ref: fa2eda237bec91d222c22459601bfc995cc2d37f
dir: /libsec/curve25519.c/
/* Copyright 2008, Google Inc. * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions are * met: * * * Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * * Redistributions in binary form must reproduce the above * copyright notice, this list of conditions and the following disclaimer * in the documentation and/or other materials provided with the * distribution. * * Neither the name of Google Inc. nor the names of its * contributors may be used to endorse or promote products derived from * this software without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. * * curve25519: Curve25519 elliptic curve, public key function * * http://code.google.com/p/curve25519-donna/ * * Adam Langley <agl@imperialviolet.org> * * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to> * * 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 "os.h" #include <libsec.h> typedef vlong felem; /* Sum two numbers: output += in */ static void fsum(felem *output, felem *in) { unsigned i; for (i = 0; i < 10; i += 2) { output[0+i] = (output[0+i] + in[0+i]); output[1+i] = (output[1+i] + in[1+i]); } } /* Find the difference of two numbers: output = in - output * (note the order of the arguments!) */ static void fdifference(felem *output, felem *in) { unsigned i; for (i = 0; i < 10; ++i) { output[i] = (in[i] - output[i]); } } /* Multiply a number my a scalar: output = in * scalar */ static void fscalar_product(felem *output, felem *in, felem scalar) { unsigned i; for (i = 0; i < 10; ++i) { output[i] = in[i] * scalar; } } /* Multiply two numbers: output = in2 * in * * output must be distinct to both inputs. The inputs are reduced coefficient * form, the output is not. */ static void fproduct(felem *output, felem *in2, felem *in) { output[0] = in2[0] * in[0]; output[1] = in2[0] * in[1] + in2[1] * in[0]; output[2] = 2 * in2[1] * in[1] + in2[0] * in[2] + in2[2] * in[0]; output[3] = in2[1] * in[2] + in2[2] * in[1] + in2[0] * in[3] + in2[3] * in[0]; output[4] = in2[2] * in[2] + 2 * (in2[1] * in[3] + in2[3] * in[1]) + in2[0] * in[4] + in2[4] * in[0]; output[5] = in2[2] * in[3] + in2[3] * in[2] + in2[1] * in[4] + in2[4] * in[1] + in2[0] * in[5] + in2[5] * in[0]; output[6] = 2 * (in2[3] * in[3] + in2[1] * in[5] + in2[5] * in[1]) + in2[2] * in[4] + in2[4] * in[2] + in2[0] * in[6] + in2[6] * in[0]; output[7] = in2[3] * in[4] + in2[4] * in[3] + in2[2] * in[5] + in2[5] * in[2] + in2[1] * in[6] + in2[6] * in[1] + in2[0] * in[7] + in2[7] * in[0]; output[8] = in2[4] * in[4] + 2 * (in2[3] * in[5] + in2[5] * in[3] + in2[1] * in[7] + in2[7] * in[1]) + in2[2] * in[6] + in2[6] * in[2] + in2[0] * in[8] + in2[8] * in[0]; output[9] = in2[4] * in[5] + in2[5] * in[4] + in2[3] * in[6] + in2[6] * in[3] + in2[2] * in[7] + in2[7] * in[2] + in2[1] * in[8] + in2[8] * in[1] + in2[0] * in[9] + in2[9] * in[0]; output[10] = 2 * (in2[5] * in[5] + in2[3] * in[7] + in2[7] * in[3] + in2[1] * in[9] + in2[9] * in[1]) + in2[4] * in[6] + in2[6] * in[4] + in2[2] * in[8] + in2[8] * in[2]; output[11] = in2[5] * in[6] + in2[6] * in[5] + in2[4] * in[7] + in2[7] * in[4] + in2[3] * in[8] + in2[8] * in[3] + in2[2] * in[9] + in2[9] * in[2]; output[12] = in2[6] * in[6] + 2 * (in2[5] * in[7] + in2[7] * in[5] + in2[3] * in[9] + in2[9] * in[3]) + in2[4] * in[8] + in2[8] * in[4]; output[13] = in2[6] * in[7] + in2[7] * in[6] + in2[5] * in[8] + in2[8] * in[5] + in2[4] * in[9] + in2[9] * in[4]; output[14] = 2 * (in2[7] * in[7] + in2[5] * in[9] + in2[9] * in[5]) + in2[6] * in[8] + in2[8] * in[6]; output[15] = in2[7] * in[8] + in2[8] * in[7] + in2[6] * in[9] + in2[9] * in[6]; output[16] = in2[8] * in[8] + 2 * (in2[7] * in[9] + in2[9] * in[7]); output[17] = in2[8] * in[9] + in2[9] * in[8]; output[18] = 