int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
BN_CTX *ctx) {
int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *,
BN_CTX *);
int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
const BIGNUM *p;
BN_CTX *new_ctx = NULL;
BIGNUM *rh, *tmp, *Z4, *Z6;
int ret = 0;
if (EC_POINT_is_at_infinity(group, point)) {
return 1;
}
field_mul = group->meth->field_mul;
field_sqr = group->meth->field_sqr;
p = &group->field;
if (ctx == NULL) {
ctx = new_ctx = BN_CTX_new();
if (ctx == NULL) {
return 0;
}
}
BN_CTX_start(ctx);
rh = BN_CTX_get(ctx);
tmp = BN_CTX_get(ctx);
Z4 = BN_CTX_get(ctx);
Z6 = BN_CTX_get(ctx);
if (Z6 == NULL) {
goto err;
}
// We have a curve defined by a Weierstrass equation
// y^2 = x^3 + a*x + b.
// The point to consider is given in Jacobian projective coordinates
// where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
// Substituting this and multiplying by Z^6 transforms the above equation
// into
// Y^2 = X^3 + a*X*Z^4 + b*Z^6.
// To test this, we add up the right-hand side in 'rh'.
// rh := X^2
if (!field_sqr(group, rh, &point->X, ctx)) {
goto err;
}
if (BN_cmp(&point->Z, &group->one) != 0) {
if (!field_sqr(group, tmp, &point->Z, ctx) ||
!field_sqr(group, Z4, tmp, ctx) ||
!field_mul(group, Z6, Z4, tmp, ctx)) {
goto err;
}
// rh := (rh + a*Z^4)*X
if (group->a_is_minus3) {
if (!bn_mod_lshift1_consttime(tmp, Z4, p, ctx) ||
!bn_mod_add_consttime(tmp, tmp, Z4, p, ctx) ||
!bn_mod_sub_consttime(rh, rh, tmp, p, ctx) ||
!field_mul(group, rh, rh, &point->X, ctx)) {
goto err;
}
} else {
if (!field_mul(group, tmp, Z4, &group->a, ctx) ||
!bn_mod_add_consttime(rh, rh, tmp, p, ctx) ||
!field_mul(group, rh, rh, &point->X, ctx)) {
goto err;
}
}
// rh := rh + b*Z^6
if (!field_mul(group, tmp, &group->b, Z6, ctx) ||
!bn_mod_add_consttime(rh, rh, tmp, p, ctx)) {
goto err;
}
} else {
// rh := (rh + a)*X
if (!bn_mod_add_consttime(rh, rh, &group->a, p, ctx) ||
!field_mul(group, rh, rh, &point->X, ctx)) {
goto err;
}
// rh := rh + b
if (!bn_mod_add_consttime(rh, rh, &group->b, p, ctx)) {
goto err;
}
}
// 'lh' := Y^2
if (!field_sqr(group, tmp, &point->Y, ctx)) {
goto err;
}
ret = (0 == BN_ucmp(tmp, rh));
err:
BN_CTX_end(ctx);
BN_CTX_free(new_ctx);
return ret;
}
int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
const EC_POINT *b, BN_CTX *ctx) {
int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *,
BN_CTX *);
int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
const BIGNUM *p;
BN_CTX *new_ctx = NULL;
BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
int ret = 0;
if (a == b) {
return EC_POINT_dbl(group, r, a, ctx);
}
if (EC_POINT_is_at_infinity(group, a)) {
return EC_POINT_copy(r, b);
}
if (EC_POINT_is_at_infinity(group, b)) {
return EC_POINT_copy(r, a);
}
field_mul = group->meth->field_mul;
field_sqr = group->meth->field_sqr;
p = &group->field;
if (ctx == NULL) {
ctx = new_ctx = BN_CTX_new();
if (ctx == NULL) {
return 0;
}
}
BN_CTX_start(ctx);
n0 = BN_CTX_get(ctx);
n1 = BN_CTX_get(ctx);
n2 = BN_CTX_get(ctx);
n3 = BN_CTX_get(ctx);
n4 = BN_CTX_get(ctx);
n5 = BN_CTX_get(ctx);
n6 = BN_CTX_get(ctx);
if (n6 == NULL) {
goto end;
}
// Note that in this function we must not read components of 'a' or 'b'
// once we have written the corresponding components of 'r'.
