar_is_zero(&sigs))
+    if (secp256k1_scalar_is_zero(&sig->s))
         return 0;
-    if (secp256k1_scalar_is_high(&sigs)) {
-        secp256k1_scalar_negate(&sigs, &sigs);
+    if (secp256k1_scalar_is_high(&sig->s)) {
+        secp256k1_scalar_negate(&sig->s, &sig->s);
         if (recid)
             *recid ^= 1;
     }
-    secp256k1_scalar_get_num(&sig->s, &sigs);
-    secp256k1_scalar_get_num(&sig->r, &sigr);
     return 1;
 }
 
-static void secp256k1_ecdsa_sig_set_rs(secp256k1_ecdsa_sig_t *sig, const secp256k1_num_t *r, const secp256k1_num_t *s) {
-    secp256k1_num_copy(&sig->r, r);
-    secp256k1_num_copy(&sig->s, s);
+static void secp256k1_ecdsa_sig_set_rs(secp256k1_ecdsa_sig_t *sig, const secp256k1_scalar_t *r, const secp256k1_scalar_t *s) {
+    sig->r = *r;
+    sig->s = *s;
 }
 
 #endif
diff --git a/src/secp256k1/src/eckey.h b/src/secp256k1/src/eckey.h
index 024c8b821b..6de5dc0a59 100644
--- a/src/secp256k1/src/eckey.h
+++ b/src/secp256k1/src/eckey.h
@@ -9,7 +9,6 @@
 
 #include "group.h"
 #include "scalar.h"
-#include "num.h"
 
 static int secp256k1_eckey_pubkey_parse(secp256k1_ge_t *elem, const unsigned char *pub, int size);
 static int secp256k1_eckey_pubkey_serialize(secp256k1_ge_t *elem, unsigned char *pub, int *size, int compressed);
@@ -18,8 +17,8 @@ static int secp256k1_eckey_privkey_parse(secp256k1_scalar_t *key, const unsigned
 static int secp256k1_eckey_privkey_serialize(unsigned char *privkey, int *privkeylen, const secp256k1_scalar_t *key, int compressed);
 
 static int secp256k1_eckey_privkey_tweak_add(secp256k1_scalar_t *key, const secp256k1_scalar_t *tweak);
-static int secp256k1_eckey_pubkey_tweak_add(secp256k1_ge_t *key, const secp256k1_num_t *tweak);
+static int secp256k1_eckey_pubkey_tweak_add(secp256k1_ge_t *key, const secp256k1_scalar_t *tweak);
 static int secp256k1_eckey_privkey_tweak_mul(secp256k1_scalar_t *key, const secp256k1_scalar_t *tweak);
-static int secp256k1_eckey_pubkey_tweak_mul(secp256k1_ge_t *key, const secp256k1_num_t *tweak);
+static int secp256k1_eckey_pubkey_tweak_mul(secp256k1_ge_t *key, const secp256k1_scalar_t *tweak);
 
 #endif
diff --git a/src/secp256k1/src/eckey_impl.h b/src/secp256k1/src/eckey_impl.h
index 290b1f0900..0f218ced9e 100644
--- a/src/secp256k1/src/eckey_impl.h
+++ b/src/secp256k1/src/eckey_impl.h
@@ -9,7 +9,7 @@
 
 #include "eckey.h"
 
-#include "num.h"
+#include "scalar.h"
 #include "field.h"
 #include "group.h"
 #include "ecmult_gen.h"
@@ -17,12 +17,12 @@
 static int secp256k1_eckey_pubkey_parse(secp256k1_ge_t *elem, const unsigned char *pub, int size) {
     if (size == 33 && (pub[0] == 0x02 || pub[0] == 0x03)) {
         secp256k1_fe_t x;
-        secp256k1_fe_set_b32(&x, pub+1);
-        return secp256k1_ge_set_xo(elem, &x, pub[0] == 0x03);
+        return secp256k1_fe_set_b32(&x, pub+1) && secp256k1_ge_set_xo(elem, &x, pub[0] == 0x03);
     } else if (size == 65 && (pub[0] == 0x04 || pub[0] == 0x06 || pub[0] == 0x07)) {
         secp256k1_fe_t x, y;
-        secp256k1_fe_set_b32(&x, pub+1);
-        secp256k1_fe_set_b32(&y, pub+33);
+        if (!secp256k1_fe_set_b32(&x, pub+1) || !secp256k1_fe_set_b32(&y, pub+33)) {
+            return 0;
+        }
         secp256k1_ge_set_xy(elem, &x, &y);
         if ((pub[0] == 0x06 || pub[0] == 0x07) && secp256k1_fe_is_odd(&y) != (pub[0] == 0x07))
             return 0;
@@ -154,17 +154,12 @@ static int secp256k1_eckey_privkey_tweak_add(secp256k1_scalar_t *key, const secp
     return 1;
 }
 
-static int secp256k1_eckey_pubkey_tweak_add(secp256k1_ge_t *key, const secp256k1_num_t *tweak) {
-    if (secp256k1_num_cmp(tweak, &secp256k1_ge_consts->order) >= 0)
-        return 0;
-
+static int secp256k1_eckey_pubkey_tweak_add(secp256k1_ge_t *key, const secp256k1_scalar_t *tweak) {
     secp256k1_gej_t pt;
     secp256k1_gej_set_ge(&pt, key);
-    secp256k1_num_t one;
-    secp256k1_num_init(&one);
-    secp256k1_num_set_int(&one, 1);
+    secp256k1_scalar_t one;
+    secp256k1_scalar_set_int(&one, 1);
     secp256k1_ecmult(&pt, &pt, &one, tweak);
-    secp256k1_num_free(&one);
 
     if (secp256k1_gej_is_infinity(&pt))
         return 0;
@@ -180,19 +175,15 @@ static int secp256k1_eckey_privkey_tweak_mul(secp256k1_scalar_t *key, const secp
     return 1;
 }
 
-static int secp256k1_eckey_pubkey_tweak_mul(secp256k1_ge_t *key, const secp256k1_num_t *tweak) {
-    if (secp256k1_num_is_zero(tweak))
-        return 0;
-    if (secp256k1_num_cmp(tweak, &secp256k1_ge_consts->order) >= 0)
+static int secp256k1_eckey_pubkey_tweak_mul(secp256k1_ge_t *key, const secp256k1_scalar_t *tweak) {
+    if (secp256k1_scalar_is_zero(tweak))
         return 0;
 
-    secp256k1_num_t zero;
-    secp256k1_num_init(&zero);
-    secp256k1_num_set_int(&zero, 0);
+    secp256k1_scalar_t zero;
+    secp256k1_scalar_set_int(&zero, 0);
     secp256k1_gej_t pt;
     secp256k1_gej_set_ge(&pt, key);
     secp256k1_ecmult(&pt, &pt, tweak, &zero);
-    secp256k1_num_free(&zero);
     secp256k1_ge_set_gej(key, &pt);
     return 1;
 }
diff --git a/src/secp256k1/src/ecmult.h b/src/secp256k1/src/ecmult.h
index e3cf18b680..15a7100a4a 100644
--- a/src/secp256k1/src/ecmult.h
+++ b/src/secp256k1/src/ecmult.h
@@ -14,6 +14,6 @@ static void secp256k1_ecmult_start(void);
 static void secp256k1_ecmult_stop(void);
 
 /** Double multiply: R = na*A + ng*G */
-static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_num_t *na, const secp256k1_num_t *ng);
+static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_scalar_t *na, const secp256k1_scalar_t *ng);
 
 #endif
diff --git a/src/secp256k1/src/ecmult_gen_impl.h b/src/secp256k1/src/ecmult_gen_impl.h
index 07859ab04b..af0ead522d 100644
--- a/src/secp256k1/src/ecmult_gen_impl.h
+++ b/src/secp256k1/src/ecmult_gen_impl.h
@@ -23,8 +23,8 @@ typedef struct {
      * precomputed (call it prec(i, n_i)). The formula now becomes sum(prec(i, n_i), i=0..63).
      * None of the resulting prec group elements have a known scalar, and neither do any of
      * the intermediate sums while computing a*G.
-     * To make memory access uniform, the bytes of prec(i, n_i) are sliced per value of n_i. */
-    unsigned char prec[64][sizeof(secp256k1_ge_t)][16]; /* prec[j][k][i] = k'th byte of (16^j * i * G + U_i) */
+     */
+    secp256k1_fe_t prec[64][16][2]; /* prec[j][i] = (16^j * i * G + U_i).{x,y} */
 } secp256k1_ecmult_gen_consts_t;
 
 static const secp256k1_ecmult_gen_consts_t *secp256k1_ecmult_gen_consts = NULL;
@@ -45,7 +45,7 @@ static void secp256k1_ecmult_gen_start(void) {
     {
         static const unsigned char nums_b32[32] = "The scalar for this x is unknown";
         secp256k1_fe_t nums_x;
-        secp256k1_fe_set_b32(&nums_x, nums_b32);
+        VERIFY_CHECK(secp256k1_fe_set_b32(&nums_x, nums_b32));
         secp256k1_ge_t nums_ge;
         VERIFY_CHECK(secp256k1_ge_set_xo(&nums_ge, &nums_x, 0));
         secp256k1_gej_set_ge(&nums_gej, &nums_ge);
@@ -81,9 +81,9 @@ static void secp256k1_ecmult_gen_start(void) {
     }
     for (int j=0; j<64; j++) {
         for (int i=0; i<16; i++) {
-            const unsigned char* raw = (const unsigned char*)(&prec[j*16 + i]);
-            for (size_t k=0; k<sizeof(secp256k1_ge_t); k++)
-                ret->prec[j][k][i] = raw[k];
+            VERIFY_CHECK(!secp256k1_ge_is_infinity(&prec[j*16 + i]));
+            ret->prec[j][i][0] = prec[j*16 + i].x;
+            ret->prec[j][i][1] = prec[j*16 + i].y;
         }
     }
 
@@ -104,11 +104,14 @@ static void secp256k1_ecmult_gen(secp256k1_gej_t *r, const secp256k1_scalar_t *g
     const secp256k1_ecmult_gen_consts_t *c = secp256k1_ecmult_gen_consts;
     secp256k1_gej_set_infinity(r);
     secp256k1_ge_t add;
+    add.infinity = 0;
     int bits;
     for (int j=0; j<64; j++) {
         bits = secp256k1_scalar_get_bits(gn, j * 4, 4);
-        for (size_t k=0; k<sizeof(secp256k1_ge_t); k++)
-            ((unsigned char*)(&add))[k] = c->prec[j][k][bits];
+        for (int i=0; i<16; i++) {
+            secp256k1_fe_cmov(&add.x, &c->prec[j][i][0], i == bits);
+            secp256k1_fe_cmov(&add.y, &c->prec[j][i][1], i == bits);
+        }
         secp256k1_gej_add_ge(r, r, &add);
     }
     bits = 0;
diff --git a/src/secp256k1/src/ecmult_impl.h b/src/secp256k1/src/ecmult_impl.h
index 508902564e..445b81593f 100644
--- a/src/secp256k1/src/ecmult_impl.h
+++ b/src/secp256k1/src/ecmult_impl.h
@@ -7,8 +7,8 @@
 #ifndef _SECP256K1_ECMULT_IMPL_H_
 #define _SECP256K1_ECMULT_IMPL_H_
 
-#include "num.h"
 #include "group.h"
+#include "scalar.h"
 #include "ecmult.h"
 
 /* optimal for 128-bit and 256-bit exponents. */
@@ -16,7 +16,11 @@
 
 /** larger numbers may result in slightly better performance, at the cost of
     exponentially larger precomputed tables. WINDOW_G == 14 results in 640 KiB. */
+#ifdef USE_ENDOMORPHISM
 #define WINDOW_G 14
+#else
+#define WINDOW_G 15
+#endif
 
 /** Fill a table 'pre' with precomputed odd multiples of a. W determines the size of the table.
  *  pre will contains the values [1*a,3*a,5*a,...,(2^(w-1)-1)*a], so it needs place for
@@ -69,7 +73,9 @@ static void secp256k1_ecmult_table_precomp_ge_var(secp256k1_ge_t *pre, const sec
 typedef struct {
     /* For accelerating the computation of a*P + b*G: */
     secp256k1_ge_t pre_g[ECMULT_TABLE_SIZE(WINDOW_G)];    /* odd multiples of the generator */
+#ifdef USE_ENDOMORPHISM
     secp256k1_ge_t pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)]; /* odd multiples of 2^128*generator */
+#endif
 } secp256k1_ecmult_consts_t;
 
 static const secp256k1_ecmult_consts_t *secp256k1_ecmult_consts = NULL;
@@ -85,14 +91,18 @@ static void secp256k1_ecmult_start(void) {
     const secp256k1_ge_t *g = &secp256k1_ge_consts->g;
     secp256k1_gej_t gj; secp256k1_gej_set_ge(&gj, g);
 
+#ifdef USE_ENDOMORPHISM
     /* calculate 2^128*generator */
     secp256k1_gej_t g_128j = gj;
     for (int i=0; i<128; i++)
         secp256k1_gej_double_var(&g_128j, &g_128j);
+#endif
 
     /* precompute the tables with odd multiples */
     secp256k1_ecmult_table_precomp_ge_var(ret->pre_g, &gj, WINDOW_G);
+#ifdef USE_ENDOMORPHISM
     secp256k1_ecmult_table_precomp_ge_var(ret->pre_g_128, &g_128j, WINDOW_G);
+#endif
 
     /* Set the global pointer to the precomputation table. */
     secp256k1_ecmult_consts = ret;
@@ -111,56 +121,62 @@ static void secp256k1_ecmult_stop(void) {
  *  with the following guarantees:
  *  - each wnaf[i] is either 0, or an odd integer between -(1<<(w-1) - 1) and (1<<(w-1) - 1)
  *  - two non-zero entries in wnaf are separated by at least w-1 zeroes.
- *  - the index of the highest non-zero entry in wnaf (=return value-1) is at most bits, where
- *    bits is the number of bits necessary to represent the absolute value of the input.
+ *  - the number of set values in wnaf is returned. This number is at most 256, and at most one more
+ *  - than the number of bits in the (absolute value) of the input.
  */
-static int secp256k1_ecmult_wnaf(int *wnaf, const secp256k1_num_t *a, int w) {
-    int ret = 0;
-    int zeroes = 0;
-    secp256k1_num_t x;
-    secp256k1_num_copy(&x, a);
+static int secp256k1_ecmult_wnaf(int *wnaf, const secp256k1_scalar_t *a, int w) {
+    secp256k1_scalar_t s = *a;
+
     int sign = 1;
-    if (secp256k1_num_is_neg(&x)) {
+    if (secp256k1_scalar_get_bits(&s, 255, 1)) {
+        secp256k1_scalar_negate(&s, &s);
         sign = -1;
-        secp256k1_num_negate(&x);
     }
-    while (!secp256k1_num_is_zero(&x)) {
-        while (!secp256k1_num_is_odd(&x)) {
-            zeroes++;
-            secp256k1_num_shift(&x, 1);
+
+    int set_bits = 0;
+    int bit = 0;
+    while (bit < 256) {
+        if (secp256k1_scalar_get_bits(&s, bit, 1) == 0) {
+            bit++;
+            continue;
+        }
+        while (set_bits < bit) {
+            wnaf[set_bits++] = 0;
         }
-        int word = secp256k1_num_shift(&x, w);
-        while (zeroes) {
-            wnaf[ret++] = 0;
-            zeroes--;
+        int now = w;
+        if (bit + now > 256) {
+            now = 256 - bit;
         }
+        int word = secp256k1_scalar_get_bits_var(&s, bit, now);
         if (word & (1 << (w-1))) {
-            secp256k1_num_inc(&x);
-            wnaf[ret++] = sign * (word - (1 << w));
+            secp256k1_scalar_add_bit(&s, bit + w);
+            wnaf[set_bits++] = sign * (word - (1 << w));
         } else {
-            wnaf[ret++] = sign * word;
+            wnaf[set_bits++] = sign * word;
         }
-        zeroes = w-1;
+        bit += now;
     }
-    return ret;
+    return set_bits;
 }
 
-static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_num_t *na, const secp256k1_num_t *ng) {
+static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_scalar_t *na, const secp256k1_scalar_t *ng) {
     const secp256k1_ecmult_consts_t *c = secp256k1_ecmult_consts;
 
 #ifdef USE_ENDOMORPHISM
-    secp256k1_num_t na_1, na_lam;
+    secp256k1_scalar_t na_1, na_lam;
     /* split na into na_1 and na_lam (where na = na_1 + na_lam*lambda, and na_1 and na_lam are ~128 bit) */
-    secp256k1_gej_split_exp_var(&na_1, &na_lam, na);
+    secp256k1_scalar_split_lambda_var(&na_1, &na_lam, na);
 
     /* build wnaf representation for na_1 and na_lam. */
-    int wnaf_na_1[129];   int bits_na_1   = secp256k1_ecmult_wnaf(wnaf_na_1,   &na_1,   WINDOW_A);
-    int wnaf_na_lam[129]; int bits_na_lam = secp256k1_ecmult_wnaf(wnaf_na_lam, &na_lam, WINDOW_A);
+    int wnaf_na_1[130];   int bits_na_1   = secp256k1_ecmult_wnaf(wnaf_na_1,   &na_1,   WINDOW_A);
+    int wnaf_na_lam[130]; int bits_na_lam = secp256k1_ecmult_wnaf(wnaf_na_lam, &na_lam, WINDOW_A);
+    VERIFY_CHECK(bits_na_1 <= 130);
+    VERIFY_CHECK(bits_na_lam <= 130);
     int bits = bits_na_1;
     if (bits_na_lam > bits) bits = bits_na_lam;
 #else
     /* build wnaf representation for na. */
-    int wnaf_na[257];     int bits_na     = secp256k1_ecmult_wnaf(wnaf_na,     na,      WINDOW_A);
+    int wnaf_na[256];     int bits_na     = secp256k1_ecmult_wnaf(wnaf_na,     na,      WINDOW_A);
     int bits = bits_na;
 #endif
 
@@ -172,19 +188,22 @@ static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const
     secp256k1_gej_t pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
     for (int i=0; i<ECMULT_TABLE_SIZE(WINDOW_A); i++)
         secp256k1_gej_mul_lambda(&pre_a_lam[i], &pre_a[i]);
-#endif
 
