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curve25519-wes.c
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curve25519-wes.c
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#include "curve25519-wes.h"
/* add, sub, mul, sqr, inv, mulC, contract, expand... & these from assembler */
extern void field25519_wes_powq(field25519_t out, const field25519_t a);
extern void curve25519_wes_loop(uint64_t a[20], const exponent_t e, int length);
/* mod p the multiplicative group has order p-1.
* p-1 = 2^255-20 = 4*3*65147*74058212732561358302231226437062788676166966415465897661863160754340907
*
* This means we can represent powers in the multiplicative group as (i, j),
* where i is [0, 3) and j is [0,r) for odd r=(p-1)/4.
*
* When you square an element you double it's power, doubling both i & j.
* To find a square-root we need to 'undouble' both i & j.
* The j term is easy to undouble; r is odd so (r+1)/2 is the inverse of 2.
*
* The i term has had information destroyed by doubling.
* If the i term of a is 0
* Then it's square-roots have i=0 or i=2.
* => we can leave it as 0, which is unaffected by *(r+1)/2
* If the i term of a is 2 (unaffected by multiplication with r, an odd #)
* Then it's square-roots have i=1 or i=3.
* We need to adjust the i term by adding +1 or +3, both generators mod 4.
* Multiplying 1 or 3 by r leaves them as 1 or 3, but kills the j term.
* Thus mul by a generator to the power r to add 1 or 3 w/o changing j.
* If the i term of a is 1 or 3
* Then a is not a square number...
*/
static const field25519_t u = { /* (s-1)/2, s=2^r (2 is a generator) */
UINT64_C(0x00070D93A507504E),
UINT64_C(0x00006AD2FE478C4E),
UINT64_C(0x0003F7AF4E5E8630),
UINT64_C(0x0007C2CAD340264F),
UINT64_C(0x00055C1924027E0E)
};
int field25519_wes_sqrt(field25519_t out, const field25519_t a) {
field25519_t b, z;
field25519_wes_powq(b, a); /* b = a^(2^252-3) */
field25519_wes_mul(z, b, a); /* z = a^(2^252-2) = a^((r+1)/2) */
field25519_wes_sqr(b, b); /* b = a^(2^253-6) */
field25519_wes_mul(b, b, a); /* b = a^(2^253-5) = a^r aka +-1 [or +-sqrt(-1)] */
/* if b = 1 then z else z * s ... want to achieve this w/o an 'if' */
/* 1*(1+b)/2 + s*(1-b)/2 = (s+1)/2 - b*(s-1)/2 = 1+u - b*u, u=(s-1)/2 */
field25519_wes_mul(b, b, u);
field25519_wes_sub(b, u, b);
b[0]++;
field25519_wes_mul(out, b, z);
return 0;
}
void curve25519_wes_mul(curve25519_t out, const curve25519_t a, const curve25519_t b) {
field25519_t l, m, x;
/* l = (Ay - By) * inv (Ax - Bx) */
field25519_wes_sub(l, &a[5], &b[5]);
field25519_wes_sub(m, a, b);
field25519_wes_inv(m, m);
field25519_wes_mul(l, l, m);
/* x = l^2 - Ax - Bx - A */
field25519_wes_sqr(m, l);
field25519_wes_sub(m, m, a);
field25519_wes_sub(x, m, b);
x[0] -= 486662;
/* y = l * (Ax - x) - Ay */
field25519_wes_sub(m, a, x);
field25519_wes_mul(m, l, m);
field25519_wes_sub(&out[5], m, &a[5]);
out[0] = x[0]; out[1] = x[1]; out[2] = x[2]; out[3] = x[3]; out[4] = x[4];
}
void curve25519_wes_power(curve25519_t out, const curve25519_t a, const exponent_t e) {
uint64_t b[20];
b[0] = a[0]; b[1] = a[1]; b[2] = a[2]; b[3] = a[3]; b[4] = a[4];
curve25519_wes_loop(b, e, 32);
field25519_wes_inv(b+ 5, b+ 5);
field25519_wes_inv(b+15, b+15);
field25519_wes_mul(b+ 0, b+ 0, b+ 5);
field25519_wes_mul(b+ 5, b+10, b+15);
/* b+0 = Rx, b+5 = RAx */
/* c = sqr (Rx - Ax) * (RAx + Rx + Ax + A) */
field25519_wes_sub(b+10, b+0, a);
field25519_wes_sqr(b+10, b+10);
field25519_wes_add(b+5, b+5, b+0);
field25519_wes_add(b+5, b+5, a);
b[5] += 486662;
field25519_wes_mul(b+5, b+5, b+10);
field25519_wes_sqr(b+15, a+5);
field25519_wes_sub(b+15, b+15, b+5);
/* b+0 = Rx, b+15 = By2 - c */
/* find Ry2 */
field25519_wes_sqr(b+10, b+0);
field25519_wes_mul(b+5, b+10, b+0);
field25519_wes_mulC(b+10, b+10, 486662);
field25519_wes_add(b+5, b+5, b+10);
field25519_wes_add(b+5, b+5, b+0);
field25519_wes_add(b+15, b+15, b+5);
/* b+0 = Rx, b+15 = Ry2 + By2 - c */
/* (Ay*2)^-1 */
b[5] = a[5] << 1;
b[6] = a[6] << 1;
b[7] = a[7] << 1;
b[8] = a[8] << 1;
b[9] = a[9] << 1;
field25519_wes_inv(b+5, b+5);
field25519_wes_mul(out+5, b+5, b+15);
out[0] = b[0]; out[1] = b[1]; out[2] = b[2]; out[3] = b[3]; out[4] = b[4];
}
void curve25519_wes_contract(raw25519_t out, const curve25519_t a) {
field25519_wes_contract(out, a);
}
int curve25519_wes_expand(curve25519_t out, const raw25519_t in) {
field25519_t tmp;
int r;
uint64_t sign;
field25519_wes_expand(out, in);
field25519_wes_sqr(tmp, out);
field25519_wes_mul(out+5, tmp, out);
field25519_wes_mulC(tmp, tmp, 486662);
field25519_wes_add(out+5, out+5, tmp);
field25519_wes_add(out+5, out+5, out);
r = field25519_wes_sqrt(out+5, out+5);
sign = 1 - ((in[31] >> 7) << 1);
out[5] *= sign; out[6] *= sign; out[7] *= sign; out[8] *= sign; out[9] *= sign;
return r;
}
void curve25519_wes_clamp(exponent_t inout) {
inout[31] = (inout[31] & 63) | 64;
inout[0] &= ~7;
}
curve25519_t generator = {
UINT64_C(0x9),
UINT64_C(0x0),
UINT64_C(0x0),
UINT64_C(0x0),
UINT64_C(0x0),
UINT64_C(0x0009C5A27ECED3D9),
UINT64_C(0x0007CDAF8C36453C),
UINT64_C(0x000523453248F535),
UINT64_C(0x00035A700F6E963B),
UINT64_C(0x00020AE19A1B8A08)
};
void curve25519_wes(raw25519_t outx, const raw25519_t inx, const exponent_t e) {
uint64_t b[20];
field25519_wes_expand(b, inx);
curve25519_wes_loop(b, e, 32);
field25519_wes_inv(b+5, b+5);
field25519_wes_mul(b+0, b+0, b+5);
field25519_wes_contract(outx, b);
}