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Improve clarity of footnotes for lift_x

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Jonas Nick 2019-12-04 19:26:06 +00:00
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@ -72,7 +72,7 @@ Using the first option would be slightly more efficient for verification (around
'''Implicit Y coordinates''' In order to support efficient verification and batch verification, the Y coordinate of ''P'' and of ''R'' cannot be ambiguous (every valid X coordinate has two possible Y coordinates). We have a choice between several options for symmetry breaking:
# Implicitly choosing the Y coordinate that is in the lower half.
# Implicitly choosing the Y coordinate that is even<ref>Since ''p'' is odd, negation modulo ''p'' will map even numbers to odd numbers and the other way around. This means that for a valid X coordinate, one of the corresponding Y coordinates will be even, and the other will be odd.</ref>.
# Implicitly choosing the Y coordinate that is a quadratic residue (has a square root modulo the field size, or "is a square" for short)<ref>A product of two numbers is a square when either both or none of the factors are squares. As ''-1'' is not a square, and the two Y coordinates corresponding to a given X coordinate are each other's negation, this means exactly one of the two must be a square.</ref>.
# Implicitly choosing the Y coordinate that is a quadratic residue (has a square root modulo the field size, or "is a square" for short)<ref>A product of two numbers is a square when either both or none of the factors are squares. As ''-1'' is not a square modulo secp256k1's field size ''p'', and the two Y coordinates corresponding to a given X coordinate are each other's negation, this means exactly one of the two must be a square.</ref>.
In the case of ''R'' the third option is slower at signing time but a bit faster to verify, as it is possible to directly compute whether the Y coordinate is a square when the points are represented in
[https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates Jacobian coordinates] (a common optimization to avoid modular inverses
@ -111,7 +111,11 @@ The following conventions are used, with constants as defined for [https://www.s
** 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 [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> &ne; 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 = &plusmn;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 = &plusmn;c<sup>((p+1)/4)((p-1)/2)</sup> mod p = &plusmn;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''<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''. The valid Y coordinates for a given candidate ''x'' are the square roots of ''c = x<sup>3</sup> + 7 mod p'' and they can be computed as ''y = &plusmn;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''.
</ref> and ''has_square_y(P)''<ref>
If ''P := lift_x(x)'' does not fail, then ''y := y(P) = c<sup>(p+1)/4</sup> mod p'' is square. Proof: If ''lift_x'' does not fail, ''y'' is a square root of ''c'' and therefore the [https://en.wikipedia.org/wiki/Legendre_symbol Legendre symbol] ''(c / p)'' is ''c<sup>(p-1)/2</sup> = 1 mod p''. Because 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<sup>((p+1)/4)</sup> mod p = 1 mod p'', ''y'' is square.
</ref>, or fails if no such point exists. 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 &ne; y<sup>2</sup> mod p''.