// ('r' might be one of 'a' or 'b'.)
// n1, n2
int b_Z_is_one = BN_cmp(&b->Z, &group->one) == 0;
if (b_Z_is_one) {
if (!BN_copy(n1, &a->X) || !BN_copy(n2, &a->Y)) {
goto end;
}
// n1 = X_a
// n2 = Y_a
} else {
if (!field_sqr(group, n0, &b->Z, ctx) ||
!field_mul(group, n1, &a->X, n0, ctx)) {
goto end;
}
// n1 = X_a * Z_b^2
if (!field_mul(group, n0, n0, &b->Z, ctx) ||
!field_mul(group, n2, &a->Y, n0, ctx)) {
goto end;
}
// n2 = Y_a * Z_b^3
}
// n3, n4
int a_Z_is_one = BN_cmp(&a->Z, &group->one) == 0;
if (a_Z_is_one) {
if (!BN_copy(n3, &b->X) || !BN_copy(n4, &b->Y)) {
goto end;
}
// n3 = X_b
// n4 = Y_b
} else {
if (!field_sqr(group, n0, &a->Z, ctx) ||
!field_mul(group, n3, &b->X, n0, ctx)) {
goto end;
}
// n3 = X_b * Z_a^2
if (!field_mul(group, n0, n0, &a->Z, ctx) ||
!field_mul(group, n4, &b->Y, n0, ctx)) {
goto end;
}
// n4 = Y_b * Z_a^3
}
// n5, n6
if (!bn_mod_sub_consttime(n5, n1, n3, p, ctx) ||
!bn_mod_sub_consttime(n6, n2, n4, p, ctx)) {
goto end;
}
// n5 = n1 - n3
// n6 = n2 - n4
if (BN_is_zero(n5)) {
if (BN_is_zero(n6)) {
// a is the same point as b
BN_CTX_end(ctx);
ret = EC_POINT_dbl(group, r, a, ctx);
ctx = NULL;
goto end;
} else {
// a is the inverse of b
BN_zero(&r->Z);
ret = 1;
goto end;
}
}
// 'n7', 'n8'
if (!bn_mod_add_consttime(n1, n1, n3, p, ctx) ||
!bn_mod_add_consttime(n2, n2, n4, p, ctx)) {
goto end;
}
// 'n7' = n1 + n3
// 'n8' = n2 + n4
// Z_r
if (a_Z_is_one && b_Z_is_one) {
if (!BN_copy(&r->Z, n5)) {
goto end;
}
} else {
if (a_Z_is_one) {
if (!BN_copy(n0, &b->Z)) {
goto end;
}
} else if (b_Z_is_one) {
if (!BN_copy(n0, &a->Z)) {
goto end;
}
} else if (!field_mul(group, n0, &a->Z, &b->Z, ctx)) {
goto end;
}
if (!field_mul(group, &r->Z, n0, n5, ctx)) {
goto end;
}
}
// Z_r = Z_a * Z_b * n5
// X_r
if (!field_sqr(group, n0, n6, ctx) ||
!field_sqr(group, n4, n5, ctx) ||
!field_mul(group, n3, n1, n4, ctx) ||
!bn_mod_sub_consttime(&r->X, n0, n3, p, ctx)) {
goto end;
}
// X_r = n6^2 - n5^2 * 'n7'
// 'n9'
if (!bn_mod_lshift1_consttime(n0, &r->X, p, ctx) ||
!bn_mod_sub_consttime(n0, n3, n0, p, ctx)) {
goto end;
}
// n9 = n5^2 * 'n7' - 2 * X_r
// Y_r
if (!field_mul(group, n0, n0, n6, ctx) ||
!field_mul(group, n5, n4, n5, ctx)) {
goto end; // now n5 is n5^3
}
if (!field_mul(group, n1, n2, n5, ctx) ||
!bn_mod_sub_consttime(n0, n0, n1, p, ctx)) {
goto end;
}
if (BN_is_odd(n0) && !BN_add(n0, n0, p)) {
goto end;
}
// now 0 <= n0 < 2*p, and n0 is even
if (!BN_rshift1(&r->Y, n0)) {
goto end;
}
// Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2
ret = 1;
end:
if (ctx) {
// otherwise we already called BN_CTX_end
BN_CTX_end(ctx);
}
BN_CTX_free(new_ctx);
return ret;
}
int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
BN_CTX *ctx) {
int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *,
BN_CTX *);
int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