     /* Splitted G factors. */
-    secp256k1_num_t ng_1, ng_128;
+    secp256k1_scalar_t ng_1, ng_128;
 
     /* split ng into ng_1 and ng_128 (where gn = gn_1 + gn_128*2^128, and gn_1 and gn_128 are ~128 bit) */
-    secp256k1_num_split(&ng_1, &ng_128, ng, 128);
+    secp256k1_scalar_split_128(&ng_1, &ng_128, ng);
 
     /* Build wnaf representation for ng_1 and ng_128 */
     int wnaf_ng_1[129];   int bits_ng_1   = secp256k1_ecmult_wnaf(wnaf_ng_1,   &ng_1,   WINDOW_G);
     int wnaf_ng_128[129]; int bits_ng_128 = secp256k1_ecmult_wnaf(wnaf_ng_128, &ng_128, WINDOW_G);
     if (bits_ng_1 > bits) bits = bits_ng_1;
     if (bits_ng_128 > bits) bits = bits_ng_128;
+#else
+    int wnaf_ng[257];     int bits_ng     = secp256k1_ecmult_wnaf(wnaf_ng,     ng,      WINDOW_G);
+    if (bits_ng > bits) bits = bits_ng;
+#endif
 
     secp256k1_gej_set_infinity(r);
     secp256k1_gej_t tmpj;
@@ -202,12 +221,6 @@ static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const
             ECMULT_TABLE_GET_GEJ(&tmpj, pre_a_lam, n, WINDOW_A);
             secp256k1_gej_add_var(r, r, &tmpj);
         }
-#else
-        if (i < bits_na && (n = wnaf_na[i])) {
-            ECMULT_TABLE_GET_GEJ(&tmpj, pre_a, n, WINDOW_A);
-            secp256k1_gej_add_var(r, r, &tmpj);
-        }
-#endif
         if (i < bits_ng_1 && (n = wnaf_ng_1[i])) {
             ECMULT_TABLE_GET_GE(&tmpa, c->pre_g, n, WINDOW_G);
             secp256k1_gej_add_ge_var(r, r, &tmpa);
@@ -216,6 +229,16 @@ static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const
             ECMULT_TABLE_GET_GE(&tmpa, c->pre_g_128, n, WINDOW_G);
             secp256k1_gej_add_ge_var(r, r, &tmpa);
         }
+#else
+        if (i < bits_na && (n = wnaf_na[i])) {
+            ECMULT_TABLE_GET_GEJ(&tmpj, pre_a, n, WINDOW_A);
+            secp256k1_gej_add_var(r, r, &tmpj);
+        }
+        if (i < bits_ng && (n = wnaf_ng[i])) {
+            ECMULT_TABLE_GET_GE(&tmpa, c->pre_g, n, WINDOW_G);
+            secp256k1_gej_add_ge_var(r, r, &tmpa);
+        }
+#endif
     }
 }
 
diff --git a/src/secp256k1/src/field.h b/src/secp256k1/src/field.h
index c7feead900..0cdf0fb479 100644
--- a/src/secp256k1/src/field.h
+++ b/src/secp256k1/src/field.h
@@ -33,7 +33,10 @@
 #endif
 
 typedef struct {
+#ifndef USE_NUM_NONE
     secp256k1_num_t p;
+#endif
+    secp256k1_fe_t order;
 } secp256k1_fe_consts_t;
 
 static const secp256k1_fe_consts_t *secp256k1_fe_consts = NULL;
@@ -59,8 +62,11 @@ static int secp256k1_fe_is_odd(const secp256k1_fe_t *a);
 /** Compare two field elements. Requires both inputs to be normalized */
 static int secp256k1_fe_equal(const secp256k1_fe_t *a, const secp256k1_fe_t *b);
 
-/** Set a field element equal to 32-byte big endian value. Resulting field element is normalized. */
-static void secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a);
+/** Compare two field elements. Requires both inputs to be normalized */
+static int secp256k1_fe_cmp_var(const secp256k1_fe_t *a, const secp256k1_fe_t *b);
+
+/** Set a field element equal to 32-byte big endian value. If succesful, the resulting field element is normalized. */
+static int secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a);
 
 /** Convert a field element to a 32-byte big endian value. Requires the input to be normalized */
 static void secp256k1_fe_get_b32(unsigned char *r, const secp256k1_fe_t *a);
@@ -78,7 +84,7 @@ static void secp256k1_fe_add(secp256k1_fe_t *r, const secp256k1_fe_t *a);
 
 /** Sets a field element to be the product of two others. Requires the inputs' magnitudes to be at most 8.
  *  The output magnitude is 1 (but not guaranteed to be normalized). */
-static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t *b);
+static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t * SECP256K1_RESTRICT b);
 
 /** Sets a field element to be the square of another. Requires the input's magnitude to be at most 8.
  *  The output magnitude is 1 (but not guaranteed to be normalized). */
@@ -104,11 +110,13 @@ static void secp256k1_fe_inv_all(size_t len, secp256k1_fe_t r[len], const secp25
 /** Potentially faster version of secp256k1_fe_inv_all, without constant-time guarantee. */
 static void secp256k1_fe_inv_all_var(size_t len, secp256k1_fe_t r[len], const secp256k1_fe_t a[len]);
 
-
 /** Convert a field element to a hexadecimal string. */
 static void secp256k1_fe_get_hex(char *r, int *rlen, const secp256k1_fe_t *a);
 
 /** Convert a hexadecimal string to a field element. */
-static void secp256k1_fe_set_hex(secp256k1_fe_t *r, const char *a, int alen);
+static int secp256k1_fe_set_hex(secp256k1_fe_t *r, const char *a, int alen);
+
+/** If flag is true, set *r equal to *a; otherwise leave it. Constant-time. */
+static void secp256k1_fe_cmov(secp256k1_fe_t *r, const secp256k1_fe_t *a, int flag);
 
 #endif
diff --git a/src/secp256k1/src/field_10x26_impl.h b/src/secp256k1/src/field_10x26_impl.h
index c0f1be0b2d..c4403fba22 100644
--- a/src/secp256k1/src/field_10x26_impl.h
+++ b/src/secp256k1/src/field_10x26_impl.h
@@ -152,7 +152,21 @@ SECP256K1_INLINE static int secp256k1_fe_equal(const secp256k1_fe_t *a, const se
           | (t[5]^u[5]) | (t[6]^u[6]) | (t[7]^u[7]) | (t[8]^u[8]) | (t[9]^u[9])) == 0;
 }
 
-static void secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
+static int secp256k1_fe_cmp_var(const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+#ifdef VERIFY
+    VERIFY_CHECK(a->normalized);
+    VERIFY_CHECK(b->normalized);
+    secp256k1_fe_verify(a);
+    secp256k1_fe_verify(b);
+#endif
+    for (int i = 9; i >= 0; i--) {
+        if (a->n[i] > b->n[i]) return 1;
+        if (a->n[i] < b->n[i]) return -1;
+    }
+    return 0;
+}
+
+static int secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
     r->n[0] = r->n[1] = r->n[2] = r->n[3] = r->n[4] = 0;
     r->n[5] = r->n[6] = r->n[7] = r->n[8] = r->n[9] = 0;
     for (int i=0; i<32; i++) {
@@ -162,11 +176,15 @@ static void secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
             r->n[limb] |= (uint32_t)((a[31-i] >> (2*j)) & 0x3) << shift;
         }
     }
+    if (r->n[9] == 0x3FFFFFUL && (r->n[8] & r->n[7] & r->n[6] & r->n[5] & r->n[4] & r->n[3] & r->n[2]) == 0x3FFFFFFUL && (r->n[1] + 0x40UL + ((r->n[0] + 0x3D1UL) >> 26)) > 0x3FFFFFFUL) {
+        return 0;
+    }
 #ifdef VERIFY
     r->magnitude = 1;
     r->normalized = 1;
     secp256k1_fe_verify(r);
 #endif
+    return 1;
 }
 
 /** Convert a field element to a 32-byte big endian value. Requires the input to be normalized */
@@ -253,7 +271,7 @@ SECP256K1_INLINE static void secp256k1_fe_add(secp256k1_fe_t *r, const secp256k1
 #define VERIFY_BITS(x, n) do { } while(0)
 #endif
 
-SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uint32_t *b, uint32_t *r) {
+SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uint32_t * SECP256K1_RESTRICT b, uint32_t *r) {
     VERIFY_BITS(a[0], 30);
     VERIFY_BITS(a[1], 30);
     VERIFY_BITS(a[2], 30);
@@ -853,12 +871,13 @@ SECP256K1_INLINE static void secp256k1_fe_sqr_inner(const uint32_t *a, uint32_t
 }
 
 
-static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t * SECP256K1_RESTRICT b) {
 #ifdef VERIFY
     VERIFY_CHECK(a->magnitude <= 8);
     VERIFY_CHECK(b->magnitude <= 8);
     secp256k1_fe_verify(a);
     secp256k1_fe_verify(b);
+    VERIFY_CHECK(r != b);
 #endif
     secp256k1_fe_mul_inner(a->n, b->n, r->n);
 #ifdef VERIFY
@@ -881,4 +900,24 @@ static void secp256k1_fe_sqr(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
 #endif
 }
 
+static void secp256k1_fe_cmov(secp256k1_fe_t *r, const secp256k1_fe_t *a, int flag) {
+    uint32_t mask0 = flag + ~((uint32_t)0), mask1 = ~mask0;
+    r->n[0] = (r->n[0] & mask0) | (a->n[0] & mask1);
+    r->n[1] = (r->n[1] & mask0) | (a->n[1] & mask1);
+    r->n[2] = (r->n[2] & mask0) | (a->n[2] & mask1);
+    r->n[3] = (r->n[3] & mask0) | (a->n[3] & mask1);
+    r->n[4] = (r->n[4] & mask0) | (a->n[4] & mask1);
+    r->n[5] = (r->n[5] & mask0) | (a->n[5] & mask1);
+    r->n[6] = (r->n[6] & mask0) | (a->n[6] & mask1);
+    r->n[7] = (r->n[7] & mask0) | (a->n[7] & mask1);
+    r->n[8] = (r->n[8] & mask0) | (a->n[8] & mask1);
+    r->n[9] = (r->n[9] & mask0) | (a->n[9] & mask1);
+#ifdef VERIFY
+    if (flag) {
+        r->magnitude = a->magnitude;
+        r->normalized = a->normalized;
+    }
+#endif
+}
+
 #endif
diff --git a/src/secp256k1/src/field_5x52_impl.h b/src/secp256k1/src/field_5x52_impl.h
index d1b06d05a4..75b210eaf6 100644
--- a/src/secp256k1/src/field_5x52_impl.h
+++ b/src/secp256k1/src/field_5x52_impl.h
@@ -150,7 +150,21 @@ SECP256K1_INLINE static int secp256k1_fe_equal(const secp256k1_fe_t *a, const se
     return ((t[0]^u[0]) | (t[1]^u[1]) | (t[2]^u[2]) | (t[3]^u[3]) | (t[4]^u[4])) == 0;
 }
 
-static void secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
+static int secp256k1_fe_cmp_var(const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+#ifdef VERIFY
+    VERIFY_CHECK(a->normalized);
+    VERIFY_CHECK(b->normalized);
+    secp256k1_fe_verify(a);
+    secp256k1_fe_verify(b);
+#endif
+    for (int i = 4; i >= 0; i--) {
+        if (a->n[i] > b->n[i]) return 1;
+        if (a->n[i] < b->n[i]) return -1;
+    }
+    return 0;
+}
+
+static int secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
     r->n[0] = r->n[1] = r->n[2] = r->n[3] = r->n[4] = 0;
     for (int i=0; i<32; i++) {
         for (int j=0; j<2; j++) {
@@ -159,11 +173,15 @@ static void secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
             r->n[limb] |= (uint64_t)((a[31-i] >> (4*j)) & 0xF) << shift;
         }
     }
+    if (r->n[4] == 0x0FFFFFFFFFFFFULL && (r->n[3] & r->n[2] & r->n[1]) == 0xFFFFFFFFFFFFFULL && r->n[0] >= 0xFFFFEFFFFFC2FULL) {
+        return 0;
+    }
 #ifdef VERIFY
     r->magnitude = 1;
     r->normalized = 1;
     secp256k1_fe_verify(r);
 #endif
+    return 1;
 }
 
 /** Convert a field element to a 32-byte big endian value. Requires the input to be normalized */
@@ -229,12 +247,13 @@ SECP256K1_INLINE static void secp256k1_fe_add(secp256k1_fe_t *r, const secp256k1
 #endif
 }
 
-static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t * SECP256K1_RESTRICT b) {
 #ifdef VERIFY
     VERIFY_CHECK(a->magnitude <= 8);
     VERIFY_CHECK(b->magnitude <= 8);
     secp256k1_fe_verify(a);
     secp256k1_fe_verify(b);
+    VERIFY_CHECK(r != b);
 #endif
     secp256k1_fe_mul_inner(a->n, b->n, r->n);
 #ifdef VERIFY
@@ -257,4 +276,19 @@ static void secp256k1_fe_sqr(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
 #endif
 }
 
+static void secp256k1_fe_cmov(secp256k1_fe_t *r, const secp256k1_fe_t *a, int flag) {
+    uint64_t mask0 = flag + ~((uint64_t)0), mask1 = ~mask0;
+    r->n[0] = (r->n[0] & mask0) | (a->n[0] & mask1);
+    r->n[1] = (r->n[1] & mask0) | (a->n[1] & mask1);
+    r->n[2] = (r->n[2] & mask0) | (a->n[2] & mask1);
+    r->n[3] = (r->n[3] & mask0) | (a->n[3] & mask1);
+    r->n[4] = (r->n[4] & mask0) | (a->n[4] & mask1);
+#ifdef VERIFY
+    if (flag) {
+        r->magnitude = a->magnitude;
+        r->normalized = a->normalized;
+    }
+#endif
+}
+
 #endif
diff --git a/src/secp256k1/src/field_5x52_int128_impl.h b/src/secp256k1/src/field_5x52_int128_impl.h
index c476428672..e552fb4319 100644
--- a/src/secp256k1/src/field_5x52_int128_impl.h
+++ b/src/secp256k1/src/field_5x52_int128_impl.h
@@ -15,7 +15,7 @@
 #define VERIFY_BITS(x, n) do { } while(0)
 #endif
 
-SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uint64_t *b, uint64_t *r) {
+SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uint64_t * SECP256K1_RESTRICT b, uint64_t *r) {
     VERIFY_BITS(a[0], 56);
     VERIFY_BITS(a[1], 56);
     VERIFY_BITS(a[2], 56);
@@ -26,6 +26,7 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uin
     VERIFY_BITS(b[2], 56);
     VERIFY_BITS(b[3], 56);
     VERIFY_BITS(b[4], 52);
+    VERIFY_CHECK(r != b);
 
     const uint64_t M = 0xFFFFFFFFFFFFFULL, R = 0x1000003D10ULL;
     /*  [... a b c] is a shorthand for ... + a<<104 + b<<52 + c<<0 mod n.
@@ -33,15 +34,17 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uin
      *  Note that [x 0 0 0 0 0] = [x*R].
      */
 
+    uint64_t a0 = a[0], a1 = a[1], a2 = a[2], a3 = a[3], a4 = a[4];
+
     __int128 c, d;
 
-    d  = (__int128)a[0] * b[3]
-       + (__int128)a[1] * b[2]
-       + (__int128)a[2] * b[1]
-       + (__int128)a[3] * b[0];
+    d  = (__int128)a0 * b[3]
+       + (__int128)a1 * b[2]
+       + (__int128)a2 * b[1]
+       + (__int128)a3 * b[0];
     VERIFY_BITS(d, 114);
     /* [d 0 0 0] = [p3 0 0 0] */
-    c  = (__int128)a[4] * b[4];
+    c  = (__int128)a4 * b[4];
     VERIFY_BITS(c, 112);
     /* [c 0 0 0 0 d 0 0 0] = [p8 0 0 0 0 p3 0 0 0] */
     d += (c & M) * R; c >>= 52;
@@ -53,11 +56,11 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uin
     VERIFY_BITS(d, 63);
     /* [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 0 p3 0 0 0] */
 
-    d += (__int128)a[0] * b[4]
-       + (__int128)a[1] * b[3]
-       + (__int128)a[2] * b[2]
-       + (__int128)a[3] * b[1]
-       + (__int128)a[4] * b[0];
+    d += (__int128)a0 * b[4]
+       + (__int128)a1 * b[3]
+       + (__int128)a2 * b[2]
+       + (__int128)a3 * b[1]
+       + (__int128)a4 * b[0];
     VERIFY_BITS(d, 115);
     /* [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0] */
     d += c * R;
@@ -72,13 +75,13 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uin
     VERIFY_BITS(t4, 48);
     /* [d t4+(tx<<48) t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0] */
 
-    c  = (__int128)a[0] * b[0];
+    c  = (__int128)a0 * b[0];
     VERIFY_BITS(c, 112);
     /* [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 0 p4 p3 0 0 p0] */
-    d += (__int128)a[1] * b[4]
-       + (__int128)a[2] * b[3]
-       + (__int128)a[3] * b[2]
-       + (__int128)a[4] * b[1];
+    d += (__int128)a1 * b[4]
+       + (__int128)a2 * b[3]
+       + (__int128)a3 * b[2]
+       + (__int128)a4 * b[1];
     VERIFY_BITS(d, 115);
     /* [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0] */
     uint64_t u0 = d & M; d >>= 52;
@@ -92,48 +95,43 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uin
     c += (__int128)u0 * (R >> 4);
     VERIFY_BITS(c, 115);
     /* [d 0 t4 t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0] */
-    uint64_t t0 = c & M; c >>= 52;
-    VERIFY_BITS(t0, 52);
+    r[0] = c & M; c >>= 52;
+    VERIFY_BITS(r[0], 52);
     VERIFY_BITS(c, 61);
-    /* [d 0 t4 t3 0 c t0] = [p8 0 0 p5 p4 p3 0 0 p0] */
+    /* [d 0 t4 t3 0 c r0] = [p8 0 0 p5 p4 p3 0 0 p0] */
 
