diff options
author | Pieter Wuille <pieter.wuille@gmail.com> | 2019-12-10 17:20:22 -0500 |
---|---|---|
committer | GitHub <noreply@github.com> | 2019-12-10 17:20:22 -0500 |
commit | 034e97bd6ea60778556fa4e0eb9252d4063f1274 (patch) | |
tree | 54d9b19ead971000456f30ca05f0d9ac10c3aee2 | |
parent | 017ca0c69ba2742bc20fa08fe9013bd93ec99375 (diff) | |
parent | 382a1d19a0fa299d599f1d4e504f95282e2ab244 (diff) |
Merge pull request #170 from jonasnick/footnote7
Fix footnote 7 and remove references to Euler's criterion
-rw-r--r-- | bip-schnorr.mediawiki | 4 |
1 files changed, 2 insertions, 2 deletions
diff --git a/bip-schnorr.mediawiki b/bip-schnorr.mediawiki index 0697902..ce5abde 100644 --- a/bip-schnorr.mediawiki +++ b/bip-schnorr.mediawiki @@ -109,9 +109,9 @@ The following conventions are used, with constants as defined for [https://www.s ** The function ''bytes(x)'', where ''x'' is an integer, returns the 32-byte encoding of ''x'', most significant byte first. ** The function ''bytes(P)'', where ''P'' is a point, returns ''bytes(x(P))''. ** The function ''int(x)'', where ''x'' is a 32-byte array, returns the 256-bit unsigned integer whose most significant byte first encoding is ''x''. -** The function ''is_square(x)'', where ''x'' is an integer, returns whether or not ''x'' is a quadratic residue modulo ''p''. Since ''p'' is prime, it is equivalent to the Legendre symbol ''(x / p) = x<sup>(p-1)/2</sup> mod p'' being equal to ''1'' (see [https://en.wikipedia.org/wiki/Euler%27s_criterion Euler's criterion])<ref>For points ''P'' on the secp256k1 curve it holds that ''x<sup>(p-1)/2</sup> ≠ 0 mod p''.</ref>. +** The function ''is_square(x)'', where ''x'' is an integer, returns whether or not ''x'' is a quadratic residue modulo ''p''. Since ''p'' is prime, it is equivalent to the [https://en.wikipedia.org/wiki/Legendre_symbol Legendre symbol] ''(x / p) = x<sup>(p-1)/2</sup> mod p'' being equal to ''1''<ref>For points ''P'' on the secp256k1 curve it holds that ''y(P)<sup>(p-1)/2</sup> ≠ 0 mod p''.</ref>. ** The function ''has_square_y(P)'', where ''P'' is a point, is defined as ''not is_infinite(P) and is_square(y(P))''<ref>For points ''P'' on the secp256k1 curve it holds that ''has_square_y(P) = not has_square_y(-P)''.</ref>. -** The function ''lift_x(x)'', where ''x'' is an integer in range ''0..p-1'', returns the point ''P'' for which ''x(P) = x'' and ''has_square_y(P)'', or fails if no such point exists<ref>Given a candidate X coordinate ''x'' in the range ''0..p-1'', there exist either exactly two or exactly zero valid Y coordinates. If no valid Y coordinate exists, then ''x'' is not a valid X coordinate either, i.e., no point ''P'' exists for which ''x(P) = x''. Given a candidate ''x'', the valid Y coordinates are the square roots of ''c = x<sup>3</sup> + 7 mod p'' and they can be computed as ''y = ±c<sup>(p+1)/4</sup> mod p'' (see [https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus Quadratic residue]) if they exist, which can be checked by squaring and comparing with ''c''. Due to [https://en.wikipedia.org/wiki/Euler%27s_criterion Euler's criterion] it then holds that ''c<sup>(p-1)/2</sup> = 1 mod p''. The same criterion applied to ''y'' results in ''y<sup>(p-1)/2</sup> mod p = ±c<sup>((p+1)/4)((p-1)/2)</sup> mod p = ±1 mod p''. Therefore ''y = +c<sup>(p+1)/4</sup> mod p'' is a quadratic residue and ''-y mod p'' is not.</ref>. The function ''lift_x(x)'' is equivalent to the following pseudocode: +** The function ''lift_x(x)'', where ''x'' is an integer in range ''0..p-1'', returns the point ''P'' for which ''x(P) = x'' and ''has_square_y(P)'', or fails if no such point exists<ref>Given a candidate X coordinate ''x'' in the range ''0..p-1'', there exist either exactly two or exactly zero valid Y coordinates. If no valid Y coordinate exists, then ''x'' is not a valid X coordinate either, i.e., no point ''P'' exists for which ''x(P) = x''. Given a candidate ''x'', the valid Y coordinates are the square roots of ''c = x<sup>3</sup> + 7 mod p'' and they can be computed as ''y = ±c<sup>(p+1)/4</sup> mod p'' (see [https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus Quadratic residue]) if they exist, which can be checked by squaring and comparing with ''c''. The [https://en.wikipedia.org/wiki/Legendre_symbol Legendre symbol] ''( c / p)'' is ''c<sup>(p-1)/2</sup> = 1 mod p''. The Legendre symbol ''( y / p )'' is ''y<sup>(p-1)/2</sup> mod p = ±c<sup>((p+1)/4)((p-1)/2)</sup> mod p = ±1 mod p''. Therefore ''y = +c<sup>(p+1)/4</sup> mod p'' is a quadratic residue and ''-y mod p'' is not.</ref>. The function ''lift_x(x)'' is equivalent to the following pseudocode: *** Let ''c = x<sup>3</sup> + 7 mod p''. *** Let ''y = c<sup>(p+1)/4</sup> mod p''. *** Fail if ''c ≠ y<sup>2</sup> mod p''. |