const BIGNUM *p;
BN_CTX *new_ctx = NULL;
BIGNUM *n0, *n1, *n2, *n3;
int ret = 0;
if (EC_POINT_is_at_infinity(group, a)) {
BN_zero(&r->Z);
return 1;
}
field_mul = group->meth->field_mul;
field_sqr = group->meth->field_sqr;
p = &group->field;
if (ctx == NULL) {
ctx = new_ctx = BN_CTX_new();
if (ctx == NULL) {
return 0;
}
}
BN_CTX_start(ctx);
n0 = BN_CTX_get(ctx);
n1 = BN_CTX_get(ctx);
n2 = BN_CTX_get(ctx);
n3 = BN_CTX_get(ctx);
if (n3 == NULL) {
goto err;
}
// Note that in this function we must not read components of 'a'
// once we have written the corresponding components of 'r'.
// ('r' might the same as 'a'.)
// n1
if (BN_cmp(&a->Z, &group->one) == 0) {
if (!field_sqr(group, n0, &a->X, ctx) ||
!bn_mod_lshift1_consttime(n1, n0, p, ctx) ||
!bn_mod_add_consttime(n0, n0, n1, p, ctx) ||
!bn_mod_add_consttime(n1, n0, &group->a, p, ctx)) {
goto err;
}
// n1 = 3 * X_a^2 + a_curve
} else if (group->a_is_minus3) {
if (!field_sqr(group, n1, &a->Z, ctx) ||
!bn_mod_add_consttime(n0, &a->X, n1, p, ctx) ||
!bn_mod_sub_consttime(n2, &a->X, n1, p, ctx) ||
!field_mul(group, n1, n0, n2, ctx) ||
!bn_mod_lshift1_consttime(n0, n1, p, ctx) ||
!bn_mod_add_consttime(n1, n0, n1, p, ctx)) {
goto err;
}
// n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
// = 3 * X_a^2 - 3 * Z_a^4
} else {
if (!field_sqr(group, n0, &a->X, ctx) ||
!bn_mod_lshift1_consttime(n1, n0, p, ctx) ||
!bn_mod_add_consttime(n0, n0, n1, p, ctx) ||
!field_sqr(group, n1, &a->Z, ctx) ||
!field_sqr(group, n1, n1, ctx) ||
!field_mul(group, n1, n1, &group->a, ctx) ||
!bn_mod_add_consttime(n1, n1, n0, p, ctx)) {
goto err;
}
// n1 = 3 * X_a^2 + a_curve * Z_a^4
}
// Z_r
if (BN_cmp(&a->Z, &group->one) == 0) {
if (!BN_copy(n0, &a->Y)) {
goto err;
}
} else if (!field_mul(group, n0, &a->Y, &a->Z, ctx)) {
goto err;
}
if (!bn_mod_lshift1_consttime(&r->Z, n0, p, ctx)) {
goto err;
}
// Z_r = 2 * Y_a * Z_a
// n2
if (!field_sqr(group, n3, &a->Y, ctx) ||
!field_mul(group, n2, &a->X, n3, ctx) ||
!bn_mod_lshift_consttime(n2, n2, 2, p, ctx)) {
goto err;
}
// n2 = 4 * X_a * Y_a^2
// X_r
if (!bn_mod_lshift1_consttime(n0, n2, p, ctx) ||
!field_sqr(group, &r->X, n1, ctx) ||
!bn_mod_sub_consttime(&r->X, &r->X, n0, p, ctx)) {
goto err;
}
// X_r = n1^2 - 2 * n2
// n3
if (!field_sqr(group, n0, n3, ctx) ||
!bn_mod_lshift_consttime(n3, n0, 3, p, ctx)) {
goto err;
}
// n3 = 8 * Y_a^4
// Y_r
if (!bn_mod_sub_consttime(n0, n2, &r->X, p, ctx) ||
!field_mul(group, n0, n1, n0, ctx) ||
!bn_mod_sub_consttime(&r->Y, n0, n3, p, ctx)) {
goto err;
}
// Y_r = n1 * (n2 - X_r) - n3
ret = 1;
err:
BN_CTX_end(ctx);
BN_CTX_free(new_ctx);
return ret;
}
BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
// Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm
// (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
// algorithm 1.5.1). |p| is assumed to be a prime.