-    c += (__int128)a[0] * b[1]
-       + (__int128)a[1] * b[0];
+    c += (__int128)a0 * b[1]
+       + (__int128)a1 * b[0];
     VERIFY_BITS(c, 114);
-    /* [d 0 t4 t3 0 c t0] = [p8 0 0 p5 p4 p3 0 p1 p0] */
-    d += (__int128)a[2] * b[4]
-       + (__int128)a[3] * b[3]
-       + (__int128)a[4] * b[2];
+    /* [d 0 t4 t3 0 c r0] = [p8 0 0 p5 p4 p3 0 p1 p0] */
+    d += (__int128)a2 * b[4]
+       + (__int128)a3 * b[3]
+       + (__int128)a4 * b[2];
     VERIFY_BITS(d, 114);
-    /* [d 0 t4 t3 0 c t0] = [p8 0 p6 p5 p4 p3 0 p1 p0] */
+    /* [d 0 t4 t3 0 c r0] = [p8 0 p6 p5 p4 p3 0 p1 p0] */
     c += (d & M) * R; d >>= 52;
     VERIFY_BITS(c, 115);
     VERIFY_BITS(d, 62);
-    /* [d 0 0 t4 t3 0 c t0] = [p8 0 p6 p5 p4 p3 0 p1 p0] */
-    uint64_t t1 = c & M; c >>= 52;
-    VERIFY_BITS(t1, 52);
+    /* [d 0 0 t4 t3 0 c r0] = [p8 0 p6 p5 p4 p3 0 p1 p0] */
+    r[1] = c & M; c >>= 52;
+    VERIFY_BITS(r[1], 52);
     VERIFY_BITS(c, 63);
-    /* [d 0 0 t4 t3 c t1 t0] = [p8 0 p6 p5 p4 p3 0 p1 p0] */
+    /* [d 0 0 t4 t3 c r1 r0] = [p8 0 p6 p5 p4 p3 0 p1 p0] */
 
-    c += (__int128)a[0] * b[2]
-       + (__int128)a[1] * b[1]
-       + (__int128)a[2] * b[0];
+    c += (__int128)a0 * b[2]
+       + (__int128)a1 * b[1]
+       + (__int128)a2 * b[0];
     VERIFY_BITS(c, 114);
-    /* [d 0 0 t4 t3 c t1 t0] = [p8 0 p6 p5 p4 p3 p2 p1 p0] */
-    d += (__int128)a[3] * b[4]
-       + (__int128)a[4] * b[3];
+    /* [d 0 0 t4 t3 c r1 r0] = [p8 0 p6 p5 p4 p3 p2 p1 p0] */
+    d += (__int128)a3 * b[4]
+       + (__int128)a4 * b[3];
     VERIFY_BITS(d, 114);
-    /* [d 0 0 t4 t3 c t1 t0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0] */
+    /* [d 0 0 t4 t3 c t1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0] */
     c += (d & M) * R; d >>= 52;
     VERIFY_BITS(c, 115);
     VERIFY_BITS(d, 62);
-    /* [d 0 0 0 t4 t3 c t1 t0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0] */
+    /* [d 0 0 0 t4 t3 c r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0] */
 
-    r[0] = t0;
-    VERIFY_BITS(r[0], 52);
-    /* [d 0 0 0 t4 t3 c t1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0] */
-    r[1] = t1;
-    VERIFY_BITS(r[1], 52);
     /* [d 0 0 0 t4 t3 c r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0] */
     r[2] = c & M; c >>= 52;
     VERIFY_BITS(r[2], 52);
diff --git a/src/secp256k1/src/field_gmp_impl.h b/src/secp256k1/src/field_gmp_impl.h
index af4728e5b4..8af7dd68f8 100644
--- a/src/secp256k1/src/field_gmp_impl.h
+++ b/src/secp256k1/src/field_gmp_impl.h
@@ -75,7 +75,15 @@ SECP256K1_INLINE static int secp256k1_fe_equal(const secp256k1_fe_t *a, const se
     return ret;
 }
 
-static void secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
+SECP256K1_INLINE static int secp256k1_fe_cmp_var(const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+    for (int i=FIELD_LIMBS; i>=0; i--) {
+        if (a->n[i] > b->n[i]) return 1;
+        if (a->n[i] < b->n[i]) return -1;
+    }
+    return 0;
+}
+
+static int secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
     for (int i=0; i<FIELD_LIMBS+1; i++)
         r->n[i] = 0;
     for (int i=0; i<256; i++) {
@@ -83,6 +91,7 @@ static void secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
         int shift = i%GMP_NUMB_BITS;
         r->n[limb] |= (mp_limb_t)((a[31-i/8] >> (i%8)) & 0x1) << shift;
     }
+    return (mpn_cmp(r->n, secp256k1_field_p, FIELD_LIMBS) < 0);
 }
 
 /** Convert a field element to a 32-byte big endian value. Requires the input to be normalized */
@@ -142,7 +151,8 @@ static void secp256k1_fe_reduce(secp256k1_fe_t *r, mp_limb_t *tmp) {
     r->n[FIELD_LIMBS] = mpn_add(r->n, tmp, FIELD_LIMBS, q, 1+(33+GMP_NUMB_BITS-1)/GMP_NUMB_BITS);
 }
 
-static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+static void secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t * SECP256K1_RESTRICT b) {
+    VERIFY_CHECK(r != b);
     secp256k1_fe_t ac = *a;
     secp256k1_fe_t bc = *b;
     secp256k1_fe_normalize(&ac);
@@ -160,4 +170,11 @@ static void secp256k1_fe_sqr(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
     secp256k1_fe_reduce(r, tmp);
 }
 
+static void secp256k1_fe_cmov(secp256k1_fe_t *r, const secp256k1_fe_t *a, int flag) {
+    mp_limb_t mask0 = flag + ~((mp_limb_t)0), mask1 = ~mask0;
+    for (int i = 0; i <= FIELD_LIMBS; i++) {
+        r->n[i] = (r->n[i] & mask0) | (a->n[i] & mask1);
+    }
+}
+
 #endif
diff --git a/src/secp256k1/src/field_impl.h b/src/secp256k1/src/field_impl.h
index 3a31e1844e..4d25e53715 100644
--- a/src/secp256k1/src/field_impl.h
+++ b/src/secp256k1/src/field_impl.h
@@ -41,7 +41,7 @@ static void secp256k1_fe_get_hex(char *r, int *rlen, const secp256k1_fe_t *a) {
     r[64] = 0x00;
 }
 
-static void secp256k1_fe_set_hex(secp256k1_fe_t *r, const char *a, int alen) {
+static int secp256k1_fe_set_hex(secp256k1_fe_t *r, const char *a, int alen) {
     unsigned char tmp[32] = {};
     static const int cvt[256] = {0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
                                  0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
@@ -63,7 +63,7 @@ static void secp256k1_fe_set_hex(secp256k1_fe_t *r, const char *a, int alen) {
         if (alen > i*2)
             tmp[32 - alen/2 + i] = (cvt[(unsigned char)a[2*i]] << 4) + cvt[(unsigned char)a[2*i+1]];
     }
-    secp256k1_fe_set_b32(r, tmp);
+    return secp256k1_fe_set_b32(r, tmp);
 }
 
 static int secp256k1_fe_sqrt(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
@@ -197,7 +197,7 @@ static void secp256k1_fe_inv(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
     for (int j=0; j<3; j++) secp256k1_fe_sqr(&t1, &t1);
     secp256k1_fe_mul(&t1, &t1, &x2);
     for (int j=0; j<2; j++) secp256k1_fe_sqr(&t1, &t1);
-    secp256k1_fe_mul(r, &t1, a);
+    secp256k1_fe_mul(r, a, &t1);
 }
 
 static void secp256k1_fe_inv_var(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
@@ -212,7 +212,7 @@ static void secp256k1_fe_inv_var(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
     secp256k1_num_set_bin(&n, b, 32);
     secp256k1_num_mod_inverse(&n, &n, &secp256k1_fe_consts->p);
     secp256k1_num_get_bin(b, 32, &n);
-    secp256k1_fe_set_b32(r, b);
+    VERIFY_CHECK(secp256k1_fe_set_b32(r, b));
 #else
 #error "Please select field inverse implementation"
 #endif
@@ -267,16 +267,20 @@ static void secp256k1_fe_inv_all_var(size_t len, secp256k1_fe_t r[len], const se
 }
 
 static void secp256k1_fe_start(void) {
+#ifndef USE_NUM_NONE
     static const unsigned char secp256k1_fe_consts_p[] = {
         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
         0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F
     };
+#endif
     if (secp256k1_fe_consts == NULL) {
         secp256k1_fe_inner_start();
         secp256k1_fe_consts_t *ret = (secp256k1_fe_consts_t*)malloc(sizeof(secp256k1_fe_consts_t));
+#ifndef USE_NUM_NONE
         secp256k1_num_set_bin(&ret->p, secp256k1_fe_consts_p, sizeof(secp256k1_fe_consts_p));
+#endif
         secp256k1_fe_consts = ret;
     }
 }
diff --git a/src/secp256k1/src/group.h b/src/secp256k1/src/group.h
index ba02549821..0f14bd25f3 100644
--- a/src/secp256k1/src/group.h
+++ b/src/secp256k1/src/group.h
@@ -27,14 +27,11 @@ typedef struct {
 
 /** Global constants related to the group */
 typedef struct {
-    secp256k1_num_t order; /* the order of the curve (= order of its generator) */
-    secp256k1_num_t half_order; /* half the order of the curve (= order of its generator) */
     secp256k1_ge_t g; /* the generator point */
 
 #ifdef USE_ENDOMORPHISM
     /* constants related to secp256k1's efficiently computable endomorphism */
     secp256k1_fe_t beta;
-    secp256k1_num_t lambda, a1b2, b1, a2;
 #endif
 } secp256k1_ge_consts_t;
 
@@ -112,10 +109,6 @@ static void secp256k1_gej_get_hex(char *r, int *rlen, const secp256k1_gej_t *a);
 #ifdef USE_ENDOMORPHISM
 /** Set r to be equal to lambda times a, where lambda is chosen in a way such that this is very fast. */
 static void secp256k1_gej_mul_lambda(secp256k1_gej_t *r, const secp256k1_gej_t *a);
-
-/** Find r1 and r2 such that r1+r2*lambda = a, and r1 and r2 are maximum 128 bits long (given that a is
-    not more than 256 bits). */
-static void secp256k1_gej_split_exp_var(secp256k1_num_t *r1, secp256k1_num_t *r2, const secp256k1_num_t *a);
 #endif
 
 /** Clear a secp256k1_gej_t to prevent leaking sensitive information. */
@@ -124,5 +117,4 @@ static void secp256k1_gej_clear(secp256k1_gej_t *r);
 /** Clear a secp256k1_ge_t to prevent leaking sensitive information. */
 static void secp256k1_ge_clear(secp256k1_ge_t *r);
 
-
 #endif
diff --git a/src/secp256k1/src/group_impl.h b/src/secp256k1/src/group_impl.h
index 1edbc6e099..cbd0d8c4fc 100644
--- a/src/secp256k1/src/group_impl.h
+++ b/src/secp256k1/src/group_impl.h
@@ -208,29 +208,25 @@ static int secp256k1_ge_is_valid(const secp256k1_ge_t *a) {
 }
 
 static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *a) {
-    if (a->infinity) {
-        r->infinity = 1;
-        return;
-    }
-
-    secp256k1_fe_t t5 = a->y;
-    secp256k1_fe_normalize(&t5);
-    if (secp256k1_fe_is_zero(&t5)) {
-        r->infinity = 1;
+    // For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity,
+    // Q must equal -Q, or that Q.y == -(Q.y), or Q.y is 0. For a point on y^2 = x^3 + 7 to have
+    // y=0, x^3 must be -7 mod p. However, -7 has no cube root mod p.
+    r->infinity = a->infinity;
+    if (r->infinity) {
         return;
     }
 
     secp256k1_fe_t t1,t2,t3,t4;
-    secp256k1_fe_mul(&r->z, &t5, &a->z);
+    secp256k1_fe_mul(&r->z, &a->z, &a->y);
     secp256k1_fe_mul_int(&r->z, 2);       /* Z' = 2*Y*Z (2) */
     secp256k1_fe_sqr(&t1, &a->x);
     secp256k1_fe_mul_int(&t1, 3);         /* T1 = 3*X^2 (3) */
     secp256k1_fe_sqr(&t2, &t1);           /* T2 = 9*X^4 (1) */
-    secp256k1_fe_sqr(&t3, &t5);
+    secp256k1_fe_sqr(&t3, &a->y);
     secp256k1_fe_mul_int(&t3, 2);         /* T3 = 2*Y^2 (2) */
     secp256k1_fe_sqr(&t4, &t3);
     secp256k1_fe_mul_int(&t4, 2);         /* T4 = 8*Y^4 (2) */
-    secp256k1_fe_mul(&t3, &a->x, &t3);    /* T3 = 2*X*Y^2 (1) */
+    secp256k1_fe_mul(&t3, &t3, &a->x);    /* T3 = 2*X*Y^2 (1) */
     r->x = t3;
     secp256k1_fe_mul_int(&r->x, 4);       /* X' = 8*X*Y^2 (4) */
     secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8*X*Y^2 (5) */
@@ -241,7 +237,6 @@ static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *
     secp256k1_fe_mul(&r->y, &t1, &t3);    /* Y' = 36*X^3*Y^2 - 27*X^6 (1) */
     secp256k1_fe_negate(&t2, &t4, 2);     /* T2 = -8*Y^4 (3) */
     secp256k1_fe_add(&r->y, &t2);         /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
-    r->infinity = 0;
 }
 
 static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_gej_t *b) {
@@ -342,7 +337,7 @@ static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, c
      *
      *  Substituting x_i = Xi / Zi^2 and yi = Yi / Zi^3, for i=1,2,3, gives:
      *    U1 = X1*Z2^2, U2 = X2*Z1^2
-     *    S1 = X1*Z2^3, S2 = X2*Z2^3
+     *    S1 = Y1*Z2^3, S2 = Y2*Z1^3
      *    Z = Z1*Z2
      *    T = U1+U2
      *    M = S1+S2
@@ -414,40 +409,9 @@ static void secp256k1_gej_mul_lambda(secp256k1_gej_t *r, const secp256k1_gej_t *
     *r = *a;
     secp256k1_fe_mul(&r->x, &r->x, beta);
 }
-
-static void secp256k1_gej_split_exp_var(secp256k1_num_t *r1, secp256k1_num_t *r2, const secp256k1_num_t *a) {
-    const secp256k1_ge_consts_t *c = secp256k1_ge_consts;
-    secp256k1_num_t bnc1, bnc2, bnt1, bnt2, bnn2;
-
-    secp256k1_num_copy(&bnn2, &c->order);
-    secp256k1_num_shift(&bnn2, 1);
-
-    secp256k1_num_mul(&bnc1, a, &c->a1b2);
-    secp256k1_num_add(&bnc1, &bnc1, &bnn2);
-    secp256k1_num_div(&bnc1, &bnc1, &c->order);
-
-    secp256k1_num_mul(&bnc2, a, &c->b1);
-    secp256k1_num_add(&bnc2, &bnc2, &bnn2);
-    secp256k1_num_div(&bnc2, &bnc2, &c->order);
-
-    secp256k1_num_mul(&bnt1, &bnc1, &c->a1b2);
-    secp256k1_num_mul(&bnt2, &bnc2, &c->a2);
-    secp256k1_num_add(&bnt1, &bnt1, &bnt2);
-    secp256k1_num_sub(r1, a, &bnt1);
-    secp256k1_num_mul(&bnt1, &bnc1, &c->b1);
-    secp256k1_num_mul(&bnt2, &bnc2, &c->a1b2);
-    secp256k1_num_sub(r2, &bnt1, &bnt2);
-}
 #endif
 
-
 static void secp256k1_ge_start(void) {
-    static const unsigned char secp256k1_ge_consts_order[] = {
-        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
-        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,
-        0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B,
-        0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41
-    };
     static const unsigned char secp256k1_ge_consts_g_x[] = {
         0x79,0xBE,0x66,0x7E,0xF9,0xDC,0xBB,0xAC,
         0x55,0xA0,0x62,0x95,0xCE,0x87,0x0B,0x07,
@@ -462,47 +426,21 @@ static void secp256k1_ge_start(void) {
     };
 #ifdef USE_ENDOMORPHISM
     /* properties of secp256k1's efficiently computable endomorphism */
-    static const unsigned char secp256k1_ge_consts_lambda[] = {
-        0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,
-        0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
-        0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,
-        0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72
-    };
     static const unsigned char secp256k1_ge_consts_beta[] = {
         0x7a,0xe9,0x6a,0x2b,0x65,0x7c,0x07,0x10,
         0x6e,0x64,0x47,0x9e,0xac,0x34,0x34,0xe9,
         0x9c,0xf0,0x49,0x75,0x12,0xf5,0x89,0x95,
         0xc1,0x39,0x6c,0x28,0x71,0x95,0x01,0xee
     };
-    static const unsigned char secp256k1_ge_consts_a1b2[] = {
-        0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,
-        0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15
-    };
-    static const unsigned char secp256k1_ge_consts_b1[] = {
-        0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,
-        0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3
-    };
-    static const unsigned char secp256k1_ge_consts_a2[] = {
-        0x01,
-        0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,
-        0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8
-    };
 #endif
     if (secp256k1_ge_consts == NULL) {
         secp256k1_ge_consts_t *ret = (secp256k1_ge_consts_t*)malloc(sizeof(secp256k1_ge_consts_t));
-        secp256k1_num_set_bin(&ret->order,  secp256k1_ge_consts_order,  sizeof(secp256k1_ge_consts_order));
-        secp256k1_num_copy(&ret->half_order, &ret->order);
-        secp256k1_num_shift(&ret->half_order, 1);
 #ifdef USE_ENDOMORPHISM
-        secp256k1_num_set_bin(&ret->lambda, secp256k1_ge_consts_lambda, sizeof(secp256k1_ge_consts_lambda));
-        secp256k1_num_set_bin(&ret->a1b2,   secp256k1_ge_consts_a1b2,   sizeof(secp256k1_ge_consts_a1b2));
-        secp256k1_num_set_bin(&ret->a2,     secp256k1_ge_consts_a2,     sizeof(secp256k1_ge_consts_a2));
-        secp256k1_num_set_bin(&ret->b1,     secp256k1_ge_consts_b1,     sizeof(secp256k1_ge_consts_b1));
-        secp256k1_fe_set_b32(&ret->beta, secp256k1_ge_consts_beta);
+        VERIFY_CHECK(secp256k1_fe_set_b32(&ret->beta, secp256k1_ge_consts_beta));
 #endif
         secp256k1_fe_t g_x, g_y;
-        secp256k1_fe_set_b32(&g_x, secp256k1_ge_consts_g_x);
-        secp256k1_fe_set_b32(&g_y, secp256k1_ge_consts_g_y);
+        VERIFY_CHECK(secp256k1_fe_set_b32(&g_x, secp256k1_ge_consts_g_x));
+        VERIFY_CHECK(secp256k1_fe_set_b32(&g_y, secp256k1_ge_consts_g_y));
         secp256k1_ge_set_xy(&ret->g, &g_x, &g_y);
         secp256k1_ge_consts = ret;
     }
diff --git a/src/secp256k1/src/num.h b/src/secp256k1/src/num.h
index c86f847858..339b6bb6ec 100644
--- a/src/secp256k1/src/num.h
+++ b/src/secp256k1/src/num.h
@@ -7,6 +7,8 @@
 #ifndef _SECP256K1_NUM_
 #define _SECP256K1_NUM_
 
