From: Mark Wooding Date: Mon, 1 May 2017 00:38:30 +0000 (+0100) Subject: math/f25519.c, utils/curve25519.sage: Slightly improve `quosqrt' algorithm. X-Git-Tag: 2.4.0~6 X-Git-Url: https://git.distorted.org.uk/~mdw/catacomb/commitdiff_plain/e830bb692041c75eb29b8c511db21af81b3aae2d math/f25519.c, utils/curve25519.sage: Slightly improve `quosqrt' algorithm. The algorithm from the Bernstein et al. paper was somewhat ugly. Replace it with a different one using the techniques I used in `fgoldi' for the main calculation, but with the same end structure. --- diff --git a/math/f25519.c b/math/f25519.c index a58a68a9..78844be6 100644 --- a/math/f25519.c +++ b/math/f25519.c @@ -1092,7 +1092,7 @@ static const piece sqrtm1_pieces[NPIECE] = { int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y) { - f25519 t, u, w, beta, xy3, t2p50m1; + f25519 t, u, v, w, t15; octet xb[32], b0[32], b1[32]; int32 rc = -1; mask32 m; @@ -1103,68 +1103,72 @@ int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y) for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \ } while (0) - /* This is a bit tricky; the algorithm is from Bernstein, Duif, Lange, - * Schwabe, and Yang, `High-speed high-security signatures', 2011-09-26, - * https://ed25519.cr.yp.to/ed25519-20110926.pdf. + /* This is a bit tricky; the algorithm is loosely based on Bernstein, Duif, + * Lange, Schwabe, and Yang, `High-speed high-security signatures', + * 2011-09-26, https://ed25519.cr.yp.to/ed25519-20110926.pdf. + */ + f25519_mul(&v, x, y); + + /* Now for an addition chain. */ /* step | value */ + f25519_sqr(&u, &v); /* 1 | 2 */ + f25519_mul(&t, &u, &v); /* 2 | 3 */ + SQRN(&u, &t, 2); /* 4 | 12 */ + f25519_mul(&t15, &u, &t); /* 5 | 15 */ + f25519_sqr(&u, &t15); /* 6 | 30 */ + f25519_mul(&t, &u, &v); /* 7 | 31 = 2^5 - 1 */ + SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */ + f25519_mul(&t, &u, &t); /* 13 | 2^10 - 1 */ + SQRN(&u, &t, 10); /* 23 | 2^20 - 2^10 */ + f25519_mul(&u, &u, &t); /* 24 | 2^20 - 1 */ + SQRN(&u, &u, 10); /* 34 | 2^30 - 2^10 */ + f25519_mul(&t, &u, &t); /* 35 | 2^30 - 1 */ + f25519_sqr(&u, &t); /* 36 | 2^31 - 2 */ + f25519_mul(&t, &u, &v); /* 37 | 2^31 - 1 */ + SQRN(&u, &t, 31); /* 68 | 2^62 - 2^31 */ + f25519_mul(&t, &u, &t); /* 69 | 2^62 - 1 */ + SQRN(&u, &t, 62); /* 131 | 2^124 - 2^62 */ + f25519_mul(&t, &u, &t); /* 132 | 2^124 - 1 */ + SQRN(&u, &t, 124); /* 256 | 2^248 - 2^124 */ + f25519_mul(&t, &u, &t); /* 257 | 2^248 - 1 */ + f25519_sqr(&u, &t); /* 258 | 2^249 - 2 */ + f25519_mul(&t, &u, &v); /* 259 | 2^249 - 1 */ + SQRN(&t, &t, 3); /* 262 | 2^252 - 8 */ + f25519_sqr(&u, &t); /* 263 | 2^253 - 16 */ + f25519_mul(&t, &u, &t); /* 264 | 3*2^252 - 24 */ + f25519_mul(&t, &t, &t15); /* 265 | 3*2^252 - 9 */ + f25519_mul(&w, &t, &v); /* 266 | 3*2^252 - 8 */ + + /* Awesome. Now let me explain. Let v be a square in GF(p), and let w = + * v^(3*2^252 - 8). In particular, let's consider * - * First of all, a complicated exponentation. The addition chain here is - * mine. We start with some preliminary values. - */ /* step | value */ - SQRN(&u, y, 1); /* 1 | 0, 2 */ - f25519_mul(&t, &u, y); /* 2 | 0, 3 */ - f25519_mul(&xy3, &t, x); /* 3 | 1, 3 */ - SQRN(&u, &u, 1); /* 4 | 0, 4 */ - f25519_mul(&w, &u, &xy3); /* 5 | 1, 7 */ - - /* And now we calculate w^((p - 5)/8) = w^(252 - 3). */ - SQRN(&u, &w, 1); /* 6 | 2 */ - f25519_mul(&t, &w, &u); /* 7 | 3 */ - SQRN(&u, &t, 1); /* 8 | 6 */ - f25519_mul(&t, &u, &w); /* 9 | 7 */ - SQRN(&u, &t, 3); /* 12 | 56 */ - f25519_mul(&t, &t, &u); /* 13 | 63 = 2^6 - 1 */ - SQRN(&u, &t, 6); /* 19 | 2^12 - 2^6 */ - f25519_mul(&t, &t, &u); /* 20 | 2^12 - 1 */ - SQRN(&u, &t, 12); /* 32 | 2^24 - 2^12 */ - f25519_mul(&t, &t, &u); /* 33 | 2^24 - 1 */ - SQRN(&u, &t, 1); /* 34 | 2^25 - 2 */ - f25519_mul(&t, &u, &w); /* 35 | 2^25 - 1 */ - SQRN(&u, &t, 25); /* 60 | 2^50 - 2^25 */ - f25519_mul(&t2p50m1, &t, &u); /* 61 | 2^50 - 1 */ - SQRN(&u, &t2p50m1, 50); /* 111 | 2^100 - 2^50 */ - f25519_mul(&t, &t2p50m1, &u); /* 112 | 2^100 - 1 */ - SQRN(&u, &t, 100); /* 212 | 2^200 - 2^100 */ - f25519_mul(&t, &t, &u); /* 213 | 2^200 - 1 */ - SQRN(&u, &t, 50); /* 263 | 2^250 - 2^50 */ - f25519_mul(&t, &t2p50m1, &u); /* 264 | 2^250 - 1 */ - SQRN(&u, &t, 2); /* 266 | 2^252 - 4 */ - f25519_mul(&t, &u, &w); /* 267 | 2^252 - 3 */ - - /* And finally... */ - f25519_mul(&beta, &t, &xy3); /* 268 | ... */ - - /* Now we have beta = (x y^3) (x y^7)^((p - 5)/8) = (x/y)^((p + 3)/8), and - * we're ready to finish the computation. Suppose that alpha^2 = u/w. - * Then beta^4 = (x/y)^((p + 3)/2) = alpha^(p + 3) = alpha^4 = (x/y)^2, so - * we have beta^2 = ±x/y. If y beta^2 = x then beta is the one we wanted; - * if -y beta^2 = x, then we want beta sqrt(-1), which we already know. Of - * course, it might not match either, in which case we fail. + * v^2 w^4 = v^2 v^{3*2^254 - 32} = (v^{2^254 - 10})^3 + * + * But 2^254 - 10 = ((2^255 - 19) - 1)/2 = (p - 1)/2. Since v is a square, + * it has order dividing (p - 1)/2, and therefore v^2 w^4 = 1 and + * + * w^4 = 1/v^2 + * + * That in turn implies that w^2 = ±1/v. Now, recall that v = x y, and let + * w' = w x. Then w'^2 = ±x^2/v = ±x/y. If y w'^2 = x then we set + * z = w', since we have z^2 = x/y; otherwise let z = i w', where i^2 = -1, + * so z^2 = -w^2 = x/y, and we're done. * * The easiest way to compare is to encode. This isn't as wasteful as it * sounds: the hard