449 lines
13 KiB
C
449 lines
13 KiB
C
/* 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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#include <string.h>
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#include <stdint.h>
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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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#undef force_inline
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#define force_inline __attribute__((always_inline))
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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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/* 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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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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/* 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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a = ((uint128_t) in[0]) * scalar;
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output[0] = ((limb)a) & 0x7ffffffffffff;
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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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a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51));
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output[2] = ((limb)a) & 0x7ffffffffffff;
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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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a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51));
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output[4] = ((limb)a) & 0x7ffffffffffff;
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output[0] += (a >> 51) * 19;
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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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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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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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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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r4 *= 19;
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r1 *= 19;
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r2 *= 19;
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r3 *= 19;
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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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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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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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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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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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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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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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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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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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/* 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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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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/* 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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/* 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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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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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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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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/* 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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t[0] += 19;
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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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/* now between 19 and 2^255-1 in both cases, and offset by 19. */
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t[0] += 0x8000000000000 - 19;
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t[1] += 0x8000000000000 - 1;
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t[2] += 0x8000000000000 - 1;
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t[3] += 0x8000000000000 - 1;
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t[4] += 0x8000000000000 - 1;
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/* now between 2^255 and 2^256-20, and offset by 2^255. */
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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[4] &= 0x7ffffffffffff;
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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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/* 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' */
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const limb *qmqp /* input Q - Q' */) {
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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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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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memcpy(origxprime, xprime, sizeof(limb) * 5);
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fsum(xprime, zprime);
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fdifference_backwards(zprime, origxprime);
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fmul(xxprime, xprime, z);
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fmul(zzprime, x, zprime);
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memcpy(origxprime, xxprime, sizeof(limb) * 5);
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fsum(xxprime, zzprime);
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fdifference_backwards(zzprime, origxprime);
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fsquare_times(x3, xxprime, 1);
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fsquare_times(zzzprime, zzprime, 1);
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fmul(z3, zzzprime, qmqp);
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fsquare_times(xx, x, 1);
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fsquare_times(zz, z, 1);
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fmul(x2, xx, zz);
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fdifference_backwards(zz, xx); // does zz = xx - zz
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fscalar_product(zzz, zz, 121665);
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fsum(zzz, xx);
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fmul(z2, zz, zzz);
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}
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// -----------------------------------------------------------------------------
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// Maybe swap the contents of two limb arrays (@a and @b), each @len elements
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// long. Perform the swap iff @swap is non-zero.
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//
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// This function performs the swap without leaking any side-channel
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// information.
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// -----------------------------------------------------------------------------
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static void
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swap_conditional(limb a[5], limb b[5], limb iswap) {
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unsigned i;
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const limb swap = -iswap;
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for (i = 0; i < 5; ++i) {
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const limb x = swap & (a[i] ^ b[i]);
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a[i] ^= x;
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b[i] ^= x;
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}
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}
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/* Calculates nQ where Q is the x-coordinate of a point on the curve
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*
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* resultx/resultz: the x coordinate of the resulting curve point (short form)
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* n: a little endian, 32-byte number
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* q: a point of the curve (short form)
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*/
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static void
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cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
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limb a[5] = {0}, b[5] = {1}, c[5] = {1}, d[5] = {0};
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limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
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limb e[5] = {0}, f[5] = {1}, g[5] = {0}, h[5] = {1};
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limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
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unsigned i, j;
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memcpy(nqpqx, q, sizeof(limb) * 5);
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for (i = 0; i < 32; ++i) {
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u8 byte = n[31 - i];
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for (j = 0; j < 8; ++j) {
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const limb bit = byte >> 7;
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swap_conditional(nqx, nqpqx, bit);
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swap_conditional(nqz, nqpqz, bit);
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fmonty(nqx2, nqz2,
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nqpqx2, nqpqz2,
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nqx, nqz,
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nqpqx, nqpqz,
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q);
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swap_conditional(nqx2, nqpqx2, bit);
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swap_conditional(nqz2, nqpqz2, bit);
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t = nqx;
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nqx = nqx2;
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nqx2 = t;
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t = nqz;
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nqz = nqz2;
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nqz2 = t;
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t = nqpqx;
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nqpqx = nqpqx2;
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nqpqx2 = t;
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t = nqpqz;
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nqpqz = nqpqz2;
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nqpqz2 = t;
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byte <<= 1;
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}
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}
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memcpy(resultx, nqx, sizeof(limb) * 5);
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memcpy(resultz, nqz, sizeof(limb) * 5);
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}
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// -----------------------------------------------------------------------------
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// Shamelessly copied from djb's code, tightened a little
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// -----------------------------------------------------------------------------
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static void
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crecip(felem out, const felem z) {
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felem a,t0,b,c;
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/* 2 */ fsquare_times(a, z, 1); // a = 2
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/* 8 */ fsquare_times(t0, a, 2);
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/* 9 */ fmul(b, t0, z); // b = 9
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/* 11 */ fmul(a, b, a); // a = 11
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/* 22 */ fsquare_times(t0, a, 1);
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/* 2^5 - 2^0 = 31 */ fmul(b, t0, b);
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/* 2^10 - 2^5 */ fsquare_times(t0, b, 5);
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/* 2^10 - 2^0 */ fmul(b, t0, b);
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/* 2^20 - 2^10 */ fsquare_times(t0, b, 10);
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/* 2^20 - 2^0 */ fmul(c, t0, b);
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/* 2^40 - 2^20 */ fsquare_times(t0, c, 20);
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/* 2^40 - 2^0 */ fmul(t0, t0, c);
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/* 2^50 - 2^10 */ fsquare_times(t0, t0, 10);
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/* 2^50 - 2^0 */ fmul(b, t0, b);
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/* 2^100 - 2^50 */ fsquare_times(t0, b, 50);
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/* 2^100 - 2^0 */ fmul(c, t0, b);
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/* 2^200 - 2^100 */ fsquare_times(t0, c, 100);
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/* 2^200 - 2^0 */ fmul(t0, t0, c);
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/* 2^250 - 2^50 */ fsquare_times(t0, t0, 50);
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|
/* 2^250 - 2^0 */ fmul(t0, t0, b);
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/* 2^255 - 2^5 */ fsquare_times(t0, t0, 5);
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/* 2^255 - 21 */ fmul(out, t0, a);
|
|
}
|
|
|
|
int curve25519_donna(u8 *, const u8 *, const u8 *);
|
|
|
|
int
|
|
curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
|
|
limb bp[5], x[5], z[5], zmone[5];
|
|
uint8_t e[32];
|
|
int i;
|
|
|
|
for (i = 0;i < 32;++i) e[i] = secret[i];
|
|
e[0] &= 248;
|
|
e[31] &= 127;
|
|
e[31] |= 64;
|
|
|
|
fexpand(bp, basepoint);
|
|
cmult(x, z, e, bp);
|
|
crecip(zmone, z);
|
|
fmul(z, x, zmone);
|
|
fcontract(mypublic, z);
|
|
return 0;
|
|
}
|