+#ifndef USE_NUM_NONE
+
 #if defined HAVE_CONFIG_H
 #include "libsecp256k1-config.h"
 #endif
@@ -17,9 +19,6 @@
 #error "Please select num implementation"
 #endif
 
-/** Clear a number to prevent the leak of sensitive data. */
-static void secp256k1_num_clear(secp256k1_num_t *r);
-
 /** Copy a number. */
 static void secp256k1_num_copy(secp256k1_num_t *r, const secp256k1_num_t *a);
 
@@ -30,15 +29,9 @@ static void secp256k1_num_get_bin(unsigned char *r, unsigned int rlen, const sec
 /** Set a number to the value of a binary big-endian string. */
 static void secp256k1_num_set_bin(secp256k1_num_t *r, const unsigned char *a, unsigned int alen);
 
-/** Set a number equal to a (signed) integer. */
-static void secp256k1_num_set_int(secp256k1_num_t *r, int a);
-
 /** Compute a modular inverse. The input must be less than the modulus. */
 static void secp256k1_num_mod_inverse(secp256k1_num_t *r, const secp256k1_num_t *a, const secp256k1_num_t *m);
 
-/** Multiply two numbers modulo another. */
-static void secp256k1_num_mod_mul(secp256k1_num_t *r, const secp256k1_num_t *a, const secp256k1_num_t *b, const secp256k1_num_t *m);
-
 /** Compare the absolute value of two numbers. */
 static int secp256k1_num_cmp(const secp256k1_num_t *a, const secp256k1_num_t *b);
 
@@ -54,47 +47,22 @@ static void secp256k1_num_sub(secp256k1_num_t *r, const secp256k1_num_t *a, cons
 /** Multiply two (signed) numbers. */
 static void secp256k1_num_mul(secp256k1_num_t *r, const secp256k1_num_t *a, const secp256k1_num_t *b);
 
-/** Divide two (signed) numbers. */
-static void secp256k1_num_div(secp256k1_num_t *r, const secp256k1_num_t *a, const secp256k1_num_t *b);
-
 /** Replace a number by its remainder modulo m. M's sign is ignored. The result is a number between 0 and m-1,
     even if r was negative. */
 static void secp256k1_num_mod(secp256k1_num_t *r, const secp256k1_num_t *m);
 
-/** Calculate the number of bits in (the absolute value of) a number. */
-static int secp256k1_num_bits(const secp256k1_num_t *a);
-
-/** Right-shift the passed number by bits bits, and return those bits. */
-static int secp256k1_num_shift(secp256k1_num_t *r, int bits);
+/** Right-shift the passed number by bits bits. */
+static void secp256k1_num_shift(secp256k1_num_t *r, int bits);
 
 /** Check whether a number is zero. */
 static int secp256k1_num_is_zero(const secp256k1_num_t *a);
 
-/** Check whether a number is odd. */
-static int secp256k1_num_is_odd(const secp256k1_num_t *a);
-
 /** Check whether a number is strictly negative. */
 static int secp256k1_num_is_neg(const secp256k1_num_t *a);
 
-/** Check whether a particular bit is set in a number. */
-static int secp256k1_num_get_bit(const secp256k1_num_t *a, int pos);
-
-/** Increase a number by 1. */
-static void secp256k1_num_inc(secp256k1_num_t *r);
-
-/** Set a number equal to the value of a hex string (unsigned). */
-static void secp256k1_num_set_hex(secp256k1_num_t *r, const char *a, int alen);
-
-/** Convert (the absolute value of) a number to a hexadecimal string. */
-static void secp256k1_num_get_hex(char *r, int rlen, const secp256k1_num_t *a);
-
-/** Split a number into a low and high part. */
-static void secp256k1_num_split(secp256k1_num_t *rl, secp256k1_num_t *rh, const secp256k1_num_t *a, int bits);
-
 /** Change a number's sign. */
 static void secp256k1_num_negate(secp256k1_num_t *r);
 
-/** Get a bunch of bits from a number. */
-static int secp256k1_num_get_bits(const secp256k1_num_t *a, int offset, int count);
+#endif
 
 #endif
diff --git a/src/secp256k1/src/num_gmp_impl.h b/src/secp256k1/src/num_gmp_impl.h
index e45a59e0cd..19d474e59f 100644
--- a/src/secp256k1/src/num_gmp_impl.h
+++ b/src/secp256k1/src/num_gmp_impl.h
@@ -22,35 +22,10 @@ static void secp256k1_num_sanity(const secp256k1_num_t *a) {
 #define secp256k1_num_sanity(a) do { } while(0)
 #endif
 
-static void secp256k1_num_init(secp256k1_num_t *r) {
-    r->neg = 0;
-    r->limbs = 1;
-    r->data[0] = 0;
-}
-
-static void secp256k1_num_clear(secp256k1_num_t *r) {
-    memset(r, 0, sizeof(*r));
-}
-
-static void secp256k1_num_free(secp256k1_num_t *r) {
-    (void)r;
-}
-
 static void secp256k1_num_copy(secp256k1_num_t *r, const secp256k1_num_t *a) {
     *r = *a;
 }
 
-static int secp256k1_num_bits(const secp256k1_num_t *a) {
-    int ret=(a->limbs-1)*GMP_NUMB_BITS;
-    mp_limb_t x=a->data[a->limbs-1];
-    while (x) {
-        x >>= 1;
-        ret++;
-    }
-    return ret;
-}
-
-
 static void secp256k1_num_get_bin(unsigned char *r, unsigned int rlen, const secp256k1_num_t *a) {
     unsigned char tmp[65];
     int len = 0;
@@ -71,18 +46,16 @@ static void secp256k1_num_set_bin(secp256k1_num_t *r, const unsigned char *a, un
     VERIFY_CHECK(alen > 0);
     VERIFY_CHECK(alen <= 64);
     int len = mpn_set_str(r->data, a, alen, 256);
+    if (len == 0) {
+        r->data[0] = 0;
+        len = 1;
+    }
     VERIFY_CHECK(len <= NUM_LIMBS*2);
     r->limbs = len;
     r->neg = 0;
     while (r->limbs > 1 && r->data[r->limbs-1]==0) r->limbs--;
 }
 
-static void secp256k1_num_set_int(secp256k1_num_t *r, int a) {
-    r->limbs = 1;
-    r->neg = (a < 0);
-    r->data[0] = (a < 0) ? -a : a;
-}
-
 static void secp256k1_num_add_abs(secp256k1_num_t *r, const secp256k1_num_t *a, const secp256k1_num_t *b) {
     mp_limb_t c = mpn_add(r->data, a->data, a->limbs, b->data, b->limbs);
     r->limbs = a->limbs;
@@ -161,10 +134,6 @@ static int secp256k1_num_is_zero(const secp256k1_num_t *a) {
     return (a->limbs == 1 && a->data[0] == 0);
 }
 
-static int secp256k1_num_is_odd(const secp256k1_num_t *a) {
-    return a->data[0] & 1;
-}
-
 static int secp256k1_num_is_neg(const secp256k1_num_t *a) {
     return (a->limbs > 1 || a->data[0] != 0) && a->neg;
 }
@@ -237,140 +206,27 @@ static void secp256k1_num_mul(secp256k1_num_t *r, const secp256k1_num_t *a, cons
     memset(tmp, 0, sizeof(tmp));
 }
 
-static void secp256k1_num_div(secp256k1_num_t *r, const secp256k1_num_t *a, const secp256k1_num_t *b) {
-    secp256k1_num_sanity(a);
-    secp256k1_num_sanity(b);
-    if (b->limbs > a->limbs) {
-        r->limbs = 1;
-        r->data[0] = 0;
-        r->neg = 0;
-        return;
-    }
-
-    mp_limb_t quo[2*NUM_LIMBS+1];
-    mp_limb_t rem[2*NUM_LIMBS+1];
-    mpn_tdiv_qr(quo, rem, 0, a->data, a->limbs, b->data, b->limbs);
-    mpn_copyi(r->data, quo, a->limbs - b->limbs + 1);
-    r->limbs = a->limbs - b->limbs + 1;
-    while (r->limbs > 1 && r->data[r->limbs - 1]==0) r->limbs--;
-    r->neg = a->neg ^ b->neg;
-}
-
-static void secp256k1_num_mod_mul(secp256k1_num_t *r, const secp256k1_num_t *a, const secp256k1_num_t *b, const secp256k1_num_t *m) {
-    secp256k1_num_mul(r, a, b);
-    secp256k1_num_mod(r, m);
-}
-
-
-static int secp256k1_num_shift(secp256k1_num_t *r, int bits) {
-    VERIFY_CHECK(bits <= GMP_NUMB_BITS);
-    mp_limb_t ret = mpn_rshift(r->data, r->data, r->limbs, bits);
-    if (r->limbs>1 && r->data[r->limbs-1]==0) r->limbs--;
-    ret >>= (GMP_NUMB_BITS - bits);
-    return ret;
-}
-
-static int secp256k1_num_get_bit(const secp256k1_num_t *a, int pos) {
-    return (a->limbs*GMP_NUMB_BITS > pos) && ((a->data[pos/GMP_NUMB_BITS] >> (pos % GMP_NUMB_BITS)) & 1);
-}
-
-static void secp256k1_num_inc(secp256k1_num_t *r) {
-    mp_limb_t ret = mpn_add_1(r->data, r->data, r->limbs, (mp_limb_t)1);
-    if (ret) {
-        VERIFY_CHECK(r->limbs < 2*NUM_LIMBS);
-        r->data[r->limbs++] = ret;
-    }
-}
-
-static void secp256k1_num_set_hex(secp256k1_num_t *r, const char *a, int alen) {
-    static const unsigned char cvt[256] = {
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 1, 2, 3, 4, 5, 6,7,8,9,0,0,0,0,0,0,
-        0,10,11,12,13,14,15,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0,10,11,12,13,14,15,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
-        0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0
-    };
-    unsigned char num[257] = {};
-    for (int i=0; i<alen; i++) {
-        num[i] = cvt[(unsigned char)a[i]];
-    }
-    r->limbs = mpn_set_str(r->data, num, alen, 16);
-    r->neg = 0;
-    while (r->limbs > 1 && r->data[r->limbs-1] == 0) r->limbs--;
-}
-
-static void secp256k1_num_get_hex(char *r, int rlen, const secp256k1_num_t *a) {
-    static const unsigned char cvt[16] = {'0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F'};
-    unsigned char *tmp = malloc(257);
-    mp_size_t len = mpn_get_str(tmp, 16, (mp_limb_t*)a->data, a->limbs);
-    VERIFY_CHECK(len <= rlen);
-    for (int i=0; i<len; i++) {
-        VERIFY_CHECK(rlen-len+i >= 0);
-        VERIFY_CHECK(rlen-len+i < rlen);
-        VERIFY_CHECK(tmp[i] < 16);
-        r[rlen-len+i] = cvt[tmp[i]];
-    }
-    for (int i=0; i<rlen-len; i++) {
-        VERIFY_CHECK(i >= 0);
-        VERIFY_CHECK(i < rlen);
-        r[i] = cvt[0];
-    }
-    free(tmp);
-}
-
-static void secp256k1_num_split(secp256k1_num_t *rl, secp256k1_num_t *rh, const secp256k1_num_t *a, int bits) {
-    VERIFY_CHECK(bits > 0);
-    rh->neg = a->neg;
-    if (bits >= a->limbs * GMP_NUMB_BITS) {
-        *rl = *a;
-        rh->limbs = 1;
-        rh->data[0] = 0;
-        return;
-    }
-    rl->limbs = 0;
-    rl->neg = a->neg;
-    int left = bits;
-    while (left >= GMP_NUMB_BITS) {
-        rl->data[rl->limbs] = a->data[rl->limbs];
-        rl->limbs++;
-        left -= GMP_NUMB_BITS;
-    }
-    if (left == 0) {
-        mpn_copyi(rh->data, a->data + rl->limbs, a->limbs - rl->limbs);
-        rh->limbs = a->limbs - rl->limbs;
-    } else {
-        mpn_rshift(rh->data, a->data + rl->limbs, a->limbs - rl->limbs, left);
-        rh->limbs = a->limbs - rl->limbs;
-        while (rh->limbs>1 && rh->data[rh->limbs-1]==0) rh->limbs--;
-    }
-    if (left > 0) {
-        rl->data[rl->limbs] = a->data[rl->limbs] & ((((mp_limb_t)1) << left) - 1);
-        rl->limbs++;
+static void secp256k1_num_shift(secp256k1_num_t *r, int bits) {
+    if (bits % GMP_NUMB_BITS) {
+        // Shift within limbs.
+        mpn_rshift(r->data, r->data, r->limbs, bits % GMP_NUMB_BITS);
+    }
+    if (bits >= GMP_NUMB_BITS) {
+        // Shift full limbs.
+        for (int i = 0; i < r->limbs; i++) {
+            int index = i + (bits / GMP_NUMB_BITS);
+            if (index < r->limbs && index < 2*NUM_LIMBS) {
+                r->data[i] = r->data[index];
+            } else {
+                r->data[i] = 0;
+            }
+        }
     }
-    while (rl->limbs>1 && rl->data[rl->limbs-1]==0) rl->limbs--;
+    while (r->limbs>1 && r->data[r->limbs-1]==0) r->limbs--;
 }
 
 static void secp256k1_num_negate(secp256k1_num_t *r) {
     r->neg ^= 1;
 }
 
-static int secp256k1_num_get_bits(const secp256k1_num_t *a, int offset, int count) {
-    int ret = 0;
-    for (int i = 0; i < count; i++) {
-        ret |= ((a->data[(offset + i) / GMP_NUMB_BITS] >> ((offset + i) % GMP_NUMB_BITS)) & 1) << i;
-    }
-    return ret;
-}
-
 #endif
diff --git a/src/secp256k1/src/num_impl.h b/src/secp256k1/src/num_impl.h
index f73d3ceea8..0b0e3a072a 100644
--- a/src/secp256k1/src/num_impl.h
+++ b/src/secp256k1/src/num_impl.h
@@ -15,6 +15,8 @@
 
 #if defined(USE_NUM_GMP)
 #include "num_gmp_impl.h"
+#elif defined(USE_NUM_NONE)
+/* Nothing. */
 #else
 #error "Please select num implementation"
 #endif
diff --git a/src/secp256k1/src/scalar.h b/src/secp256k1/src/scalar.h
index 3baacb3721..2f5ba0d447 100644
--- a/src/secp256k1/src/scalar.h
+++ b/src/secp256k1/src/scalar.h
@@ -21,20 +21,32 @@
 #error "Please select scalar implementation"
 #endif
 
+static void secp256k1_scalar_start(void);
+static void secp256k1_scalar_stop(void);
+
 /** Clear a scalar to prevent the leak of sensitive data. */
 static void secp256k1_scalar_clear(secp256k1_scalar_t *r);
 
-/** Access bits from a scalar. */
-static int secp256k1_scalar_get_bits(const secp256k1_scalar_t *a, int offset, int count);
+/** Access bits from a scalar. All requested bits must belong to the same 32-bit limb. */
+static unsigned int secp256k1_scalar_get_bits(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count);
+
+/** Access bits from a scalar. Not constant time. */
+static unsigned int secp256k1_scalar_get_bits_var(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count);
 
 /** Set a scalar from a big endian byte array. */
 static void secp256k1_scalar_set_b32(secp256k1_scalar_t *r, const unsigned char *bin, int *overflow);
 
+/** Set a scalar to an unsigned integer. */
+static void secp256k1_scalar_set_int(secp256k1_scalar_t *r, unsigned int v);
+
 /** Convert a scalar to a byte array. */
 static void secp256k1_scalar_get_b32(unsigned char *bin, const secp256k1_scalar_t* a);
 
-/** Add two scalars together (modulo the group order). */
-static void secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b);
+/** Add two scalars together (modulo the group order). Returns whether it overflowed. */
+static int secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b);
+
+/** Add a power of two to a scalar. The result is not allowed to overflow. */
+static void secp256k1_scalar_add_bit(secp256k1_scalar_t *r, unsigned int bit);
 
 /** Multiply two scalars (modulo the group order). */
 static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b);
@@ -45,6 +57,9 @@ static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t
 /** Compute the inverse of a scalar (modulo the group order). */
 static void secp256k1_scalar_inverse(secp256k1_scalar_t *r, const secp256k1_scalar_t *a);
 
+/** Compute the inverse of a scalar (modulo the group order), without constant-time guarantee. */
+static void secp256k1_scalar_inverse_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *a);
+
 /** Compute the complement of a scalar (modulo the group order). */
 static void secp256k1_scalar_negate(secp256k1_scalar_t *r, const secp256k1_scalar_t *a);
 
@@ -57,7 +72,25 @@ static int secp256k1_scalar_is_one(const secp256k1_scalar_t *a);
 /** Check whether a scalar is higher than the group order divided by 2. */
 static int secp256k1_scalar_is_high(const secp256k1_scalar_t *a);
 
