/* Copyright 2008, Google Inc. * All rights reserved. * * Code released into the public domain. * * curve25519-donna: Curve25519 elliptic curve, public key function * * http://code.google.com/p/curve25519-donna/ * * Adam Langley * * Derived from public domain C code by Daniel J. Bernstein * * More information about curve25519 can be found here * http://cr.yp.to/ecdh.html * * djb's sample implementation of curve25519 is written in a special assembly * language called qhasm and uses the floating point registers. * * This is, almost, a clean room reimplementation from the curve25519 paper. It * uses many of the tricks described therein. Only the crecip function is taken * from the sample implementation. */ #include #include typedef uint8_t u8; typedef uint64_t limb; typedef limb felem[5]; // This is a special gcc mode for 128-bit integers. It's implemented on 64-bit // platforms only as far as I know. typedef unsigned uint128_t __attribute__((mode(TI))); #undef force_inline #define force_inline __attribute__((always_inline)) /* Sum two numbers: output += in */ static inline void force_inline fsum (limb *output, const limb *in) { output[0] += in[0]; output[1] += in[1]; output[2] += in[2]; output[3] += in[3]; output[4] += in[4]; } /* Find the difference of two numbers: output = in - output * (note the order of the arguments!) * * Assumes that out[i] < 2**52 * On return, out[i] < 2**55 */ static inline void force_inline fdifference_backwards (felem out, const felem in) { /* 152 is 19 << 3 */ static const limb two54m152 = (((limb) 1) << 54) - 152; static const limb two54m8 = (((limb) 1) << 54) - 8; out[0] = in[0] + two54m152 - out[0]; out[1] = in[1] + two54m8 - out[1]; out[2] = in[2] + two54m8 - out[2]; out[3] = in[3] + two54m8 - out[3]; out[4] = in[4] + two54m8 - out[4]; } /* Multiply a number by a scalar: output = in * scalar */ static inline void force_inline fscalar_product (felem output, const felem in, const limb scalar) { uint128_t a; a = ((uint128_t) in[0]) * scalar; output[0] = ((limb) a) & 0x7ffffffffffff; a = ((uint128_t) in[1]) * scalar + ((limb) (a >> 51)); output[1] = ((limb) a) & 0x7ffffffffffff; a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51)); output[2] = ((limb) a) & 0x7ffffffffffff; a = ((uint128_t) in[3]) * scalar + ((limb) (a >> 51)); output[3] = ((limb) a) & 0x7ffffffffffff; a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51)); output[4] = ((limb) a) & 0x7ffffffffffff; output[0] += (a >> 51) * 19; } /* Multiply two numbers: output = in2 * in * * output must be distinct to both inputs. The inputs are reduced coefficient * form, the output is not. * * Assumes that in[i] < 2**55 and likewise for in2. * On return, output[i] < 2**52 */ static inline void force_inline fmul (felem output, const felem in2, const felem in) { uint128_t t[5]; limb r0, r1, r2, r3, r4, s0, s1, s2, s3, s4, c; r0 = in[0]; r1 = in[1]; r2 = in[2]; r3 = in[3]; r4 = in[4]; s0 = in2[0]; s1 = in2[1]; s2 = in2[2]; s3 = in2[3]; s4 = in2[4]; t[0] = ((uint128_t) r0) * s0; t[1] = ((uint128_t) r0) * s1 + ((uint128_t) r1) * s0; t[2] = ((uint128_t) r0) * s2 + ((uint128_t) r2) * s0 + ((uint128_t) r1) * s1; t[3] = ((uint128_t) r0) * s3 + ((uint128_t) r3) * s0 + ((uint128_t) r1) * s2 + ((uint128_t) r2) * s1; t[4] = ((uint128_t) r0) * s4 + ((uint128_t) r4) * s0 + ((uint128_t) r3) * s1 + ((uint128_t) r1) * s3 + ((uint128_t) r2) * s2; r4 *= 19; r1 *= 19; r2 *= 19; r3 *= 19; t[0] += ((uint128_t) r4) * s1 + ((uint128_t) r1) * s4 + ((uint128_t) r2) * s3 + ((uint128_t) r3) * s2; t[1] += ((uint128_t) r4) * s2 + ((uint128_t) r2) * s4 + ((uint128_t) r3) * s3; t[2] += ((uint128_t) r4) * s3 + ((uint128_t) r3) * s4; t[3] += ((uint128_t) r4) * s4; r0 = (limb) t[0] & 0x7ffffffffffff; c = (limb) (t[0] >> 51); t[1] += c; r1 = (limb) t[1] & 0x7ffffffffffff; c = (limb) (t[1] >> 51); t[2] += c; r2 = (limb) t[2] & 0x7ffffffffffff; c = (limb) (t[2] >> 51); t[3] += c; r3 = (limb) t[3] & 0x7ffffffffffff; c = (limb) (t[3] >> 51); t[4] += c; r4 = (limb) t[4] & 0x7ffffffffffff; c = (limb) (t[4] >> 51); r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff; r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff; r2 += c; output[0] = r0; output[1] = r1; output[2] = r2; output[3] = r3; output[4] = r4; } static inline void force_inline fsquare_times (felem output, const felem in, limb count) { uint128_t t[5]; limb r0, r1, r2, r3, r4, c; limb d0, d1, d2, d4, d419; r0 = in[0]; r1 = in[1]; r2 = in[2]; r3 = in[3]; r4 = in[4]; do { d0 = r0 * 2; d1 = r1 * 2; d2 = r2 * 2 * 19; d419 = r4 * 19; d4 = d419 * 2; t[0] = ((uint128_t) r0) * r0 + ((uint128_t) d4) * r1 + (((uint128_t) d2) * (r3)); t[1] = ((uint128_t) d0) * r1 + ((uint128_t) d4) * r2 + (((uint128_t) r3) * (r3 * 19)); t[2] = ((uint128_t) d0) * r2 + ((uint128_t) r1) * r1 + (((uint128_t) d4) * (r3)); t[3] = ((uint128_t) d0) * r3 + ((uint128_t) d1) * r2 + (((uint128_t) r4) * (d419)); t[4] = ((uint128_t) d0) * r4 + ((uint128_t) d1) * r3 + (((uint128_t) r2) * (r2)); r0 = (limb) t[0] & 0x7ffffffffffff; c = (limb) (t[0] >> 51); t[1] += c; r1 = (limb) t[1] & 0x7ffffffffffff; c = (limb) (t[1] >> 51); t[2] += c; r2 = (limb) t[2] & 0x7ffffffffffff; c = (limb) (t[2] >> 51); t[3] += c; r3 = (limb) t[3] & 0x7ffffffffffff; c = (limb) (t[3] >> 51); t[4] += c; r4 = (limb) t[4] & 0x7ffffffffffff; c = (limb) (t[4] >> 51); r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff; r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff; r2 += c; } while (--count); output[0] = r0; output[1] = r1; output[2] = r2; output[3] = r3; output[4] = r4; } /* Load a little-endian 64-bit number */ static limb load_limb (const u8 *in) { return ((limb) in[0]) | (((limb) in[1]) << 8) | (((limb) in[2]) << 16) | (((limb) in[3]) << 24) | (((limb) in[4]) << 32) | (((limb) in[5]) << 40) | (((limb) in[6]) << 48) | (((limb) in[7]) << 56); } static void store_limb (u8 *out, limb in) { out[0] = in & 0xff; out[1] = (in >> 8) & 0xff; out[2] = (in >> 16) & 0xff; out[3] = (in >> 24) & 0xff; out[4] = (in >> 32) & 0xff; out[5] = (in >> 40) & 0xff; out[6] = (in >> 48) & 0xff; out[7] = (in >> 56) & 0xff; } /* Take a little-endian, 32-byte number and expand it into polynomial form */ static void fexpand (limb *output, const u8 *in) { output[0] = load_limb (in) & 0x7ffffffffffff; output[1] = (load_limb (in + 6) >> 3) & 0x7ffffffffffff; output[2] = (load_limb (in + 12) >> 6) & 0x7ffffffffffff; output[3] = (load_limb (in + 19) >> 1) & 0x7ffffffffffff; output[4] = (load_limb (in + 24) >> 12) & 0x7ffffffffffff; } /* Take a fully reduced polynomial form number and contract it into a * little-endian, 32-byte array */ static void fcontract (u8 *output, const felem input) { uint128_t t[5]; t[0] = input[0]; t[1] = input[1]; t[2] = input[2]; t[3] = input[3]; t[4] = input[4]; t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff; t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff; /* now t is between 0 and 2^255-1, properly carried. */ /* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */ t[0] += 19; t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff; /* now between 19 and 2^255-1 in both cases, and offset by 19. */ t[0] += 0x8000000000000 - 19; t[1] += 0x8000000000000 - 1; t[2] += 0x8000000000000 - 1; t[3] += 0x8000000000000 - 1; t[4] += 0x8000000000000 - 1; /* now between 2^255 and 2^256-20, and offset by 2^255. */ t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; t[4] &= 0x7ffffffffffff; store_limb (output, t[0] | (t[1] << 51)); store_limb (output + 8, (t[1] >> 13) | (t[2] << 38)); store_limb (output + 16, (t[2] >> 26) | (t[3] << 25)); store_limb (output + 24, (t[3] >> 39) | (t[4] << 12)); } /* Input: Q, Q', Q-Q' * Output: 2Q, Q+Q' * * x2 z3: long form * x3 z3: long form * x z: short form, destroyed * xprime zprime: short form, destroyed * qmqp: short form, preserved */ static void fmonty (limb *x2, limb *z2, /* output 2Q */ limb *x3, limb *z3, /* output Q + Q' */ limb *x, limb *z, /* input Q */ limb *xprime, limb *zprime, /* input Q' */ const limb *qmqp /* input Q - Q' */) { limb origx[5], origxprime[5], zzz[5], xx[5], zz[5], xxprime[5], zzprime[5], zzzprime[5]; memcpy (origx, x, 5 * sizeof(limb)); fsum (x, z); fdifference_backwards (z, origx); // does x - z memcpy (origxprime, xprime, sizeof(limb) * 5); fsum (xprime, zprime); fdifference_backwards (zprime, origxprime); fmul (xxprime, xprime, z); fmul (zzprime, x, zprime); memcpy (origxprime, xxprime, sizeof(limb) * 5); fsum (xxprime, zzprime); fdifference_backwards (zzprime, origxprime); fsquare_times (x3, xxprime, 1); fsquare_times (zzzprime, zzprime, 1); fmul (z3, zzzprime, qmqp); fsquare_times (xx, x, 1); fsquare_times (zz, z, 1); fmul (x2, xx, zz); fdifference_backwards (zz, xx); // does zz = xx - zz fscalar_product (zzz, zz, 121665); fsum (zzz, xx); fmul (z2, zz, zzz); } // ----------------------------------------------------------------------------- // Maybe swap the contents of two limb arrays (@a and @b), each @len elements // long. Perform the swap iff @swap is non-zero. // // This function performs the swap without leaking any side-channel // information. // ----------------------------------------------------------------------------- static void swap_conditional (limb a[5], limb b[5], limb iswap) { unsigned i; const limb swap = -iswap; for (i = 0; i < 5; ++i) { const limb x = swap & (a[i] ^ b[i]); a[i] ^= x; b[i] ^= x; } } /* Calculates nQ where Q is the x-coordinate of a point on the curve * * resultx/resultz: the x coordinate of the resulting curve point (short form) * n: a little endian, 32-byte number * q: a point of the curve (short form) */ static void cmult (limb *resultx, limb *resultz, const u8 *n, const limb *q) { limb a[5] = { 0 }, b[5] = { 1 }, c[5] = { 1 }, d[5] = { 0 }; limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t; limb e[5] = { 0 }, f[5] = { 1 }, g[5] = { 0 }, h[5] = { 1 }; limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h; unsigned i, j; memcpy (nqpqx, q, sizeof(limb) * 5); for (i = 0; i < 32; ++i) { u8 byte = n[31 - i]; for (j = 0; j < 8; ++j) { const limb bit = byte >> 7; swap_conditional (nqx, nqpqx, bit); swap_conditional (nqz, nqpqz, bit); fmonty (nqx2, nqz2, nqpqx2, nqpqz2, nqx, nqz, nqpqx, nqpqz, q); swap_conditional (nqx2, nqpqx2, bit); swap_conditional (nqz2, nqpqz2, bit); t = nqx; nqx = nqx2; nqx2 = t; t = nqz; nqz = nqz2; nqz2 = t; t = nqpqx; nqpqx = nqpqx2; nqpqx2 = t; t = nqpqz; nqpqz = nqpqz2; nqpqz2 = t; byte <<= 1; } } memcpy (resultx, nqx, sizeof(limb) * 5); memcpy (resultz, nqz, sizeof(limb) * 5); } // ----------------------------------------------------------------------------- // Shamelessly copied from djb's code, tightened a little // ----------------------------------------------------------------------------- static void crecip (felem out, const felem z) { felem a, t0, b, c; /* 2 */fsquare_times (a, z, 1); // a = 2 /* 8 */fsquare_times (t0, a, 2); /* 9 */fmul (b, t0, z); // b = 9 /* 11 */fmul (a, b, a); // a = 11 /* 22 */fsquare_times (t0, a, 1); /* 2^5 - 2^0 = 31 */fmul (b, t0, b); /* 2^10 - 2^5 */fsquare_times (t0, b, 5); /* 2^10 - 2^0 */fmul (b, t0, b); /* 2^20 - 2^10 */fsquare_times (t0, b, 10); /* 2^20 - 2^0 */fmul (c, t0, b); /* 2^40 - 2^20 */fsquare_times (t0, c, 20); /* 2^40 - 2^0 */fmul (t0, t0, c); /* 2^50 - 2^10 */fsquare_times (t0, t0, 10); /* 2^50 - 2^0 */fmul (b, t0, b); /* 2^100 - 2^50 */fsquare_times (t0, b, 50); /* 2^100 - 2^0 */fmul (c, t0, b); /* 2^200 - 2^100 */fsquare_times (t0, c, 100); /* 2^200 - 2^0 */fmul (t0, t0, c); /* 2^250 - 2^50 */fsquare_times (t0, t0, 50); /* 2^250 - 2^0 */fmul (t0, t0, b); /* 2^255 - 2^5 */fsquare_times (t0, t0, 5); /* 2^255 - 21 */fmul (out, t0, a); } int curve25519 (u8 *, const u8 *, const u8 *); int curve25519 (u8 *mypublic, const u8 *secret, const u8 *basepoint) { limb bp[5], x[5], z[5], zmone[5]; int i; fexpand (bp, basepoint); cmult (x, z, secret, bp); crecip (zmone, z); fmul (z, x, zmone); fcontract (mypublic, z); return 0; }