BIGNUM *ret = in;
int err = 1;
int r;
BIGNUM *A, *b, *q, *t, *x, *y;
int e, i, j;
if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
if (BN_abs_is_word(p, 2)) {
if (ret == NULL) {
ret = BN_new();
}
if (ret == NULL) {
goto end;
}
if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
if (ret != in) {
BN_free(ret);
}
return NULL;
}
return ret;
}
OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
return (NULL);
}
if (BN_is_zero(a) || BN_is_one(a)) {
if (ret == NULL) {
ret = BN_new();
}
if (ret == NULL) {
goto end;
}
if (!BN_set_word(ret, BN_is_one(a))) {
if (ret != in) {
BN_free(ret);
}
return NULL;
}
return ret;
}
BN_CTX_start(ctx);
A = BN_CTX_get(ctx);
b = BN_CTX_get(ctx);
q = BN_CTX_get(ctx);
t = BN_CTX_get(ctx);
x = BN_CTX_get(ctx);
y = BN_CTX_get(ctx);
if (y == NULL) {
goto end;
}
if (ret == NULL) {
ret = BN_new();
}
if (ret == NULL) {
goto end;
}
// A = a mod p
if (!BN_nnmod(A, a, p, ctx)) {
goto end;
}
// now write |p| - 1 as 2^e*q where q is odd
e = 1;
while (!BN_is_bit_set(p, e)) {
e++;
}
// we'll set q later (if needed)
if (e == 1) {
// The easy case: (|p|-1)/2 is odd, so 2 has an inverse
// modulo (|p|-1)/2, and square roots can be computed
// directly by modular exponentiation.
// We have
// 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
// so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
if (!BN_rshift(q, p, 2)) {
goto end;
}
q->neg = 0;
if (!BN_add_word(q, 1) ||
!BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
goto end;
}
err = 0;
goto vrfy;
}
if (e == 2) {
// |p| == 5 (mod 8)
//
// In this case 2 is always a non-square since
// Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
// So if a really is a square, then 2*a is a non-square.
// Thus for
// b := (2*a)^((|p|-5)/8),
// i := (2*a)*b^2
// we have
// i^2 = (2*a)^((1 + (|p|-5)/4)*2)
// = (2*a)^((p-1)/2)
// = -1;
// so if we set
// x := a*b*(i-1),
// then
// x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
// = a^2 * b^2 * (-2*i)
// = a*(-i)*(2*a*b^2)
// = a*(-i)*i
// = a.
//
// (This is due to A.O.L. Atkin,
// <URL:
//http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
// November 1992.)
// t := 2*a
if (!bn_mod_lshift1_consttime(t, A, p, ctx)) {
goto end;
}
// b := (2*a)^((|p|-5)/8)
if (!BN_rshift(q, p, 3)) {
goto end;
}
q->neg = 0;
if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
goto end;
}
// y := b^2
if (!BN_mod_sqr(y, b, p, ctx)) {
goto end;
}
// t := (2*a)*b^2 - 1
if (!BN_mod_mul(t, t, y, p, ctx) ||
!BN_sub_word(t, 1)) {
goto end;
}
// x = a*b*t
if (!BN_mod_mul(x, A, b, p, ctx) ||
!BN_mod_mul(x, x, t, p, ctx)) {
goto end;
}
if (!BN_copy(ret, x)) {
goto end;
}
err = 0;
goto vrfy;
}
// e > 2, so we really have to use the Tonelli/Shanks algorithm.