+#ifndef USE_NUM_NONE
 /** Convert a scalar to a number. */
 static void secp256k1_scalar_get_num(secp256k1_num_t *r, const secp256k1_scalar_t *a);
 
+/** Get the order of the group as a number. */
+static void secp256k1_scalar_order_get_num(secp256k1_num_t *r);
+#endif
+
+/** Compare two scalars. */
+static int secp256k1_scalar_eq(const secp256k1_scalar_t *a, const secp256k1_scalar_t *b);
+
+static void secp256k1_scalar_split_128(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a);
+
+#ifdef USE_ENDOMORPHISM
+/** Find r1 and r2 such that r1+r2*lambda = a, and r1 and r2 are maximum 128 bits long (see secp256k1_gej_mul_lambda). */
+static void secp256k1_scalar_split_lambda_var(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a);
+#endif
+
+/** Multiply a and b (without taking the modulus!), divide by 2**shift, and round to the nearest integer. Shift must be at least 256. */
+static void secp256k1_scalar_mul_shift_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b, unsigned int shift);
+
 #endif
diff --git a/src/secp256k1/src/scalar_4x64_impl.h b/src/secp256k1/src/scalar_4x64_impl.h
index f78718234f..d144775220 100644
--- a/src/secp256k1/src/scalar_4x64_impl.h
+++ b/src/secp256k1/src/scalar_4x64_impl.h
@@ -33,9 +33,27 @@ SECP256K1_INLINE static void secp256k1_scalar_clear(secp256k1_scalar_t *r) {
     r->d[3] = 0;
 }
 
-SECP256K1_INLINE static int secp256k1_scalar_get_bits(const secp256k1_scalar_t *a, int offset, int count) {
-    VERIFY_CHECK((offset + count - 1) / 64 == offset / 64);
-    return (a->d[offset / 64] >> (offset % 64)) & ((((uint64_t)1) << count) - 1);
+SECP256K1_INLINE static void secp256k1_scalar_set_int(secp256k1_scalar_t *r, unsigned int v) {
+    r->d[0] = v;
+    r->d[1] = 0;
+    r->d[2] = 0;
+    r->d[3] = 0;
+}
+
+SECP256K1_INLINE static unsigned int secp256k1_scalar_get_bits(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count) {
+    VERIFY_CHECK((offset + count - 1) >> 6 == offset >> 6);
+    return (a->d[offset >> 6] >> (offset & 0x3F)) & ((((uint64_t)1) << count) - 1);
+}
+
+SECP256K1_INLINE static unsigned int secp256k1_scalar_get_bits_var(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count) {
+    VERIFY_CHECK(count < 32);
+    VERIFY_CHECK(offset + count <= 256);
+    if ((offset + count - 1) >> 6 == offset >> 6) {
+        return secp256k1_scalar_get_bits(a, offset, count);
+    } else {
+        VERIFY_CHECK((offset >> 6) + 1 < 4);
+        return ((a->d[offset >> 6] >> (offset & 0x3F)) | (a->d[(offset >> 6) + 1] << (64 - (offset & 0x3F)))) & ((((uint64_t)1) << count) - 1);
+    }
 }
 
 SECP256K1_INLINE static int secp256k1_scalar_check_overflow(const secp256k1_scalar_t *a) {
@@ -63,7 +81,7 @@ SECP256K1_INLINE static int secp256k1_scalar_reduce(secp256k1_scalar_t *r, unsig
     return overflow;
 }
 
-static void secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+static int secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
     uint128_t t = (uint128_t)a->d[0] + b->d[0];
     r->d[0] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
     t += (uint128_t)a->d[1] + b->d[1];
@@ -72,7 +90,26 @@ static void secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t
     r->d[2] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
     t += (uint128_t)a->d[3] + b->d[3];
     r->d[3] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
-    secp256k1_scalar_reduce(r, t + secp256k1_scalar_check_overflow(r));
+    int overflow = t + secp256k1_scalar_check_overflow(r);
+    VERIFY_CHECK(overflow == 0 || overflow == 1);
+    secp256k1_scalar_reduce(r, overflow);
+    return overflow;
+}
+
+static void secp256k1_scalar_add_bit(secp256k1_scalar_t *r, unsigned int bit) {
+    VERIFY_CHECK(bit < 256);
+    uint128_t t = (uint128_t)r->d[0] + (((uint64_t)((bit >> 6) == 0)) << (bit & 0x3F));
+    r->d[0] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
+    t += (uint128_t)r->d[1] + (((uint64_t)((bit >> 6) == 1)) << (bit & 0x3F));
+    r->d[1] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
+    t += (uint128_t)r->d[2] + (((uint64_t)((bit >> 6) == 2)) << (bit & 0x3F));
+    r->d[2] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
+    t += (uint128_t)r->d[3] + (((uint64_t)((bit >> 6) == 3)) << (bit & 0x3F));
+    r->d[3] = t & 0xFFFFFFFFFFFFFFFFULL;
+#ifdef VERIFY
+    VERIFY_CHECK((t >> 64) == 0);
+    VERIFY_CHECK(secp256k1_scalar_check_overflow(r) == 0);
+#endif
 }
 
 static void secp256k1_scalar_set_b32(secp256k1_scalar_t *r, const unsigned char *b32, int *overflow) {
@@ -280,13 +317,11 @@ static void secp256k1_scalar_reduce_512(secp256k1_scalar_t *r, const uint64_t *l
     secp256k1_scalar_reduce(r, c + secp256k1_scalar_check_overflow(r));
 }
 
-static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+static void secp256k1_scalar_mul_512(uint64_t l[8], const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
     /* 160 bit accumulator. */
     uint64_t c0 = 0, c1 = 0;
     uint32_t c2 = 0;
 
-    uint64_t l[8];
-
     /* l[0..7] = a[0..3] * b[0..3]. */
     muladd_fast(a->d[0], b->d[0]);
     extract_fast(l[0]);
@@ -313,17 +348,13 @@ static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t
     extract_fast(l[6]);
     VERIFY_CHECK(c1 <= 0);
     l[7] = c0;
-
-    secp256k1_scalar_reduce_512(r, l);
 }
 
-static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t *a) {
+static void secp256k1_scalar_sqr_512(uint64_t l[8], const secp256k1_scalar_t *a) {
     /* 160 bit accumulator. */
     uint64_t c0 = 0, c1 = 0;
     uint32_t c2 = 0;
 
-    uint64_t l[8];
-
     /* l[0..7] = a[0..3] * b[0..3]. */
     muladd_fast(a->d[0], a->d[0]);
     extract_fast(l[0]);
@@ -344,8 +375,6 @@ static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t
     extract_fast(l[6]);
     VERIFY_CHECK(c1 == 0);
     l[7] = c0;
-
-    secp256k1_scalar_reduce_512(r, l);
 }
 
 #undef sumadd
@@ -356,4 +385,47 @@ static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t
 #undef extract
 #undef extract_fast
 
+static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+    uint64_t l[8];
+    secp256k1_scalar_mul_512(l, a, b);
+    secp256k1_scalar_reduce_512(r, l);
+}
+
+static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t *a) {
+    uint64_t l[8];
+    secp256k1_scalar_sqr_512(l, a);
+    secp256k1_scalar_reduce_512(r, l);
+}
+
+static void secp256k1_scalar_split_128(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) {
+    r1->d[0] = a->d[0];
+    r1->d[1] = a->d[1];
+    r1->d[2] = 0;
+    r1->d[3] = 0;
+    r2->d[0] = a->d[2];
+    r2->d[1] = a->d[3];
+    r2->d[2] = 0;
+    r2->d[3] = 0;
+}
+
+SECP256K1_INLINE static int secp256k1_scalar_eq(const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+    return ((a->d[0] ^ b->d[0]) | (a->d[1] ^ b->d[1]) | (a->d[2] ^ b->d[2]) | (a->d[3] ^ b->d[3])) == 0;
+}
+
+SECP256K1_INLINE static void secp256k1_scalar_mul_shift_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b, unsigned int shift) {
+    VERIFY_CHECK(shift >= 256);
+    uint64_t l[8];
+    secp256k1_scalar_mul_512(l, a, b);
+    unsigned int shiftlimbs = shift >> 6;
+    unsigned int shiftlow = shift & 0x3F;
+    unsigned int shifthigh = 64 - shiftlow;
+    r->d[0] = shift < 512 ? (l[0 + shiftlimbs] >> shiftlow | (shift < 448 && shiftlow ? (l[1 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[1] = shift < 448 ? (l[1 + shiftlimbs] >> shiftlow | (shift < 384 && shiftlow ? (l[2 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[2] = shift < 384 ? (l[2 + shiftlimbs] >> shiftlow | (shift < 320 && shiftlow ? (l[3 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[3] = shift < 320 ? (l[3 + shiftlimbs] >> shiftlow) : 0;
+    if ((l[(shift - 1) >> 6] >> ((shift - 1) & 0x3f)) & 1) {
+        secp256k1_scalar_add_bit(r, 0);
+    }
+}
+
 #endif
diff --git a/src/secp256k1/src/scalar_8x32_impl.h b/src/secp256k1/src/scalar_8x32_impl.h
index e58be1365f..915cbcddbe 100644
--- a/src/secp256k1/src/scalar_8x32_impl.h
+++ b/src/secp256k1/src/scalar_8x32_impl.h
@@ -45,9 +45,31 @@ SECP256K1_INLINE static void secp256k1_scalar_clear(secp256k1_scalar_t *r) {
     r->d[7] = 0;
 }
 
-SECP256K1_INLINE static int secp256k1_scalar_get_bits(const secp256k1_scalar_t *a, int offset, int count) {
-    VERIFY_CHECK((offset + count - 1) / 32 == offset / 32);
-    return (a->d[offset / 32] >> (offset % 32)) & ((1 << count) - 1);
+SECP256K1_INLINE static void secp256k1_scalar_set_int(secp256k1_scalar_t *r, unsigned int v) {
+    r->d[0] = v;
+    r->d[1] = 0;
+    r->d[2] = 0;
+    r->d[3] = 0;
+    r->d[4] = 0;
+    r->d[5] = 0;
+    r->d[6] = 0;
+    r->d[7] = 0;
+}
+
+SECP256K1_INLINE static unsigned int secp256k1_scalar_get_bits(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count) {
+    VERIFY_CHECK((offset + count - 1) >> 5 == offset >> 5);
+    return (a->d[offset >> 5] >> (offset & 0x1F)) & ((1 << count) - 1);
+}
+
+SECP256K1_INLINE static unsigned int secp256k1_scalar_get_bits_var(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count) {
+    VERIFY_CHECK(count < 32);
+    VERIFY_CHECK(offset + count <= 256);
+    if ((offset + count - 1) >> 5 == offset >> 5) {
+        return secp256k1_scalar_get_bits(a, offset, count);
+    } else {
+        VERIFY_CHECK((offset >> 5) + 1 < 8);
+        return ((a->d[offset >> 5] >> (offset & 0x1F)) | (a->d[(offset >> 5) + 1] << (32 - (offset & 0x1F)))) & ((((uint32_t)1) << count) - 1);
+    }
 }
 
 SECP256K1_INLINE static int secp256k1_scalar_check_overflow(const secp256k1_scalar_t *a) {
@@ -89,7 +111,7 @@ SECP256K1_INLINE static int secp256k1_scalar_reduce(secp256k1_scalar_t *r, uint3
     return overflow;
 }
 
-static void secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+static int secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
     uint64_t t = (uint64_t)a->d[0] + b->d[0];
     r->d[0] = t & 0xFFFFFFFFULL; t >>= 32;
     t += (uint64_t)a->d[1] + b->d[1];
@@ -106,7 +128,34 @@ static void secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t
     r->d[6] = t & 0xFFFFFFFFULL; t >>= 32;
     t += (uint64_t)a->d[7] + b->d[7];
     r->d[7] = t & 0xFFFFFFFFULL; t >>= 32;
-    secp256k1_scalar_reduce(r, t + secp256k1_scalar_check_overflow(r));
+    int overflow = t + secp256k1_scalar_check_overflow(r);
+    VERIFY_CHECK(overflow == 0 || overflow == 1);
+    secp256k1_scalar_reduce(r, overflow);
+    return overflow;
+}
+
+static void secp256k1_scalar_add_bit(secp256k1_scalar_t *r, unsigned int bit) {
+    VERIFY_CHECK(bit < 256);
+    uint64_t t = (uint64_t)r->d[0] + (((uint32_t)((bit >> 5) == 0)) << (bit & 0x1F));
+    r->d[0] = t & 0xFFFFFFFFULL; t >>= 32;
+    t += (uint64_t)r->d[1] + (((uint32_t)((bit >> 5) == 1)) << (bit & 0x1F));
+    r->d[1] = t & 0xFFFFFFFFULL; t >>= 32;
+    t += (uint64_t)r->d[2] + (((uint32_t)((bit >> 5) == 2)) << (bit & 0x1F));
+    r->d[2] = t & 0xFFFFFFFFULL; t >>= 32;
+    t += (uint64_t)r->d[3] + (((uint32_t)((bit >> 5) == 3)) << (bit & 0x1F));
+    r->d[3] = t & 0xFFFFFFFFULL; t >>= 32;
+    t += (uint64_t)r->d[4] + (((uint32_t)((bit >> 5) == 4)) << (bit & 0x1F));
+    r->d[4] = t & 0xFFFFFFFFULL; t >>= 32;
+    t += (uint64_t)r->d[5] + (((uint32_t)((bit >> 5) == 5)) << (bit & 0x1F));
+    r->d[5] = t & 0xFFFFFFFFULL; t >>= 32;
+    t += (uint64_t)r->d[6] + (((uint32_t)((bit >> 5) == 6)) << (bit & 0x1F));
+    r->d[6] = t & 0xFFFFFFFFULL; t >>= 32;
+    t += (uint64_t)r->d[7] + (((uint32_t)((bit >> 5) == 7)) << (bit & 0x1F));
+    r->d[7] = t & 0xFFFFFFFFULL;
+#ifdef VERIFY
+    VERIFY_CHECK((t >> 32) == 0);
+    VERIFY_CHECK(secp256k1_scalar_check_overflow(r) == 0);
+#endif
 }
 
 static void secp256k1_scalar_set_b32(secp256k1_scalar_t *r, const unsigned char *b32, int *overflow) {
@@ -405,12 +454,10 @@ static void secp256k1_scalar_reduce_512(secp256k1_scalar_t *r, const uint32_t *l
     secp256k1_scalar_reduce(r, c + secp256k1_scalar_check_overflow(r));
 }
 
-static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+static void secp256k1_scalar_mul_512(uint32_t l[16], const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
     /* 96 bit accumulator. */
     uint32_t c0 = 0, c1 = 0, c2 = 0;
 
-    uint32_t l[16];
-
     /* l[0..15] = a[0..7] * b[0..7]. */
     muladd_fast(a->d[0], b->d[0]);
     extract_fast(l[0]);
@@ -493,16 +540,12 @@ static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t
     extract_fast(l[14]);
     VERIFY_CHECK(c1 == 0);
     l[15] = c0;
-
-    secp256k1_scalar_reduce_512(r, l);
 }
 
-static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t *a) {
+static void secp256k1_scalar_sqr_512(uint32_t l[16], const secp256k1_scalar_t *a) {
     /* 96 bit accumulator. */
     uint32_t c0 = 0, c1 = 0, c2 = 0;
 
-    uint32_t l[16];
-
     /* l[0..15] = a[0..7]^2. */
     muladd_fast(a->d[0], a->d[0]);
     extract_fast(l[0]);
@@ -557,8 +600,6 @@ static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t
     extract_fast(l[14]);
     VERIFY_CHECK(c1 == 0);
     l[15] = c0;
-
-    secp256k1_scalar_reduce_512(r, l);
 }
 
 #undef sumadd
@@ -569,4 +610,59 @@ static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t
 #undef extract
 #undef extract_fast
 
+static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+    uint32_t l[16];
+    secp256k1_scalar_mul_512(l, a, b);
+    secp256k1_scalar_reduce_512(r, l);
+}
+
+static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t *a) {
+    uint32_t l[16];
+    secp256k1_scalar_sqr_512(l, a);
+    secp256k1_scalar_reduce_512(r, l);
+}
+
+static void secp256k1_scalar_split_128(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) {
+    r1->d[0] = a->d[0];
+    r1->d[1] = a->d[1];
+    r1->d[2] = a->d[2];
+    r1->d[3] = a->d[3];
+    r1->d[4] = 0;
+    r1->d[5] = 0;
+    r1->d[6] = 0;
+    r1->d[7] = 0;
+    r2->d[0] = a->d[4];
+    r2->d[1] = a->d[5];
+    r2->d[2] = a->d[6];
+    r2->d[3] = a->d[7];
+    r2->d[4] = 0;
+    r2->d[5] = 0;
+    r2->d[6] = 0;
+    r2->d[7] = 0;
+}
+
+SECP256K1_INLINE static int secp256k1_scalar_eq(const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
+    return ((a->d[0] ^ b->d[0]) | (a->d[1] ^ b->d[1]) | (a->d[2] ^ b->d[2]) | (a->d[3] ^ b->d[3]) | (a->d[4] ^ b->d[4]) | (a->d[5] ^ b->d[5]) | (a->d[6] ^ b->d[6]) | (a->d[7] ^ b->d[7])) == 0;
+}
+
+SECP256K1_INLINE static void secp256k1_scalar_mul_shift_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b, unsigned int shift) {
+    VERIFY_CHECK(shift >= 256);
+    uint32_t l[16];
+    secp256k1_scalar_mul_512(l, a, b);
+    unsigned int shiftlimbs = shift >> 5;
+    unsigned int shiftlow = shift & 0x1F;
+    unsigned int shifthigh = 32 - shiftlow;
+    r->d[0] = shift < 512 ? (l[0 + shiftlimbs] >> shiftlow | (shift < 480 && shiftlow ? (l[1 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[1] = shift < 480 ? (l[1 + shiftlimbs] >> shiftlow | (shift < 448 && shiftlow ? (l[2 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[2] = shift < 448 ? (l[2 + shiftlimbs] >> shiftlow | (shift < 416 && shiftlow ? (l[3 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[3] = shift < 416 ? (l[3 + shiftlimbs] >> shiftlow | (shift < 384 && shiftlow ? (l[4 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[4] = shift < 384 ? (l[4 + shiftlimbs] >> shiftlow | (shift < 352 && shiftlow ? (l[5 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[5] = shift < 352 ? (l[5 + shiftlimbs] >> shiftlow | (shift < 320 && shiftlow ? (l[6 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[6] = shift < 320 ? (l[6 + shiftlimbs] >> shiftlow | (shift < 288 && shiftlow ? (l[7 + shiftlimbs] << shifthigh) : 0)) : 0;
+    r->d[7] = shift < 288 ? (l[7 + shiftlimbs] >> shiftlow)  : 0;
+    if ((l[(shift - 1) >> 5] >> ((shift - 1) & 0x1f)) & 1) {
+        secp256k1_scalar_add_bit(r, 0);
+    }
+}
+
 #endif
diff --git a/src/secp256k1/src/scalar_impl.h b/src/secp256k1/src/scalar_impl.h
index ddc5061c76..7fc159df77 100644
--- a/src/secp256k1/src/scalar_impl.h
+++ b/src/secp256k1/src/scalar_impl.h
@@ -9,6 +9,7 @@
 
