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/* Copyright 2008, Google Inc. |
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* All rights reserved. |
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* |
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* Code released into the public domain. |
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* |
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* curve25519-donna: Curve25519 elliptic curve, public key function |
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* |
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* http://code.google.com/p/curve25519-donna/ |
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* |
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* Adam Langley <agl@imperialviolet.org> |
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* |
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* Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to> |
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* |
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* More information about curve25519 can be found here |
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* http://cr.yp.to/ecdh.html |
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* |
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* djb's sample implementation of curve25519 is written in a special assembly |
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* language called qhasm and uses the floating point registers. |
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* |
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* This is, almost, a clean room reimplementation from the curve25519 paper. It |
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* uses many of the tricks described therein. Only the crecip function is taken |
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* from the sample implementation. |
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*/ |
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|
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#include <string.h> |
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#include <stdint.h> |
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|
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typedef uint8_t u8; |
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typedef uint64_t limb; |
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typedef limb felem[5]; |
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// This is a special gcc mode for 128-bit integers. It's implemented on 64-bit |
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// platforms only as far as I know. |
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typedef unsigned uint128_t __attribute__((mode(TI))); |
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|
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#undef force_inline |
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#define force_inline __attribute__((always_inline)) |
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|
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/* Sum two numbers: output += in */ |
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static inline void force_inline |
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fsum(limb *output, const limb *in) { |
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output[0] += in[0]; |
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output[1] += in[1]; |
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output[2] += in[2]; |
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output[3] += in[3]; |
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output[4] += in[4]; |
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} |
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|
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/* Find the difference of two numbers: output = in - output |
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* (note the order of the arguments!) |
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* |
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* Assumes that out[i] < 2**52 |
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* On return, out[i] < 2**55 |
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*/ |
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static inline void force_inline |
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fdifference_backwards(felem out, const felem in) { |
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/* 152 is 19 << 3 */ |
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static const limb two54m152 = (((limb)1) << 54) - 152; |
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static const limb two54m8 = (((limb)1) << 54) - 8; |
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|
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out[0] = in[0] + two54m152 - out[0]; |
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out[1] = in[1] + two54m8 - out[1]; |
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out[2] = in[2] + two54m8 - out[2]; |
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out[3] = in[3] + two54m8 - out[3]; |
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out[4] = in[4] + two54m8 - out[4]; |
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} |
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|
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/* Multiply a number by a scalar: output = in * scalar */ |
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static inline void force_inline |
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fscalar_product(felem output, const felem in, const limb scalar) { |
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uint128_t a; |
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|
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a = ((uint128_t) in[0]) * scalar; |
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output[0] = ((limb)a) & 0x7ffffffffffff; |
