Description:
Mark Wooden and Franck Rondepierre noted that the square-root-mod-p
operations used in the EdDSA RFC (RFC 8032) can be simplified. For
Ed25519, instead of computing u*v^3 * (u * v^7)^((p-5)/8), we can
compute u * (u*v)^((p-5)/8). This saves 3 multiplications and 2
squarings. For more details (including a proof), see the following
message from the CFRG mailing list:
https://mailarchive.ietf.org/arch/msg/cfrg/qlKpMBqxXZYmDpXXIx6LO3Oznv4/
Note that the Ed448 implementation (see
ossl_curve448_point_decode_like_eddsa_and_mul_by_ratio() in
./crypto/ec/curve448/curve448.c) appears to already use this simpler
method (i.e. it does not follow the method suggested in RFC 8032).
Testing:
Build and then run the test suite:
./Configure -Werror --strict-warnings
make update
make
make test
Numerical testing of the square-root computation can be done using the
following sage script:
def legendre(x,p):
return kronecker(x,p)
#
Ed25519
p = 2**255-19
# -1 is a square
if legendre(-1,p)==1:
print("-1 is a square")
# suppose u/v is a square.
# to compute one of its square roots, find x such that
# x**4 == (u/v)**2 .
# this implies
# x**2 == u/v, or
# x**2 == -(u/v) ,
# which implies either x or i*x is a square-root of u/v (where i is a square root of -1).
# we can take x equal to u * (u*v)**((p-5)/8).
# 2 is a generator
# this can be checked by factoring p-1
# and then showing 2**((p-1)/q) != 1 (mod p)
# for all primes q dividing p-1.
g = 2
s = p>>2 # s = (p-1)/4
i = power_mod(g, s, p)
t = p>>3 # t = (p-5)/8
COUNT = 1<<18
while COUNT > 0:
COUNT -= 1
r = randint(0,p-1) # r = u/v
v = randint(1,p-1)
u = mod(r*v,p)
# compute x = u * (u*v)**((p-5)/8)
w = mod(u*v,p)
x = mod(u*power_mod(w, t, p), p)
# check that x**2 == r, or (i*x)**2 == r, or r is not a square
rr = power_mod(x, 2, p)
if rr==r:
continue
rr = power_mod(mod(i*x,p), 2, p)
if rr==r:
continue
if legendre(r,p) != 1:
continue
print("failure!")
exit()
print("passed!")
Reviewed-by: Paul Dale <pauli@openssl.org>
Reviewed-by: Tomas Mraz <tomas@openssl.org>
(Merged from https://github.com/openssl/openssl/pull/17544)
{
fe u;
fe v;
- fe v3;
+ fe w;
fe vxx;
fe check;
fe_sub(u, u, h->Z); /* u = y^2-1 */
fe_add(v, v, h->Z); /* v = dy^2+1 */
- fe_sq(v3, v);
- fe_mul(v3, v3, v); /* v3 = v^3 */
- fe_sq(h->X, v3);
- fe_mul(h->X, h->X, v);
- fe_mul(h->X, h->X, u); /* x = uv^7 */
+ fe_mul(w, u, v); /* w = u*v */
- fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */
- fe_mul(h->X, h->X, v3);
- fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */
+ fe_pow22523(h->X, w); /* x = w^((q-5)/8) */
+ fe_mul(h->X, h->X, u); /* x = u * w^((q-5)/8) */
fe_sq(vxx, h->X);
fe_mul(vxx, vxx, v);