 #include <string.h>
 
+#include "group.h"
 #include "scalar.h"
 
 #if defined HAVE_CONFIG_H
@@ -23,12 +24,132 @@
 #error "Please select scalar implementation"
 #endif
 
+typedef struct {
+#ifndef USE_NUM_NONE
+    secp256k1_num_t order;
+#endif
+#ifdef USE_ENDOMORPHISM
+    secp256k1_scalar_t minus_lambda, minus_b1, minus_b2, g1, g2;
+#endif
+} secp256k1_scalar_consts_t;
+
+static const secp256k1_scalar_consts_t *secp256k1_scalar_consts = NULL;
+
+static void secp256k1_scalar_start(void) {
+    if (secp256k1_scalar_consts != NULL)
+        return;
+
+    /* Allocate. */
+    secp256k1_scalar_consts_t *ret = (secp256k1_scalar_consts_t*)malloc(sizeof(secp256k1_scalar_consts_t));
+
+#ifndef USE_NUM_NONE
+    static const unsigned char secp256k1_scalar_consts_order[] = {
+        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
+        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,
+        0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B,
+        0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41
+    };
+    secp256k1_num_set_bin(&ret->order, secp256k1_scalar_consts_order, sizeof(secp256k1_scalar_consts_order));
+#endif
+#ifdef USE_ENDOMORPHISM
+    /**
+     * Lambda is a scalar which has the property for secp256k1 that point multiplication by
+     * it is efficiently computable (see secp256k1_gej_mul_lambda). */
+    static const unsigned char secp256k1_scalar_consts_lambda[32] = {
+         0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,
+         0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
+         0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,
+         0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72
+    };
+    /**
+     * "Guide to Elliptic Curve Cryptography" (Hankerson, Menezes, Vanstone) gives an algorithm
+     * (algorithm 3.74) to find k1 and k2 given k, such that k1 + k2 * lambda == k mod n, and k1
+     * and k2 have a small size.
+     * It relies on constants a1, b1, a2, b2. These constants for the value of lambda above are:
+     *
+     * - a1 =      {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
+     * - b1 =     -{0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3}
+     * - a2 = {0x01,0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8}
+     * - b2 =      {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
+     *
+     * The algorithm then computes c1 = round(b1 * k / n) and c2 = round(b2 * k / n), and gives
+     * k1 = k - (c1*a1 + c2*a2) and k2 = -(c1*b1 + c2*b2). Instead, we use modular arithmetic, and
+     * compute k1 as k - k2 * lambda, avoiding the need for constants a1 and a2.
+     *
+     * g1, g2 are precomputed constants used to replace division with a rounded multiplication
+     * when decomposing the scalar for an endomorphism-based point multiplication.
+     *
+     * The possibility of using precomputed estimates is mentioned in "Guide to Elliptic Curve
+     * Cryptography" (Hankerson, Menezes, Vanstone) in section 3.5.
+     *
+     * The derivation is described in the paper "Efficient Software Implementation of Public-Key
+     * Cryptography on Sensor Networks Using the MSP430X Microcontroller" (Gouvea, Oliveira, Lopez),
+     * Section 4.3 (here we use a somewhat higher-precision estimate):
+     * d = a1*b2 - b1*a2
+     * g1 = round((2^272)*b2/d)
+     * g2 = round((2^272)*b1/d)
+     *
+     * (Note that 'd' is also equal to the curve order here because [a1,b1] and [a2,b2] are found
+     * as outputs of the Extended Euclidean Algorithm on inputs 'order' and 'lambda').
+     */
+    static const unsigned char secp256k1_scalar_consts_minus_b1[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,
+        0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3
+    };
+    static const unsigned char secp256k1_scalar_consts_b2[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,
+        0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15
+    };
+    static const unsigned char secp256k1_scalar_consts_g1[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0x30,0x86,
+        0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,
+        0x90,0xe4,0x92,0x84,0xeb,0x15,0x3d,0xab
+    };
+    static const unsigned char secp256k1_scalar_consts_g2[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0xe4,0x43,
+        0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,
+        0x7f,0xa9,0x0a,0xbf,0xe4,0xc4,0x22,0x12
+    };
+
+    secp256k1_scalar_set_b32(&ret->minus_lambda, secp256k1_scalar_consts_lambda, NULL);
+    secp256k1_scalar_negate(&ret->minus_lambda, &ret->minus_lambda);
+    secp256k1_scalar_set_b32(&ret->minus_b1, secp256k1_scalar_consts_minus_b1, NULL);
+    secp256k1_scalar_set_b32(&ret->minus_b2, secp256k1_scalar_consts_b2, NULL);
+    secp256k1_scalar_negate(&ret->minus_b2, &ret->minus_b2);
+    secp256k1_scalar_set_b32(&ret->g1, secp256k1_scalar_consts_g1, NULL);
+    secp256k1_scalar_set_b32(&ret->g2, secp256k1_scalar_consts_g2, NULL);
+#endif
+
+    /* Set the global pointer. */
+    secp256k1_scalar_consts = ret;
+}
+
+static void secp256k1_scalar_stop(void) {
+    if (secp256k1_scalar_consts == NULL)
+        return;
+
+    secp256k1_scalar_consts_t *c = (secp256k1_scalar_consts_t*)secp256k1_scalar_consts;
+    secp256k1_scalar_consts = NULL;
+    free(c);
+}
+
+#ifndef USE_NUM_NONE
 static void secp256k1_scalar_get_num(secp256k1_num_t *r, const secp256k1_scalar_t *a) {
     unsigned char c[32];
     secp256k1_scalar_get_b32(c, a);
     secp256k1_num_set_bin(r, c, 32);
 }
 
+static void secp256k1_scalar_order_get_num(secp256k1_num_t *r) {
+    *r = secp256k1_scalar_consts->order;
+}
+#endif
 
 static void secp256k1_scalar_inverse(secp256k1_scalar_t *r, const secp256k1_scalar_t *x) {
     /* First compute x ^ (2^N - 1) for some values of N. */
@@ -181,4 +302,35 @@ static void secp256k1_scalar_inverse(secp256k1_scalar_t *r, const secp256k1_scal
     secp256k1_scalar_mul(r, t, &x6); /* 111111 */
 }
 
+static void secp256k1_scalar_inverse_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *x) {
+#if defined(USE_SCALAR_INV_BUILTIN)
+    secp256k1_scalar_inverse(r, x);
+#elif defined(USE_SCALAR_INV_NUM)
+    unsigned char b[32];
+    secp256k1_scalar_get_b32(b, x);
+    secp256k1_num_t n;
+    secp256k1_num_set_bin(&n, b, 32);
+    secp256k1_num_mod_inverse(&n, &n, &secp256k1_scalar_consts->order);
+    secp256k1_num_get_bin(b, 32, &n);
+    secp256k1_scalar_set_b32(r, b, NULL);
+#else
+#error "Please select scalar inverse implementation"
+#endif
+}
+
+#ifdef USE_ENDOMORPHISM
+static void secp256k1_scalar_split_lambda_var(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) {
+    VERIFY_CHECK(r1 != a);
+    VERIFY_CHECK(r2 != a);
+    secp256k1_scalar_t c1, c2;
+    secp256k1_scalar_mul_shift_var(&c1, a, &secp256k1_scalar_consts->g1, 272);
+    secp256k1_scalar_mul_shift_var(&c2, a, &secp256k1_scalar_consts->g2, 272);
+    secp256k1_scalar_mul(&c1, &c1, &secp256k1_scalar_consts->minus_b1);
+    secp256k1_scalar_mul(&c2, &c2, &secp256k1_scalar_consts->minus_b2);
+    secp256k1_scalar_add(r2, &c1, &c2);
+    secp256k1_scalar_mul(r1, r2, &secp256k1_scalar_consts->minus_lambda);
+    secp256k1_scalar_add(r1, r1, a);
+}
+#endif
+
 #endif
diff --git a/src/secp256k1/src/secp256k1.c b/src/secp256k1/src/secp256k1.c
index 1ab5b3722c..20fc27df74 100644
--- a/src/secp256k1/src/secp256k1.c
+++ b/src/secp256k1/src/secp256k1.c
@@ -21,6 +21,8 @@
 void secp256k1_start(unsigned int flags) {
     secp256k1_fe_start();
     secp256k1_ge_start();
+    secp256k1_scalar_start();
+    secp256k1_ecdsa_start();
     if (flags & SECP256K1_START_SIGN) {
         secp256k1_ecmult_gen_start();
     }
@@ -32,6 +34,8 @@ void secp256k1_start(unsigned int flags) {
 void secp256k1_stop(void) {
     secp256k1_ecmult_stop();
     secp256k1_ecmult_gen_stop();
+    secp256k1_ecdsa_stop();
+    secp256k1_scalar_stop();
     secp256k1_ge_stop();
     secp256k1_fe_stop();
 }
@@ -43,11 +47,13 @@ int secp256k1_ecdsa_verify(const unsigned char *msg, int msglen, const unsigned
     DEBUG_CHECK(sig != NULL);
     DEBUG_CHECK(pubkey != NULL);
 
+    unsigned char msg32[32] = {0};
+    memcpy(msg32 + 32 - msglen, msg, msglen);
     int ret = -3;
-    secp256k1_num_t m;
+    secp256k1_scalar_t m;
     secp256k1_ecdsa_sig_t s;
     secp256k1_ge_t q;
-    secp256k1_num_set_bin(&m, msg, msglen);
+    secp256k1_scalar_set_b32(&m, msg32, NULL);
 
     if (!secp256k1_eckey_pubkey_parse(&q, pubkey, pubkeylen)) {
         ret = -1;
@@ -123,8 +129,8 @@ int secp256k1_ecdsa_sign_compact(const unsigned char *message, int messagelen, u
         ret = secp256k1_ecdsa_sig_sign(&sig, &sec, &msg, &non, recid);
     }
     if (ret) {
-        secp256k1_num_get_bin(sig64, 32, &sig.r);
-        secp256k1_num_get_bin(sig64 + 32, 32, &sig.s);
+        secp256k1_scalar_get_b32(sig64, &sig.r);
+        secp256k1_scalar_get_b32(sig64 + 32, &sig.s);
     }
     secp256k1_scalar_clear(&msg);
     secp256k1_scalar_clear(&non);
@@ -142,11 +148,20 @@ int secp256k1_ecdsa_recover_compact(const unsigned char *msg, int msglen, const
     DEBUG_CHECK(recid >= 0 && recid <= 3);
 
     int ret = 0;
-    secp256k1_num_t m;
+    unsigned char msg32[32] = {0};
+    memcpy(msg32 + 32 - msglen, msg, msglen);
+    secp256k1_scalar_t m;
     secp256k1_ecdsa_sig_t sig;
-    secp256k1_num_set_bin(&sig.r, sig64, 32);
-    secp256k1_num_set_bin(&sig.s, sig64 + 32, 32);
-    secp256k1_num_set_bin(&m, msg, msglen);
+    int overflow = 0;
+    secp256k1_scalar_set_b32(&sig.r, sig64, &overflow);
+    if (overflow) {
+        return 0;
+    }
+    secp256k1_scalar_set_b32(&sig.s, sig64 + 32, &overflow);
+    if (overflow) {
+        return 0;
+    }
+    secp256k1_scalar_set_b32(&m, msg32, NULL);
 
     secp256k1_ge_t q;
     if (secp256k1_ecdsa_sig_recover(&sig, &q, &m, recid)) {
@@ -224,8 +239,12 @@ int secp256k1_ec_pubkey_tweak_add(unsigned char *pubkey, int pubkeylen, const un
     DEBUG_CHECK(pubkey != NULL);
     DEBUG_CHECK(tweak != NULL);
 
-    secp256k1_num_t term;
-    secp256k1_num_set_bin(&term, tweak, 32);
+    secp256k1_scalar_t term;
+    int overflow = 0;
+    secp256k1_scalar_set_b32(&term, tweak, &overflow);
+    if (overflow) {
+        return 0;
+    }
     secp256k1_ge_t p;
     int ret = secp256k1_eckey_pubkey_parse(&p, pubkey, pubkeylen);
     if (ret) {
@@ -264,8 +283,12 @@ int secp256k1_ec_pubkey_tweak_mul(unsigned char *pubkey, int pubkeylen, const un
     DEBUG_CHECK(pubkey != NULL);
     DEBUG_CHECK(tweak != NULL);
 
-    secp256k1_num_t factor;
-    secp256k1_num_set_bin(&factor, tweak, 32);
+    secp256k1_scalar_t factor;
+    int overflow = 0;
+    secp256k1_scalar_set_b32(&factor, tweak, &overflow);
+    if (overflow) {
+        return 0;
+    }
     secp256k1_ge_t p;
     int ret = secp256k1_eckey_pubkey_parse(&p, pubkey, pubkeylen);
     if (ret) {
diff --git a/src/secp256k1/src/tests.c b/src/secp256k1/src/tests.c
index 5d9b8344d9..78cdd67f27 100644
--- a/src/secp256k1/src/tests.c
+++ b/src/secp256k1/src/tests.c
@@ -23,23 +23,13 @@
 
 static int count = 64;
 
-/***** NUM TESTS *****/
-
-void random_num_negate(secp256k1_num_t *num) {
-    if (secp256k1_rand32() & 1)
-        secp256k1_num_negate(num);
-}
-
 void random_field_element_test(secp256k1_fe_t *fe) {
     do {
         unsigned char b32[32];
         secp256k1_rand256_test(b32);
-        secp256k1_num_t num;
-        secp256k1_num_set_bin(&num, b32, 32);
-        if (secp256k1_num_cmp(&num, &secp256k1_fe_consts->p) >= 0)
-            continue;
-        secp256k1_fe_set_b32(fe, b32);
-        break;
+        if (secp256k1_fe_set_b32(fe, b32)) {
+            break;
+        }
     } while(1);
 }
 
@@ -75,19 +65,6 @@ void random_group_element_jacobian_test(secp256k1_gej_t *gej, const secp256k1_ge
     gej->infinity = ge->infinity;
 }
 
-void random_num_order_test(secp256k1_num_t *num) {
-    do {
-        unsigned char b32[32];
-        secp256k1_rand256_test(b32);
-        secp256k1_num_set_bin(num, b32, 32);
-        if (secp256k1_num_is_zero(num))
-            continue;
-        if (secp256k1_num_cmp(num, &secp256k1_ge_consts->order) >= 0)
-            continue;
-        break;
-    } while(1);
-}
-
 void random_scalar_order_test(secp256k1_scalar_t *num) {
     do {
         unsigned char b32[32];
@@ -100,82 +77,36 @@ void random_scalar_order_test(secp256k1_scalar_t *num) {
     } while(1);
 }
 
-void random_num_order(secp256k1_num_t *num) {
+void random_scalar_order(secp256k1_scalar_t *num) {
     do {
         unsigned char b32[32];
         secp256k1_rand256(b32);
-        secp256k1_num_set_bin(num, b32, 32);
-        if (secp256k1_num_is_zero(num))
-            continue;
-        if (secp256k1_num_cmp(num, &secp256k1_ge_consts->order) >= 0)
+        int overflow = 0;
+        secp256k1_scalar_set_b32(num, b32, &overflow);
+        if (overflow || secp256k1_scalar_is_zero(num))
             continue;
         break;
     } while(1);
 }
 
-void test_num_copy_inc_cmp(void) {
-    secp256k1_num_t n1,n2;
-    random_num_order(&n1);
-    secp256k1_num_copy(&n2, &n1);
-    CHECK(secp256k1_num_eq(&n1, &n2));
-    CHECK(secp256k1_num_eq(&n2, &n1));
-    secp256k1_num_inc(&n2);
-    CHECK(!secp256k1_num_eq(&n1, &n2));
-    CHECK(!secp256k1_num_eq(&n2, &n1));
-}
-
+/***** NUM TESTS *****/
 
-void test_num_get_set_hex(void) {
-    secp256k1_num_t n1,n2;
-    random_num_order_test(&n1);
-    char c[64];
-    secp256k1_num_get_hex(c, 64, &n1);
-    secp256k1_num_set_hex(&n2, c, 64);
-    CHECK(secp256k1_num_eq(&n1, &n2));
-    for (int i=0; i<64; i++) {
-        /* check whether the lower 4 bits correspond to the last hex character */
-        int low1 = secp256k1_num_shift(&n1, 4);
-        int lowh = c[63];
-        int low2 = ((lowh>>6)*9+(lowh-'0'))&15;
-        CHECK(low1 == low2);
-        /* shift bits off the hex representation, and compare */
-        memmove(c+1, c, 63);
-        c[0] = '0';
-        secp256k1_num_set_hex(&n2, c, 64);
-        CHECK(secp256k1_num_eq(&n1, &n2));
-    }
+#ifndef USE_NUM_NONE
+void random_num_negate(secp256k1_num_t *num) {
+    if (secp256k1_rand32() & 1)
+        secp256k1_num_negate(num);
 }
 
