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| -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 f63e3ae..dbc474b 100644 --- a/bip-schnorr.mediawiki +++ b/bip-schnorr.mediawiki @@ -96,7 +96,7 @@ The following conventions are used, with constants as defined for [https://www.s ** The constant ''p'' refers to the field size, ''0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F''. ** The constant ''n'' refers to the curve order, ''0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141''. * Uppercase variables refer to points on the curve with equation ''y<sup>2</sup> = x<sup>3</sup> + 7'' over the integers modulo ''p''. -** ''infinite(P)'' returns whether or not ''P'' is the point at infinity. +** ''is_infinite(P)'' returns whether or not ''P'' is the point at infinity. ** ''x(P)'' and ''y(P)'' are integers in the range ''0..p-1'' and refer to the X and Y coordinates of a point ''P'' (assuming it is not infinity). ** The constant ''G'' refers to the generator, for which ''x(G) = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798'' and ''y(G) = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8''. ** Addition of points refers to the usual [https://en.wikipedia.org/wiki/Elliptic_curve#The_group_law elliptic curve group operation]. @@ -108,7 +108,7 @@ The following conventions are used, with constants as defined for [https://www.s ** 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_positive(P)'', where ''P'' is a point, is defined as ''not infinite(P) and is_square(y(P))''<ref>For points ''P'' on the secp256k1 curve it holds that ''is_positive(P) = not is_positive(-P)''.</ref>. +** The function ''is_positive(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 ''is_positive(P) = not is_positive(-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 ''is_positive(P)'', or fails if no such point exists<ref>Given an 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: *** Let ''c = x<sup>3</sup> + 7 mod p''. *** Let ''y = c<sup>(p+1)/4</sup> mod p''. |