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|
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a = ((uint128_t) in[1]) * scalar + ((limb) (a >> 51)); |
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output[1] = ((limb)a) & 0x7ffffffffffff; |
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|
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a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51)); |
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output[2] = ((limb)a) & 0x7ffffffffffff; |
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|
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a = ((uint128_t) in[3]) * scalar + ((limb) (a >> 51)); |
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output[3] = ((limb)a) & 0x7ffffffffffff; |
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|
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a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51)); |
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output[4] = ((limb)a) & 0x7ffffffffffff; |
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|
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output[0] += (a >> 51) * 19; |
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} |
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|
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/* Multiply two numbers: output = in2 * in |
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* |
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* output must be distinct to both inputs. The inputs are reduced coefficient |
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* form, the output is not. |
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* |
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* Assumes that in[i] < 2**55 and likewise for in2. |
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* On return, output[i] < 2**52 |
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*/ |
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static inline void force_inline |
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fmul(felem output, const felem in2, const felem in) { |
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uint128_t t[5]; |
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limb r0,r1,r2,r3,r4,s0,s1,s2,s3,s4,c; |
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|
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r0 = in[0]; |
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r1 = in[1]; |
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r2 = in[2]; |
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r3 = in[3]; |
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r4 = in[4]; |
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|
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s0 = in2[0]; |
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s1 = in2[1]; |
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s2 = in2[2]; |
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s3 = in2[3]; |
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s4 = in2[4]; |
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|
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t[0] = ((uint128_t) r0) * s0; |
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t[1] = ((uint128_t) r0) * s1 + ((uint128_t) r1) * s0; |
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t[2] = ((uint128_t) r0) * s2 + ((uint128_t) r2) * s0 + ((uint128_t) r1) * s1; |
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t[3] = ((uint128_t) r0) * s3 + ((uint128_t) r3) * s0 + ((uint128_t) r1) * s2 + ((uint128_t) r2) * s1; |
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t[4] = ((uint128_t) r0) * s4 + ((uint128_t) r4) * s0 + ((uint128_t) r3) * s1 + ((uint128_t) r1) * s3 + ((uint128_t) r2) * s2; |
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|
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r4 *= 19; |
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r1 *= 19; |
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r2 *= 19; |
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r3 *= 19; |
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|
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t[0] += ((uint128_t) r4) * s1 + ((uint128_t) r1) * s4 + ((uint128_t) r2) * s3 + ((uint128_t) r3) * s2; |
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t[1] += ((uint128_t) r4) * s2 + ((uint128_t) r2) * s4 + ((uint128_t) r3) * s3; |
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t[2] += ((uint128_t) r4) * s3 + ((uint128_t) r3) * s4; |
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t[3] += ((uint128_t) r4) * s4; |
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|
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r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51); |
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t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51); |
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t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51); |
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t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51); |
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t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51); |
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r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff; |
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r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff; |
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r2 += c; |
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|
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output[0] = r0; |
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output[1] = r1; |
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output[2] = r2; |
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output[3] = r3; |
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output[4] = r4; |
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} |