-void test_num_get_set_bin(void) {
-    secp256k1_num_t n1,n2;
-    random_num_order_test(&n1);
-    unsigned char c[32];
-    secp256k1_num_get_bin(c, 32, &n1);
-    secp256k1_num_set_bin(&n2, c, 32);
-    CHECK(secp256k1_num_eq(&n1, &n2));
-    for (int i=0; i<32; i++) {
-        /* check whether the lower 8 bits correspond to the last byte */
-        int low1 = secp256k1_num_shift(&n1, 8);
-        int low2 = c[31];
-        CHECK(low1 == low2);
-        /* shift bits off the byte representation, and compare */
-        memmove(c+1, c, 31);
-        c[0] = 0;
-        secp256k1_num_set_bin(&n2, c, 32);
-        CHECK(secp256k1_num_eq(&n1, &n2));
-    }
+void random_num_order_test(secp256k1_num_t *num) {
+    secp256k1_scalar_t sc;
+    random_scalar_order_test(&sc);
+    secp256k1_scalar_get_num(num, &sc);
 }
 
-void run_num_int(void) {
-    secp256k1_num_t n1;
-    for (int i=-255; i<256; i++) {
-        unsigned char c1[3] = {};
-        c1[2] = abs(i);
-        unsigned char c2[3] = {0x11,0x22,0x33};
-        secp256k1_num_set_int(&n1, i);
-        secp256k1_num_get_bin(c2, 3, &n1);
-        CHECK(memcmp(c1, c2, 3) == 0);
-    }
+void random_num_order(secp256k1_num_t *num) {
+    secp256k1_scalar_t sc;
+    random_scalar_order(&sc);
+    secp256k1_scalar_get_num(num, &sc);
 }
 
 void test_num_negate(void) {
@@ -229,82 +160,84 @@ void test_num_add_sub(void) {
 
 void run_num_smalltests(void) {
     for (int i=0; i<100*count; i++) {
-        test_num_copy_inc_cmp();
-        test_num_get_set_hex();
-        test_num_get_set_bin();
         test_num_negate();
         test_num_add_sub();
     }
-    run_num_int();
 }
+#endif
 
 /***** SCALAR TESTS *****/
 
-int secp256k1_scalar_eq(const secp256k1_scalar_t *s1, const secp256k1_scalar_t *s2) {
-    secp256k1_scalar_t t;
-    secp256k1_scalar_negate(&t, s2);
-    secp256k1_scalar_add(&t, &t, s1);
-    int ret = secp256k1_scalar_is_zero(&t);
-    return ret;
-}
-
 void scalar_test(void) {
     unsigned char c[32];
 
     /* Set 's' to a random scalar, with value 'snum'. */
-    secp256k1_rand256_test(c);
     secp256k1_scalar_t s;
-    secp256k1_scalar_set_b32(&s, c, NULL);
-    secp256k1_num_t snum;
-    secp256k1_num_set_bin(&snum, c, 32);
-    secp256k1_num_mod(&snum, &secp256k1_ge_consts->order);
+    random_scalar_order_test(&s);
 
     /* Set 's1' to a random scalar, with value 's1num'. */
-    secp256k1_rand256_test(c);
     secp256k1_scalar_t s1;
-    secp256k1_scalar_set_b32(&s1, c, NULL);
-    secp256k1_num_t s1num;
-    secp256k1_num_set_bin(&s1num, c, 32);
-    secp256k1_num_mod(&s1num, &secp256k1_ge_consts->order);
+    random_scalar_order_test(&s1);
 
     /* Set 's2' to a random scalar, with value 'snum2', and byte array representation 'c'. */
-    secp256k1_rand256_test(c);
     secp256k1_scalar_t s2;
-    int overflow = 0;
-    secp256k1_scalar_set_b32(&s2, c, &overflow);
-    secp256k1_num_t s2num;
-    secp256k1_num_set_bin(&s2num, c, 32);
-    secp256k1_num_mod(&s2num, &secp256k1_ge_consts->order);
+    random_scalar_order_test(&s2);
+    secp256k1_scalar_get_b32(c, &s2);
+
+#ifndef USE_NUM_NONE
+    secp256k1_num_t snum, s1num, s2num;
+    secp256k1_scalar_get_num(&snum, &s);
+    secp256k1_scalar_get_num(&s1num, &s1);
+    secp256k1_scalar_get_num(&s2num, &s2);
+
+    secp256k1_num_t order;
+    secp256k1_scalar_order_get_num(&order);
+    secp256k1_num_t half_order = order;
+    secp256k1_num_shift(&half_order, 1);
+#endif
 
     {
         /* Test that fetching groups of 4 bits from a scalar and recursing n(i)=16*n(i-1)+p(i) reconstructs it. */
-        secp256k1_num_t n, t, m;
-        secp256k1_num_set_int(&n, 0);
-        secp256k1_num_set_int(&m, 16);
+        secp256k1_scalar_t n;
+        secp256k1_scalar_set_int(&n, 0);
         for (int i = 0; i < 256; i += 4) {
-            secp256k1_num_set_int(&t, secp256k1_scalar_get_bits(&s, 256 - 4 - i, 4));
-            secp256k1_num_mul(&n, &n, &m);
-            secp256k1_num_add(&n, &n, &t);
+            secp256k1_scalar_t t;
+            secp256k1_scalar_set_int(&t, secp256k1_scalar_get_bits(&s, 256 - 4 - i, 4));
+            for (int j = 0; j < 4; j++) {
+                secp256k1_scalar_add(&n, &n, &n);
+            }
+            secp256k1_scalar_add(&n, &n, &t);
         }
-        CHECK(secp256k1_num_eq(&n, &snum));
+        CHECK(secp256k1_scalar_eq(&n, &s));
     }
 
     {
-        /* Test that get_b32 returns the same as get_bin on the number. */
-        unsigned char r1[32];
-        secp256k1_scalar_get_b32(r1, &s2);
-        unsigned char r2[32];
-        secp256k1_num_get_bin(r2, 32, &s2num);
-        CHECK(memcmp(r1, r2, 32) == 0);
-        /* If no overflow occurred when assigning, it should also be equal to the original byte array. */
-        CHECK((memcmp(r1, c, 32) == 0) == (overflow == 0));
+        /* Test that fetching groups of randomly-sized bits from a scalar and recursing n(i)=b*n(i-1)+p(i) reconstructs it. */
+        secp256k1_scalar_t n;
+        secp256k1_scalar_set_int(&n, 0);
+        int i = 0;
+        while (i < 256) {
+            int now = (secp256k1_rand32() % 15) + 1;
+            if (now + i > 256) {
+                now = 256 - i;
+            }
+            secp256k1_scalar_t t;
+            secp256k1_scalar_set_int(&t, secp256k1_scalar_get_bits_var(&s, 256 - now - i, now));
+            for (int j = 0; j < now; j++) {
+                secp256k1_scalar_add(&n, &n, &n);
+            }
+            secp256k1_scalar_add(&n, &n, &t);
+            i += now;
+        }
+        CHECK(secp256k1_scalar_eq(&n, &s));
     }
 
+#ifndef USE_NUM_NONE
     {
         /* Test that adding the scalars together is equal to adding their numbers together modulo the order. */
         secp256k1_num_t rnum;
         secp256k1_num_add(&rnum, &snum, &s2num);
-        secp256k1_num_mod(&rnum, &secp256k1_ge_consts->order);
+        secp256k1_num_mod(&rnum, &order);
         secp256k1_scalar_t r;
         secp256k1_scalar_add(&r, &s, &s2);
         secp256k1_num_t r2num;
@@ -316,7 +249,7 @@ void scalar_test(void) {
         /* Test that multipying the scalars is equal to multiplying their numbers modulo the order. */
         secp256k1_num_t rnum;
         secp256k1_num_mul(&rnum, &snum, &s2num);
-        secp256k1_num_mod(&rnum, &secp256k1_ge_consts->order);
+        secp256k1_num_mod(&rnum, &order);
         secp256k1_scalar_t r;
         secp256k1_scalar_mul(&r, &s, &s2);
         secp256k1_num_t r2num;
@@ -333,14 +266,14 @@ void scalar_test(void) {
         /* Check that comparison with zero matches comparison with zero on the number. */
         CHECK(secp256k1_num_is_zero(&snum) == secp256k1_scalar_is_zero(&s));
         /* Check that comparison with the half order is equal to testing for high scalar. */
-        CHECK(secp256k1_scalar_is_high(&s) == (secp256k1_num_cmp(&snum, &secp256k1_ge_consts->half_order) > 0));
+        CHECK(secp256k1_scalar_is_high(&s) == (secp256k1_num_cmp(&snum, &half_order) > 0));
         secp256k1_scalar_t neg;
         secp256k1_scalar_negate(&neg, &s);
         secp256k1_num_t negnum;
-        secp256k1_num_sub(&negnum, &secp256k1_ge_consts->order, &snum);
-        secp256k1_num_mod(&negnum, &secp256k1_ge_consts->order);
+        secp256k1_num_sub(&negnum, &order, &snum);
+        secp256k1_num_mod(&negnum, &order);
         /* Check that comparison with the half order is equal to testing for high scalar after negation. */
-        CHECK(secp256k1_scalar_is_high(&neg) == (secp256k1_num_cmp(&negnum, &secp256k1_ge_consts->half_order) > 0));
+        CHECK(secp256k1_scalar_is_high(&neg) == (secp256k1_num_cmp(&negnum, &half_order) > 0));
         /* Negating should change the high property, unless the value was already zero. */
         CHECK((secp256k1_scalar_is_high(&s) == secp256k1_scalar_is_high(&neg)) == secp256k1_scalar_is_zero(&s));
         secp256k1_num_t negnum2;
@@ -356,15 +289,36 @@ void scalar_test(void) {
     }
 
     {
+        /* Test secp256k1_scalar_mul_shift_var. */
+        secp256k1_scalar_t r;
+        unsigned int shift = 256 + (secp256k1_rand32() % 257);
+        secp256k1_scalar_mul_shift_var(&r, &s1, &s2, shift);
+        secp256k1_num_t rnum;
+        secp256k1_num_mul(&rnum, &s1num, &s2num);
+        secp256k1_num_shift(&rnum, shift - 1);
+        secp256k1_num_t one;
+        unsigned char cone[1] = {0x01};
+        secp256k1_num_set_bin(&one, cone, 1);
+        secp256k1_num_add(&rnum, &rnum, &one);
+        secp256k1_num_shift(&rnum, 1);
+        secp256k1_num_t rnum2;
+        secp256k1_scalar_get_num(&rnum2, &r);
+        CHECK(secp256k1_num_eq(&rnum, &rnum2));
+    }
+#endif
+
+    {
         /* Test that scalar inverses are equal to the inverse of their number modulo the order. */
         if (!secp256k1_scalar_is_zero(&s)) {
             secp256k1_scalar_t inv;
             secp256k1_scalar_inverse(&inv, &s);
+#ifndef USE_NUM_NONE
             secp256k1_num_t invnum;
-            secp256k1_num_mod_inverse(&invnum, &snum, &secp256k1_ge_consts->order);
+            secp256k1_num_mod_inverse(&invnum, &snum, &order);
             secp256k1_num_t invnum2;
             secp256k1_scalar_get_num(&invnum2, &inv);
             CHECK(secp256k1_num_eq(&invnum, &invnum2));
+#endif
             secp256k1_scalar_mul(&inv, &inv, &s);
             /* Multiplying a scalar with its inverse must result in one. */
             CHECK(secp256k1_scalar_is_one(&inv));
@@ -383,6 +337,23 @@ void scalar_test(void) {
     }
 
     {
+        /* Test add_bit. */
+        int bit = secp256k1_rand32() % 256;
+        secp256k1_scalar_t b;
+        secp256k1_scalar_set_int(&b, 1);
+        CHECK(secp256k1_scalar_is_one(&b));
+        for (int i = 0; i < bit; i++) {
+            secp256k1_scalar_add(&b, &b, &b);
+        }
+        secp256k1_scalar_t r1 = s1, r2 = s1;
+        if (!secp256k1_scalar_add(&r1, &r1, &b)) {
+            /* No overflow happened. */
+            secp256k1_scalar_add_bit(&r2, bit);
+            CHECK(secp256k1_scalar_eq(&r1, &r2));
+        }
+    }
+
+    {
         /* Test commutativity of mul. */
         secp256k1_scalar_t r1, r2;
         secp256k1_scalar_mul(&r1, &s1, &s2);
@@ -428,20 +399,49 @@ void scalar_test(void) {
         secp256k1_scalar_mul(&r2, &s1, &s1);
         CHECK(secp256k1_scalar_eq(&r1, &r2));
     }
+
 }
 
 void run_scalar_tests(void) {
     for (int i = 0; i < 128 * count; i++) {
         scalar_test();
     }
+
+    {
+        /* (-1)+1 should be zero. */
+        secp256k1_scalar_t s, o;
+        secp256k1_scalar_set_int(&s, 1);
+        secp256k1_scalar_negate(&o, &s);
+        secp256k1_scalar_add(&o, &o, &s);
+        CHECK(secp256k1_scalar_is_zero(&o));
+    }
+
+#ifndef USE_NUM_NONE
+    {
+        /* A scalar with value of the curve order should be 0. */
+        secp256k1_num_t order;
+        secp256k1_scalar_order_get_num(&order);
+        unsigned char bin[32];
+        secp256k1_num_get_bin(bin, 32, &order);
+        secp256k1_scalar_t zero;
+        int overflow = 0;
+        secp256k1_scalar_set_b32(&zero, bin, &overflow);
+        CHECK(overflow == 1);
+        CHECK(secp256k1_scalar_is_zero(&zero));
+    }
+#endif
 }
 
 /***** FIELD TESTS *****/
 
 void random_fe(secp256k1_fe_t *x) {
     unsigned char bin[32];
-    secp256k1_rand256(bin);
-    secp256k1_fe_set_b32(x, bin);
+    do {
+        secp256k1_rand256(bin);
+        if (secp256k1_fe_set_b32(x, bin)) {
+            return;
+        }
+    } while(1);
 }
 
 void random_fe_non_zero(secp256k1_fe_t *nz) {
@@ -617,9 +617,17 @@ void gej_equals_gej(const secp256k1_gej_t *a, const secp256k1_gej_t *b) {
 }
 
 void test_ge(void) {
+    char ca[135];
+    char cb[68];
+    int rlen;
     secp256k1_ge_t a, b, i, n;
     random_group_element_test(&a);
     random_group_element_test(&b);
+    rlen = sizeof(ca);
+    secp256k1_ge_get_hex(ca,&rlen,&a);
+    CHECK(rlen > 4 && rlen <= (int)sizeof(ca));
+    rlen = sizeof(cb);
+    secp256k1_ge_get_hex(cb,&rlen,&b); /* Intentionally undersized buffer. */
     n = a;
     secp256k1_fe_normalize(&a.y);
     secp256k1_fe_negate(&n.y, &a.y, 1);
@@ -697,39 +705,51 @@ void run_ge(void) {
 
 void run_ecmult_chain(void) {
     /* random starting point A (on the curve) */
-    secp256k1_fe_t ax; secp256k1_fe_set_hex(&ax, "8b30bbe9ae2a990696b22f670709dff3727fd8bc04d3362c6c7bf458e2846004", 64);
-    secp256k1_fe_t ay; secp256k1_fe_set_hex(&ay, "a357ae915c4a65281309edf20504740f0eb3343990216b4f81063cb65f2f7e0f", 64);
+    secp256k1_fe_t ax; VERIFY_CHECK(secp256k1_fe_set_hex(&ax, "8b30bbe9ae2a990696b22f670709dff3727fd8bc04d3362c6c7bf458e2846004", 64));
+    secp256k1_fe_t ay; VERIFY_CHECK(secp256k1_fe_set_hex(&ay, "a357ae915c4a65281309edf20504740f0eb3343990216b4f81063cb65f2f7e0f", 64));
     secp256k1_gej_t a; secp256k1_gej_set_xy(&a, &ax, &ay);
     /* two random initial factors xn and gn */
-    secp256k1_num_t xn;
-    secp256k1_num_set_hex(&xn, "84cc5452f7fde1edb4d38a8ce9b1b84ccef31f146e569be9705d357a42985407", 64);
-    secp256k1_num_t gn;
-    secp256k1_num_set_hex(&gn, "a1e58d22553dcd42b23980625d4c57a96e9323d42b3152e5ca2c3990edc7c9de", 64);
+    static const unsigned char xni[32] = {
+        0x84, 0xcc, 0x54, 0x52, 0xf7, 0xfd, 0xe1, 0xed,
+        0xb4, 0xd3, 0x8a, 0x8c, 0xe9, 0xb1, 0xb8, 0x4c,
+        0xce, 0xf3, 0x1f, 0x14, 0x6e, 0x56, 0x9b, 0xe9,
+        0x70, 0x5d, 0x35, 0x7a, 0x42, 0x98, 0x54, 0x07
+    };
+    secp256k1_scalar_t xn;
+    secp256k1_scalar_set_b32(&xn, xni, NULL);
+    static const unsigned char gni[32] = {
+        0xa1, 0xe5, 0x8d, 0x22, 0x55, 0x3d, 0xcd, 0x42,
+        0xb2, 0x39, 0x80, 0x62, 0x5d, 0x4c, 0x57, 0xa9,
+        0x6e, 0x93, 0x23, 0xd4, 0x2b, 0x31, 0x52, 0xe5,
+        0xca, 0x2c, 0x39, 0x90, 0xed, 0xc7, 0xc9, 0xde
+    };
+    secp256k1_scalar_t gn;
+    secp256k1_scalar_set_b32(&gn, gni, NULL);
     /* two small multipliers to be applied to xn and gn in every iteration: */
-    secp256k1_num_t xf;
-    secp256k1_num_set_hex(&xf, "1337", 4);
-    secp256k1_num_t gf;
-    secp256k1_num_set_hex(&gf, "7113", 4);
+    static const unsigned char xfi[32] = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0x13,0x37};
+    secp256k1_scalar_t xf;
+    secp256k1_scalar_set_b32(&xf, xfi, NULL);
+    static const unsigned char gfi[32] = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0x71,0x13};
+    secp256k1_scalar_t gf;
+    secp256k1_scalar_set_b32(&gf, gfi, NULL);
     /* accumulators with the resulting coefficients to A and G */
-    secp256k1_num_t ae;
-    secp256k1_num_set_int(&ae, 1);
-    secp256k1_num_t ge;
-    secp256k1_num_set_int(&ge, 0);
+    secp256k1_scalar_t ae;
+    secp256k1_scalar_set_int(&ae, 1);
+    secp256k1_scalar_t ge;
+    secp256k1_scalar_set_int(&ge, 0);
     /* the point being computed */
     secp256k1_gej_t x = a;
-    const secp256k1_num_t *order = &secp256k1_ge_consts->order;
     for (int i=0; i<200*count; i++) {
         /* in each iteration, compute X = xn*X + gn*G; */
         secp256k1_ecmult(&x, &x, &xn, &gn);
         /* also compute ae and ge: the actual accumulated factors for A and G */
         /* if X was (ae*A+ge*G), xn*X + gn*G results in (xn*ae*A + (xn*ge+gn)*G) */
-        secp256k1_num_mod_mul(&ae, &ae, &xn, order);
-        secp256k1_num_mod_mul(&ge, &ge, &xn, order);
-        secp256k1_num_add(&ge, &ge, &gn);
-        secp256k1_num_mod(&ge, order);
+        secp256k1_scalar_mul(&ae, &ae, &xn);
+        secp256k1_scalar_mul(&ge, &ge, &xn);
+        secp256k1_scalar_add(&ge, &ge, &gn);
         /* modify xn and gn */
-        secp256k1_num_mod_mul(&xn, &xn, &xf, order);
-        secp256k1_num_mod_mul(&gn, &gn, &gf, order);
+        secp256k1_scalar_mul(&xn, &xn, &xf);
+        secp256k1_scalar_mul(&gn, &gn, &gf);
 