// First, find some y that is not a square.
if (!BN_copy(q, p)) {
goto end; // use 'q' as temp
}
q->neg = 0;
i = 2;
do {
// For efficiency, try small numbers first;
// if this fails, try random numbers.
if (i < 22) {
if (!BN_set_word(y, i)) {
goto end;
}
} else {
if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
goto end;
}
if (BN_ucmp(y, p) >= 0) {
if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
goto end;
}
}
// now 0 <= y < |p|
if (BN_is_zero(y)) {
if (!BN_set_word(y, i)) {
goto end;
}
}
}
r = bn_jacobi(y, q, ctx); // here 'q' is |p|
if (r < -1) {
goto end;
}
if (r == 0) {
// m divides p
OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
goto end;
}
} while (r == 1 && ++i < 82);
if (r != -1) {
// Many rounds and still no non-square -- this is more likely
// a bug than just bad luck.
// Even if p is not prime, we should have found some y
// such that r == -1.
OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
goto end;
}
// Here's our actual 'q':
if (!BN_rshift(q, q, e)) {
goto end;
}
// Now that we have some non-square, we can find an element
// of order 2^e by computing its q'th power.
if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
goto end;
}
if (BN_is_one(y)) {
OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
goto end;
}
// Now we know that (if p is indeed prime) there is an integer
// k, 0 <= k < 2^e, such that
//
// a^q * y^k == 1 (mod p).
//
// As a^q is a square and y is not, k must be even.
// q+1 is even, too, so there is an element
//
// X := a^((q+1)/2) * y^(k/2),
//
// and it satisfies
//
// X^2 = a^q * a * y^k
// = a,
//
// so it is the square root that we are looking for.
// t := (q-1)/2 (note that q is odd)
if (!BN_rshift1(t, q)) {
goto end;
}
// x := a^((q-1)/2)
if (BN_is_zero(t)) // special case: p = 2^e + 1
{
if (!BN_nnmod(t, A, p, ctx)) {
goto end;
}
if (BN_is_zero(t)) {
// special case: a == 0 (mod p)
BN_zero(ret);
err = 0;
goto end;
} else if (!BN_one(x)) {
goto end;
}
} else {
if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
goto end;
}
if (BN_is_zero(x)) {
// special case: a == 0 (mod p)
BN_zero(ret);
err = 0;
goto end;
}
}
// b := a*x^2 (= a^q)
if (!BN_mod_sqr(b, x, p, ctx) ||
!BN_mod_mul(b, b, A, p, ctx)) {
goto end;
}
// x := a*x (= a^((q+1)/2))
if (!BN_mod_mul(x, x, A, p, ctx)) {
goto end;
}
while (1) {
// Now b is a^q * y^k for some even k (0 <= k < 2^E
// where E refers to the original value of e, which we
// don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
//
// We have a*b = x^2,
// y^2^(e-1) = -1,
// b^2^(e-1) = 1.
if (BN_is_one(b)) {
if (!BN_copy(ret, x)) {
goto end;
}
err = 0;
goto vrfy;
}
// find smallest i such that b^(2^i) = 1
i = 1;
if (!BN_mod_sqr(t, b, p, ctx)) {
goto end;
}
while (!BN_is_one(t)) {
i++;
if (i == e) {
OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
goto end;
}
if (!BN_mod_mul(t, t, t, p, ctx)) {
goto end;
}
}
// t := y^2^(e - i - 1)
if (!BN_copy(t, y)) {
goto end;
}
for (j = e - i - 1; j > 0; j--) {
if (!BN_mod_sqr(t, t, p, ctx)) {
goto end;
}
}
if (!BN_mod_mul(y, t, t, p, ctx) ||
!BN_mod_mul(x, x, t, p, ctx) ||
!BN_mod_mul(b, b, y, p, ctx)) {
goto end;
}
e = i;
}
vrfy:
if (!err) {
// verify the result -- the input might have been not a square
// (test added in 0.9.8)
if (!BN_mod_sqr(x, ret, p, ctx)) {
err = 1;
}
if (!err && 0 != BN_cmp(x, A)) {
OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
err = 1;
}
}
end:
if (err) {
if (ret != in) {
BN_clear_free(ret);
}
ret = NULL;
}
BN_CTX_end(ctx);
return ret;
}