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|
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static inline void force_inline |
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fsquare_times(felem output, const felem in, limb count) { |
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uint128_t t[5]; |
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limb r0,r1,r2,r3,r4,c; |
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limb d0,d1,d2,d4,d419; |
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|
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r0 = in[0]; |
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r1 = in[1]; |
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r2 = in[2]; |
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r3 = in[3]; |
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r4 = in[4]; |
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|
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do { |
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d0 = r0 * 2; |
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d1 = r1 * 2; |
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d2 = r2 * 2 * 19; |
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d419 = r4 * 19; |
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d4 = d419 * 2; |
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|
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t[0] = ((uint128_t) r0) * r0 + ((uint128_t) d4) * r1 + (((uint128_t) d2) * (r3 )); |
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t[1] = ((uint128_t) d0) * r1 + ((uint128_t) d4) * r2 + (((uint128_t) r3) * (r3 * 19)); |
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t[2] = ((uint128_t) d0) * r2 + ((uint128_t) r1) * r1 + (((uint128_t) d4) * (r3 )); |
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t[3] = ((uint128_t) d0) * r3 + ((uint128_t) d1) * r2 + (((uint128_t) r4) * (d419 )); |
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t[4] = ((uint128_t) d0) * r4 + ((uint128_t) d1) * r3 + (((uint128_t) r2) * (r2 )); |
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|
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r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51); |
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t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51); |
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t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51); |
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t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51); |
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t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51); |
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r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff; |
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r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff; |
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r2 += c; |
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} while(--count); |
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|
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output[0] = r0; |
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output[1] = r1; |
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output[2] = r2; |
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output[3] = r3; |
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output[4] = r4; |
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} |
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|
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/* Load a little-endian 64-bit number */ |
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static limb |
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load_limb(const u8 *in) { |
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return |
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((limb)in[0]) | |
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(((limb)in[1]) << 8) | |
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(((limb)in[2]) << 16) | |
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(((limb)in[3]) << 24) | |
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(((limb)in[4]) << 32) | |
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(((limb)in[5]) << 40) | |
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(((limb)in[6]) << 48) | |
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(((limb)in[7]) << 56); |
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} |
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|
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static void |
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store_limb(u8 *out, limb in) { |
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out[0] = in & 0xff; |
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out[1] = (in >> 8) & 0xff; |
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out[2] = (in >> 16) & 0xff; |
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out[3] = (in >> 24) & 0xff; |
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out[4] = (in >> 32) & 0xff; |
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out[5] = (in >> 40) & 0xff; |
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out[6] = (in >> 48) & 0xff; |
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out[7] = (in >> 56) & 0xff; |
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} |
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|
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/* Take a little-endian, 32-byte number and expand it into polynomial form */ |
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static void |
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fexpand(limb *output, const u8 *in) { |
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output[0] = load_limb(in) & 0x7ffffffffffff; |
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output[1] = (load_limb(in+6) >> 3) & 0x7ffffffffffff; |