         /* verify */
         if (i == 19999) {
@@ -749,17 +769,25 @@ void run_ecmult_chain(void) {
 }
 
 void test_point_times_order(const secp256k1_gej_t *point) {
-    /* multiplying a point by the order results in O */
-    const secp256k1_num_t *order = &secp256k1_ge_consts->order;
-    secp256k1_num_t zero;
-    secp256k1_num_set_int(&zero, 0);
-    secp256k1_gej_t res;
-    secp256k1_ecmult(&res, point, order, order); /* calc res = order * point + order * G; */
-    CHECK(secp256k1_gej_is_infinity(&res));
+    /* X * (point + G) + (order-X) * (pointer + G) = 0 */
+    secp256k1_scalar_t x;
+    random_scalar_order_test(&x);
+    secp256k1_scalar_t nx;
+    secp256k1_scalar_negate(&nx, &x);
+    secp256k1_gej_t res1, res2;
+    secp256k1_ecmult(&res1, point, &x, &x); /* calc res1 = x * point + x * G; */
+    secp256k1_ecmult(&res2, point, &nx, &nx); /* calc res2 = (order - x) * point + (order - x) * G; */
+    secp256k1_gej_add_var(&res1, &res1, &res2);
+    CHECK(secp256k1_gej_is_infinity(&res1));
+    CHECK(secp256k1_gej_is_valid(&res1) == 0);
+    secp256k1_ge_t res3;
+    secp256k1_ge_set_gej(&res3, &res1);
+    CHECK(secp256k1_ge_is_infinity(&res3));
+    CHECK(secp256k1_ge_is_valid(&res3) == 0);
 }
 
 void run_point_times_order(void) {
-    secp256k1_fe_t x; secp256k1_fe_set_hex(&x, "02", 2);
+    secp256k1_fe_t x; VERIFY_CHECK(secp256k1_fe_set_hex(&x, "02", 2));
     for (int i=0; i<500; i++) {
         secp256k1_ge_t p;
         if (secp256k1_ge_set_xo(&p, &x, 1)) {
@@ -776,15 +804,16 @@ void run_point_times_order(void) {
     CHECK(strcmp(c, "7603CB59B0EF6C63FE6084792A0C378CDB3233A80F8A9A09A877DEAD31B38C45") == 0);
 }
 
-void test_wnaf(const secp256k1_num_t *number, int w) {
-    secp256k1_num_t x, two, t;
-    secp256k1_num_set_int(&x, 0);
-    secp256k1_num_set_int(&two, 2);
-    int wnaf[257];
+void test_wnaf(const secp256k1_scalar_t *number, int w) {
+    secp256k1_scalar_t x, two, t;
+    secp256k1_scalar_set_int(&x, 0);
+    secp256k1_scalar_set_int(&two, 2);
+    int wnaf[256];
     int bits = secp256k1_ecmult_wnaf(wnaf, number, w);
+    CHECK(bits <= 256);
     int zeroes = -1;
     for (int i=bits-1; i>=0; i--) {
-        secp256k1_num_mul(&x, &x, &two);
+        secp256k1_scalar_mul(&x, &x, &two);
         int v = wnaf[i];
         if (v) {
             CHECK(zeroes == -1 || zeroes >= w-1); /* check that distance between non-zero elements is at least w-1 */
@@ -796,18 +825,23 @@ void test_wnaf(const secp256k1_num_t *number, int w) {
             CHECK(zeroes != -1); /* check that no unnecessary zero padding exists */
             zeroes++;
         }
-        secp256k1_num_set_int(&t, v);
-        secp256k1_num_add(&x, &x, &t);
+        if (v >= 0) {
+            secp256k1_scalar_set_int(&t, v);
+        } else {
+            secp256k1_scalar_set_int(&t, -v);
+            secp256k1_scalar_negate(&t, &t);
+        }
+        secp256k1_scalar_add(&x, &x, &t);
     }
-    CHECK(secp256k1_num_eq(&x, number)); /* check that wnaf represents number */
+    CHECK(secp256k1_scalar_eq(&x, number)); /* check that wnaf represents number */
 }
 
 void run_wnaf(void) {
-    secp256k1_num_t n;
+    secp256k1_scalar_t n;
     for (int i=0; i<count; i++) {
-        random_num_order(&n);
+        random_scalar_order(&n);
         if (i % 1)
-            secp256k1_num_negate(&n);
+            secp256k1_scalar_negate(&n, &n);
         test_wnaf(&n, 4+(i%10));
     }
 }
@@ -820,18 +854,22 @@ void random_sign(secp256k1_ecdsa_sig_t *sig, const secp256k1_scalar_t *key, cons
 }
 
 void test_ecdsa_sign_verify(void) {
+    int recid;
+    int getrec;
     secp256k1_scalar_t msg, key;
     random_scalar_order_test(&msg);
     random_scalar_order_test(&key);
     secp256k1_gej_t pubj; secp256k1_ecmult_gen(&pubj, &key);
     secp256k1_ge_t pub; secp256k1_ge_set_gej(&pub, &pubj);
     secp256k1_ecdsa_sig_t sig;
-    random_sign(&sig, &key, &msg, NULL);
-    secp256k1_num_t msg_num;
-    secp256k1_scalar_get_num(&msg_num, &msg);
-    CHECK(secp256k1_ecdsa_sig_verify(&sig, &pub, &msg_num));
-    secp256k1_num_inc(&msg_num);
-    CHECK(!secp256k1_ecdsa_sig_verify(&sig, &pub, &msg_num));
+    getrec = secp256k1_rand32()&1;
+    random_sign(&sig, &key, &msg, getrec?&recid:NULL);
+    if (getrec) CHECK(recid >= 0 && recid < 4);
+    CHECK(secp256k1_ecdsa_sig_verify(&sig, &pub, &msg));
+    secp256k1_scalar_t one;
+    secp256k1_scalar_set_int(&one, 1);
+    secp256k1_scalar_add(&msg, &msg, &one);
+    CHECK(!secp256k1_ecdsa_sig_verify(&sig, &pub, &msg));
 }
 
 void run_ecdsa_sign_verify(void) {
@@ -846,11 +884,11 @@ void test_ecdsa_end_to_end(void) {
 
     /* Generate a random key and message. */
     {
-        secp256k1_num_t msg, key;
-        random_num_order_test(&msg);
-        random_num_order_test(&key);
-        secp256k1_num_get_bin(privkey, 32, &key);
-        secp256k1_num_get_bin(message, 32, &msg);
+        secp256k1_scalar_t msg, key;
+        random_scalar_order_test(&msg);
+        random_scalar_order_test(&key);
+        secp256k1_scalar_get_b32(privkey, &key);
+        secp256k1_scalar_get_b32(message, &msg);
     }
 
     /* Construct and verify corresponding public key. */
@@ -935,7 +973,8 @@ void run_ecdsa_end_to_end(void) {
     }
 }
 
-void test_ecdsa_infinity(void) {
+/* Tests several edge cases. */
+void test_ecdsa_edge_cases(void) {
     const unsigned char msg32[32] = {
         'T', 'h', 'i', 's', ' ', 'i', 's', ' ',
         'a', ' ', 'v', 'e', 'r', 'y', ' ', 's',
@@ -943,8 +982,8 @@ void test_ecdsa_infinity(void) {
         's', 's', 'a', 'g', 'e', '.', '.', '.'
     };
     const unsigned char sig64[64] = {
-        // Generated by signing the above message with nonce 'This is the nonce we will use...'
-        // and secret key 0 (which is not valid), resulting in recid 0.
+        /* Generated by signing the above message with nonce 'This is the nonce we will use...'
+         * and secret key 0 (which is not valid), resulting in recid 0. */
         0x67, 0xCB, 0x28, 0x5F, 0x9C, 0xD1, 0x94, 0xE8,
         0x40, 0xD6, 0x29, 0x39, 0x7A, 0xF5, 0x56, 0x96,
         0x62, 0xFD, 0xE4, 0x46, 0x49, 0x99, 0x59, 0x63,
@@ -960,10 +999,93 @@ void test_ecdsa_infinity(void) {
     CHECK(secp256k1_ecdsa_recover_compact(msg32, 32, sig64, pubkey, &pubkeylen, 0, 1));
     CHECK(!secp256k1_ecdsa_recover_compact(msg32, 32, sig64, pubkey, &pubkeylen, 0, 2));
     CHECK(!secp256k1_ecdsa_recover_compact(msg32, 32, sig64, pubkey, &pubkeylen, 0, 3));
+
+    /* signature (r,s) = (4,4), which can be recovered with all 4 recids. */
+    const unsigned char sigb64[64] = {
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x04,
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x04,
+    };
+    unsigned char pubkeyb[33];
+    int pubkeyblen = 33;
+    for (int recid = 0; recid < 4; recid++) {
+        /* (4,4) encoded in DER. */
+        unsigned char sigbder[8] = {0x30, 0x06, 0x02, 0x01, 0x04, 0x02, 0x01, 0x04};
+        /* (order + r,4) encoded in DER. */
+        unsigned char sigbderlong[40] = {
+            0x30, 0x26, 0x02, 0x21, 0x00, 0xFF, 0xFF, 0xFF,
+            0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
+            0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xBA, 0xAE, 0xDC,
+            0xE6, 0xAF, 0x48, 0xA0, 0x3B, 0xBF, 0xD2, 0x5E,
+            0x8C, 0xD0, 0x36, 0x41, 0x45, 0x02, 0x01, 0x04
+        };
+        CHECK(secp256k1_ecdsa_recover_compact(msg32, 32, sigb64, pubkeyb, &pubkeyblen, 1, recid));
+        CHECK(secp256k1_ecdsa_verify(msg32, 32, sigbder, sizeof(sigbder), pubkeyb, pubkeyblen) == 1);
+        for (int recid2 = 0; recid2 < 4; recid2++) {
+            unsigned char pubkey2b[33];
+            int pubkey2blen = 33;
+            CHECK(secp256k1_ecdsa_recover_compact(msg32, 32, sigb64, pubkey2b, &pubkey2blen, 1, recid2));
+            /* Verifying with (order + r,4) should always fail. */
+            CHECK(secp256k1_ecdsa_verify(msg32, 32, sigbderlong, sizeof(sigbderlong), pubkey2b, pubkey2blen) != 1);
+        }
+        /* Damage signature. */
+        sigbder[7]++;
+        CHECK(secp256k1_ecdsa_verify(msg32, 32, sigbder, sizeof(sigbder), pubkeyb, pubkeyblen) == 0);
+    }
+
+    /* Test the case where ECDSA recomputes a point that is infinity. */
+    {
+        secp256k1_ecdsa_sig_t sig;
+        secp256k1_scalar_set_int(&sig.s, 1);
+        secp256k1_scalar_negate(&sig.s, &sig.s);
+        secp256k1_scalar_inverse(&sig.s, &sig.s);
+        secp256k1_scalar_set_int(&sig.r, 1);
+        secp256k1_gej_t keyj;
+        secp256k1_ecmult_gen(&keyj, &sig.r);
+        secp256k1_ge_t key;
+        secp256k1_ge_set_gej(&key, &keyj);
+        secp256k1_scalar_t msg = sig.s;
+        CHECK(secp256k1_ecdsa_sig_verify(&sig, &key, &msg) == 0);
+    }
+
+    /* Test r/s equal to zero */
+    {
+        /* (1,1) encoded in DER. */
+        unsigned char sigcder[8] = {0x30, 0x06, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01};
+        unsigned char sigc64[64] = {
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01,
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+            0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01,
+        };
+        unsigned char pubkeyc[65];
+        int pubkeyclen = 65;
+        CHECK(secp256k1_ecdsa_recover_compact(msg32, 32, sigc64, pubkeyc, &pubkeyclen, 0, 0) == 1);
+        CHECK(secp256k1_ecdsa_verify(msg32, 32, sigcder, sizeof(sigcder), pubkeyc, pubkeyclen) == 1);
+        sigcder[4] = 0;
+        sigc64[31] = 0;
+        CHECK(secp256k1_ecdsa_recover_compact(msg32, 32, sigc64, pubkeyb, &pubkeyblen, 1, 0) == 0);
+        CHECK(secp256k1_ecdsa_verify(msg32, 32, sigcder, sizeof(sigcder), pubkeyc, pubkeyclen) == 0);
+        sigcder[4] = 1;
+        sigcder[7] = 0;
+        sigc64[31] = 1;
+        sigc64[63] = 0;
+        CHECK(secp256k1_ecdsa_recover_compact(msg32, 32, sigc64, pubkeyb, &pubkeyblen, 1, 0) == 0);
+        CHECK(secp256k1_ecdsa_verify(msg32, 32, sigcder, sizeof(sigcder), pubkeyc, pubkeyclen) == 0);
+    }
 }
 
-void run_ecdsa_infinity(void) {
-    test_ecdsa_infinity();
+void run_ecdsa_edge_cases(void) {
+    test_ecdsa_edge_cases();
 }
 
 #ifdef ENABLE_OPENSSL_TESTS
@@ -996,11 +1118,12 @@ void test_ecdsa_openssl(void) {
     CHECK(ECDSA_sign(0, message, sizeof(message), signature, &sigsize, ec_key));
     secp256k1_ecdsa_sig_t sig;
     CHECK(secp256k1_ecdsa_sig_parse(&sig, signature, sigsize));
-    secp256k1_num_t msg_num;
-    secp256k1_scalar_get_num(&msg_num, &msg);
-    CHECK(secp256k1_ecdsa_sig_verify(&sig, &q, &msg_num));
-    secp256k1_num_inc(&sig.r);
-    CHECK(!secp256k1_ecdsa_sig_verify(&sig, &q, &msg_num));
+    CHECK(secp256k1_ecdsa_sig_verify(&sig, &q, &msg));
+    secp256k1_scalar_t one;
+    secp256k1_scalar_set_int(&one, 1);
+    secp256k1_scalar_t msg2;
+    secp256k1_scalar_add(&msg2, &msg, &one);
+    CHECK(!secp256k1_ecdsa_sig_verify(&sig, &q, &msg2));
 
     random_sign(&sig, &key, &msg, NULL);
     int secp_sigsize = 80;
@@ -1042,8 +1165,19 @@ int main(int argc, char **argv) {
     /* initialize */
     secp256k1_start(SECP256K1_START_SIGN | SECP256K1_START_VERIFY);
 
+    /* initializing a second time shouldn't cause any harm or memory leaks. */
+    secp256k1_start(SECP256K1_START_SIGN | SECP256K1_START_VERIFY);
+
+    /* Likewise, re-running the internal init functions should be harmless. */
+    secp256k1_fe_start();
+    secp256k1_ge_start();
+    secp256k1_scalar_start();
+    secp256k1_ecdsa_start();
+
+#ifndef USE_NUM_NONE
     /* num tests */
     run_num_smalltests();
+#endif
 
     /* scalar tests */
     run_scalar_tests();
@@ -1067,7 +1201,7 @@ int main(int argc, char **argv) {
     /* ecdsa tests */
     run_ecdsa_sign_verify();
     run_ecdsa_end_to_end();
-    run_ecdsa_infinity();
+    run_ecdsa_edge_cases();
 #ifdef ENABLE_OPENSSL_TESTS
     run_ecdsa_openssl();
 #endif
@@ -1076,5 +1210,14 @@ int main(int argc, char **argv) {
 
     /* shutdown */
     secp256k1_stop();
+
+    /* shutting down twice shouldn't cause any double frees. */
+    secp256k1_stop();
+
+    /* Same for the internal shutdown functions. */
+    secp256k1_fe_stop();
+    secp256k1_ge_stop();
+    secp256k1_scalar_stop();
+    secp256k1_ecdsa_stop();
     return 0;
 }
diff --git a/src/secp256k1/src/util.h b/src/secp256k1/src/util.h
index 96b47057c0..08b23a9d38 100644
--- a/src/secp256k1/src/util.h
+++ b/src/secp256k1/src/util.h
@@ -61,4 +61,21 @@
 #define VERIFY_CHECK(cond) do { (void)(cond); } while(0)
 #endif
 
+/* Macro for restrict, when available and not in a VERIFY build. */
+#if defined(SECP256K1_BUILD) && defined(VERIFY)
+# define SECP256K1_RESTRICT
+#else
+# if (!defined(__STDC_VERSION__) || (__STDC_VERSION__ < 199901L) )
+#  if SECP256K1_GNUC_PREREQ(3,0)
+#   define SECP256K1_RESTRICT __restrict__
+#  elif (defined(_MSC_VER) && _MSC_VER >= 1400)
+#   define SECP256K1_RESTRICT __restrict
+#  else
+#   define SECP256K1_RESTRICT
+#  endif
+# else
+#  define SECP256K1_RESTRICT restrict
+# endif
+#endif
+
 #endif