part is normalizing the representations, which we have * to do anyway. */ - f25519_sqr(&t, &beta); + f25519_mul(&w, &w, x); + f25519_sqr(&t, &w); f25519_mul(&t, &t, y); f25519_neg(&u, &t); f25519_store(xb, x); f25519_store(b0, &t); f25519_store(b1, &u); - f25519_mul(&u, &beta, SQRTM1); + f25519_mul(&u, &w, SQRTM1); m = -ct_memeq(b0, xb, 32); rc = PICK2(0, rc, m); - f25519_pick2(z, &beta, &u, m); + f25519_pick2(z, &w, &u, m); m = -ct_memeq(b1, xb, 32); rc = PICK2(0, rc, m); diff --git a/utils/curve25519.sage b/utils/curve25519.sage index 416f059f..24e74b12 100644 --- a/utils/curve25519.sage +++ b/utils/curve25519.sage @@ -115,7 +115,7 @@ def inv(x): t = u*t11 # 265 | 2^255 - 21 return t -def quosqrt(x, y): +def quosqrt_djb(x, y): ## First, some preliminary values. y2 = sqrn(y, 1) # 1 | 0, 2 @@ -162,6 +162,60 @@ def quosqrt(x, y): elif t == -x: return beta*sqrtm1 else: raise ValueError, 'not a square' +def quosqrt_mdw(x, y): + v = x*y + + ## Now we calculate w = v^{3*2^252 - 8}. This will be explained later. + u = sqrn(v, 1) # 1 | 2 + t = u*v # 2 | 3 + u = sqrn(t, 2) # 4 | 12 + t15 = u*t # 5 | 15 + u = sqrn(t15, 1) # 6 | 30 + t = u*v # 7 | 31 = 2^5 - 1 + u = sqrn(t, 5) # 12 | 2^10 - 2^5 + t = u*t # 13 | 2^10 - 1 + u = sqrn(t, 10) # 23 | 2^20 - 2^10 + u = u*t # 24 | 2^20 - 1 + u = sqrn(u, 10) # 34 | 2^30 - 2^10 + t = u*t # 35 | 2^30 - 1 + u = sqrn(t, 1) # 36 | 2^31 - 2 + t = u*v # 37 | 2^31 - 1 + u = sqrn(t, 31) # 68 | 2^62 - 2^31 + t = u*t # 69 | 2^62 - 1 + u = sqrn(t, 62) # 131 | 2^124 - 2^62 + t = u*t # 132 | 2^124 - 1 + u = sqrn(t, 124) # 256 | 2^248 - 2^124 + t = u*t # 257 | 2^248 - 1 + u = sqrn(t, 1) # 258 | 2^249 - 2 + t = u*v # 259 | 2^249 - 1 + t = sqrn(t, 3) # 262 | 2^252 - 8 + u = sqrn(t, 1) # 263 | 2^253 - 16 + t = u*t # 264 | 3*2^252 - 24 + t = t*t15 # 265 | 3*2^252 - 9 + w = t*v # 266 | 3*2^252 - 8 + + ## Awesome. Now let me explain. Let v be a square in GF(p), and let w = + ## v^(3*2^252 - 8). In particular, let's consider + ## + ## v^2 w^4 = v^2 v^{3*2^254 - 32} = (v^{2^254 - 10})^3 + ## + ## But 2^254 - 10 = ((2^255 - 19) - 1)/2 = (p - 1)/2. Since v is a square, + ## it has order dividing (p - 1)/2, and therefore v^2 w^4 = 1 and + ## + ## w^4 = 1/v^2 + ## + ## That in turn implies that w^2 = ±1/v. Now, recall that v = x y, and let + ## w' = w x. Then w'^2 = ±x^2/v = ±x/y. If y w'^2 = x then we set + ## z = w', since we have z^2 = x/y; otherwise let z = i w', where i^2 = -1, + ## so z^2 = -w^2 = x/y, and we're done. + t = w*x + u = y*t^2 + if u == x: return t + elif u == -x: return t*sqrtm1 + else: raise ValueError, 'not a square' + +quosqrt = quosqrt_mdw + assert inv(k(9))*9 == 1 assert 5*quosqrt(k(4), k(5))^2 == 4