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output[2] = (load_limb(in+12) >> 6) & 0x7ffffffffffff; |
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output[3] = (load_limb(in+19) >> 1) & 0x7ffffffffffff; |
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output[4] = (load_limb(in+24) >> 12) & 0x7ffffffffffff; |
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} |
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|
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/* Take a fully reduced polynomial form number and contract it into a |
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* little-endian, 32-byte array |
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*/ |
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static void |
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fcontract(u8 *output, const felem input) { |
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uint128_t t[5]; |
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|
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t[0] = input[0]; |
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t[1] = input[1]; |
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t[2] = input[2]; |
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t[3] = input[3]; |
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t[4] = input[4]; |
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|
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t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; |
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t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; |
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t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; |
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t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; |
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t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff; |
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|
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t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; |
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t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; |
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t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; |
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t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; |
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t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff; |
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|
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/* now t is between 0 and 2^255-1, properly carried. */ |
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/* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */ |
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|
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t[0] += 19; |
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|
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t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; |
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t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; |
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t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; |
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t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; |
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t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff; |
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|
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/* now between 19 and 2^255-1 in both cases, and offset by 19. */ |
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|
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t[0] += 0x8000000000000 - 19; |
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t[1] += 0x8000000000000 - 1; |
| 265 |
t[2] += 0x8000000000000 - 1; |
| 266 |
t[3] += 0x8000000000000 - 1; |
| 267 |
t[4] += 0x8000000000000 - 1; |
| 268 |
|
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/* now between 2^255 and 2^256-20, and offset by 2^255. */ |
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|
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t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff; |
| 272 |
t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff; |
| 273 |
t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff; |
| 274 |
t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff; |
| 275 |
t[4] &= 0x7ffffffffffff; |
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|
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store_limb(output, t[0] | (t[1] << 51)); |
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store_limb(output+8, (t[1] >> 13) | (t[2] << 38)); |
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store_limb(output+16, (t[2] >> 26) | (t[3] << 25)); |
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store_limb(output+24, (t[3] >> 39) | (t[4] << 12)); |
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} |
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|
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/* Input: Q, Q', Q-Q' |
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* Output: 2Q, Q+Q' |
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* |
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* x2 z3: long form |
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* x3 z3: long form |
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* x z: short form, destroyed |
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* xprime zprime: short form, destroyed |
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* qmqp: short form, preserved |
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*/ |
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static void |
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fmonty(limb *x2, limb *z2, /* output 2Q */ |
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limb *x3, limb *z3, /* output Q + Q' */ |
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limb *x, limb *z, /* input Q */ |
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limb *xprime, limb *zprime, /* input Q' */ |
| 297 |
const limb *qmqp /* input Q - Q' */) { |
| 298 |
limb origx[5], origxprime[5], zzz[5], xx[5], zz[5], xxprime[5], |
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zzprime[5], zzzprime[5]; |
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|
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memcpy(origx, x, 5 * sizeof(limb)); |
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fsum(x, z); |
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fdifference_backwards(z, origx); // does x - z |
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|
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memcpy(origxprime, xprime, sizeof(limb) * 5); |
| 306 |
fsum(xprime, zprime); |
| 307 |
fdifference_backwards(zprime, origxprime); |
| 308 |
fmul(xxprime, xprime, z); |
| 309 |
fmul(zzprime, x, zprime); |
| 310 |
memcpy(origxprime, xxprime, sizeof(limb) * 5); |
| 311 |
fsum(xxprime, zzprime); |
| 312 |
fdifference_backwards(zzprime, origxprime); |
| 313 |
fsquare_times(x3, xxprime, 1); |
| 314 |
fsquare_times(zzzprime, zzprime, 1); |
| 315 |
fmul(z3, zzzprime, qmqp); |
| 316 |
|
| 317 |
fsquare_times(xx, x, 1); |
| 318 |
fsquare_times(zz, z, 1); |
| 319 |
fmul(x2, xx, zz); |
| 320 |
fdifference_backwards(zz, xx); // does zz = xx - zz |
| 321 |
fscalar_product(zzz, zz, 121665); |
| 322 |
fsum(zzz, xx); |
| 323 |
fmul(z2, zz, zzz); |
| 324 |
} |
| 325 |
|
| 326 |
// ----------------------------------------------------------------------------- |
| 327 |
// Maybe swap the contents of two limb arrays (@a and @b), each @len elements |
| 328 |
// long. Perform the swap iff @swap is non-zero. |
| 329 |
// |
| 330 |
// This function performs the swap without leaking any side-channel |
| 331 |
// information. |
| 332 |
// ----------------------------------------------------------------------------- |
| 333 |
static void |
| 334 |
swap_conditional(limb a[5], limb b[5], limb iswap) { |
| 335 |
unsigned i; |
| 336 |
const limb swap = -iswap; |
| 337 |
|
| 338 |
for (i = 0; i < 5; ++i) { |
| 339 |
const limb x = swap & (a[i] ^ b[i]); |
| 340 |
a[i] ^= x; |
| 341 |
b[i] ^= x; |
| 342 |
} |
| 343 |
} |
| 344 |
|
| 345 |
/* Calculates nQ where Q is the x-coordinate of a point on the curve |
| 346 |
* |
| 347 |
* resultx/resultz: the x coordinate of the resulting curve point (short form) |
| 348 |
* n: a little endian, 32-byte number |
| 349 |
* q: a point of the curve (short form) |
| 350 |
*/ |
| 351 |
static void |
| 352 |
cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) { |
| 353 |
limb a[5] = {0}, b[5] = {1}, c[5] = {1}, d[5] = {0}; |
| 354 |
limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t; |
| 355 |
limb e[5] = {0}, f[5] = {1}, g[5] = {0}, h[5] = {1}; |
| 356 |
limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h; |
| 357 |
|
| 358 |
unsigned i, j; |
| 359 |
|
| 360 |
memcpy(nqpqx, q, sizeof(limb) * 5); |
| 361 |
|
| 362 |
for (i = 0; i < 32; ++i) { |
| 363 |
u8 byte = n[31 - i]; |
| 364 |
for (j = 0; j < 8; ++j) { |
| 365 |
const limb bit = byte >> 7; |
| 366 |
|
| 367 |
swap_conditional(nqx, nqpqx, bit); |
| 368 |
swap_conditional(nqz, nqpqz, bit); |
| 369 |
fmonty(nqx2, nqz2, |
| 370 |
nqpqx2, nqpqz2, |
| 371 |
nqx, nqz, |
| 372 |
nqpqx, nqpqz, |
| 373 |
q); |
| 374 |
swap_conditional(nqx2, nqpqx2, bit); |
| 375 |
swap_conditional(nqz2, nqpqz2, bit); |
| 376 |
|
| 377 |
t = nqx; |
| 378 |
nqx = nqx2; |
| 379 |
nqx2 = t; |
| 380 |
t = nqz; |
| 381 |
nqz = nqz2; |
| 382 |
nqz2 = t; |
| 383 |
t = nqpqx; |
| 384 |
nqpqx = nqpqx2; |
| 385 |
nqpqx2 = t; |
| 386 |
t = nqpqz; |
| 387 |
nqpqz = nqpqz2; |
| 388 |
nqpqz2 = t; |
| 389 |
|
| 390 |
byte <<= 1; |
| 391 |
} |
| 392 |
} |
| 393 |
|
| 394 |
memcpy(resultx, nqx, sizeof(limb) * 5); |
| 395 |
memcpy(resultz, nqz, sizeof(limb) * 5); |
| 396 |
} |
| 397 |
|
| 398 |
|
| 399 |
// ----------------------------------------------------------------------------- |
| 400 |
// Shamelessly copied from djb's code, tightened a little |
| 401 |
// ----------------------------------------------------------------------------- |
| 402 |
static void |
| 403 |
crecip(felem out, const felem z) { |
| 404 |
felem a,t0,b,c; |
| 405 |
|
| 406 |
/* 2 */ fsquare_times(a, z, 1); // a = 2 |
| 407 |
/* 8 */ fsquare_times(t0, a, 2); |
| 408 |
/* 9 */ fmul(b, t0, z); // b = 9 |
| 409 |
/* 11 */ fmul(a, b, a); // a = 11 |
| 410 |
/* 22 */ fsquare_times(t0, a, 1); |
| 411 |
/* 2^5 - 2^0 = 31 */ fmul(b, t0, b); |
| 412 |
/* 2^10 - 2^5 */ fsquare_times(t0, b, 5); |
| 413 |
/* 2^10 - 2^0 */ fmul(b, t0, b); |
| 414 |
/* 2^20 - 2^10 */ fsquare_times(t0, b, 10); |
| 415 |
/* 2^20 - 2^0 */ fmul(c, t0, b); |
| 416 |
/* 2^40 - 2^20 */ fsquare_times(t0, c, 20); |
| 417 |
/* 2^40 - 2^0 */ fmul(t0, t0, c); |
| 418 |
/* 2^50 - 2^10 */ fsquare_times(t0, t0, 10); |
| 419 |
/* 2^50 - 2^0 */ fmul(b, t0, b); |
| 420 |
/* 2^100 - 2^50 */ fsquare_times(t0, b, 50); |
| 421 |
/* 2^100 - 2^0 */ fmul(c, t0, b); |
| 422 |
/* 2^200 - 2^100 */ fsquare_times(t0, c, 100); |
| 423 |
/* 2^200 - 2^0 */ fmul(t0, t0, c); |
| 424 |
/* 2^250 - 2^50 */ fsquare_times(t0, t0, 50); |
| 425 |
/* 2^250 - 2^0 */ fmul(t0, t0, b); |
| 426 |
/* 2^255 - 2^5 */ fsquare_times(t0, t0, 5); |
| 427 |
/* 2^255 - 21 */ fmul(out, t0, a); |
| 428 |
} |
| 429 |
|
| 430 |
int curve25519_donna(u8 *, const u8 *, const u8 *); |
| 431 |
|
| 432 |
int |
| 433 |
curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) { |
| 434 |
limb bp[5], x[5], z[5], zmone[5]; |
| 435 |
uint8_t e[32]; |
| 436 |
int i; |
| 437 |
|
| 438 |
for (i = 0;i < 32;++i) e[i] = secret[i]; |
| 439 |
e[0] &= 248; |
| 440 |
e[31] &= 127; |
| 441 |
e[31] |= 64; |
| 442 |
|
| 443 |
fexpand(bp, basepoint); |
| 444 |
cmult(x, z, e, bp); |
| 445 |
crecip(zmone, z); |
| 446 |
fmul(z, x, zmone); |
| 447 |
fcontract(mypublic, z); |
| 448 |
return 0; |
| 449 |
} |