2 * in2[9] * in[9]; } /* Reduce a long form to a short form by taking the input mod 2^255 - 19. */ static void freduce_degree(felem *output) { output[8] += 19 * output[18]; output[7] += 19 * output[17]; output[6] += 19 * output[16]; output[5] += 19 * output[15]; output[4] += 19 * output[14]; output[3] += 19 * output[13]; output[2] += 19 * output[12]; output[1] += 19 * output[11]; output[0] += 19 * output[10]; } /* Reduce all coefficients of the short form input to be -2**25 <= x <= 2**25 */ static void freduce_coefficients(felem *output) { unsigned i; do { output[10] = 0; for (i = 0; i < 10; i += 2) { felem over = output[i] / 0x2000000l; felem over2 = (over + ((over >> 63) * 2) + 1) / 2; output[i+1] += over2; output[i] -= over2 * 0x4000000l; over = output[i+1] / 0x2000000; output[i+2] += over; output[i+1] -= over * 0x2000000; } output[0] += 19 * output[10]; } while (output[10]); } /* A helpful wrapper around fproduct: output = in * in2. * * output must be distinct to both inputs. The output is reduced degree and * reduced coefficient. */ static void fmul(felem *output, felem *in, felem *in2) { felem t[19]; fproduct(t, in, in2); freduce_degree(t); freduce_coefficients(t); memcpy(output, t, sizeof(felem) * 10); } static void fsquare_inner(felem *output, felem *in) { felem tmp; output[0] = in[0] * in[0]; output[1] = 2 * in[0] * in[1]; output[2] = 2 * (in[1] * in[1] + in[0] * in[2]); output[3] = 2 * (in[1] * in[2] + in[0] * in[3]); output[4] = in[2] * in[2] + 4 * in[1] * in[3] + 2 * in[0] * in[4]; output[5] = 2 * (in[2] * in[3] + in[1] * in[4] + in[0] * in[5]); output[6] = 2 * (in[3] * in[3] + in[2] * in[4] + in[0] * in[6] + 2 * in[1] * in[5]); output[7] = 2 * (in[3] * in[4] + in[2] * in[5] + in[1] * in[6] + in[0] * in[7]); tmp = in[1] * in[7] + in[3] * in[5]; output[8] = in[4] * in[4] + 2 * (in[2] * in[6] + in[0] * in[8] + 2 * tmp); output[9] = 2 * (in[4] * in[5] + in[3] * in[6] + in[2] * in[7] + in[1] * in[8] + in[0] * in[9]); tmp = in[3] * in[7] + in[1] * in[9]; output[10] = 2 * (in[5] * in[5] + in[4] * in[6] + in[2] * in[8] + 2 * tmp); output[11] = 2 * (in[5] * in[6] + in[4] * in[7] + in[3] * in[8] + in[2] * in[9]); output[12] = in[6] * in[6] + 2 * (in[4] * in[8] + 2 * (in[5] * in[7] + in[3] * in[9])); output[13] = 2 * (in[6] * in[7] + in[5] * in[8] + in[4] * in[9]); output[14] = 2 * (in[7] * in[7] + in[6] * in[8] + 2 * in[5] * in[9]); output[15] = 2 * (in[7] * in[8] + in[6] * in[9]); output[16] = in[8] * in[8] + 4 * in[7] * in[9]; output[17] = 2 * in[8] * in[9]; output[18] = 2 * in[9] * in[9]; } static void fsquare(felem *output, felem *in) { felem t[19]; fsquare_inner(t, in); freduce_degree(t); freduce_coefficients(t); memcpy(output, t, sizeof(felem) * 10); } /* Take a little-endian, 32-byte number and expand it into polynomial form */ static void fexpand(felem *output, uchar *input) { #define F(n,start,shift,mask) \ output[n] = ((((felem) input[start + 0]) | \ ((felem) input[start + 1]) << 8 | \ ((felem) input[start + 2]) << 16 | \ ((felem) input[start + 3]) << 24) >> shift) & mask; F(0, 0, 0, 0x3ffffff); F(1, 3, 2, 0x1ffffff); F(2, 6, 3, 0x3ffffff); F(3, 9, 5, 0x1ffffff); F(4, 12, 6, 0x3ffffff); F(5, 16, 0, 0x1ffffff); F(6, 19, 1, 0x3ffffff); F(7, 22, 3, 0x1ffffff); F(8, 25, 4, 0x3ffffff); F(9, 28, 6, 0x1ffffff); #undef F } /* Take a fully reduced polynomial form number and contract it into a * little-endian, 32-byte array */ static void fcontract(uchar *output, felem *input) { int i; do { for (i = 0; i < 9; ++i) { if ((i & 1) == 1) { while (input[i] < 0) { input[i] += 0x2000000; input[i + 1]--; } } else { while (input[i] < 0) { input[i] += 0x4000000; input[i + 1]--; } } } while (input[9] < 0) { input[9] += 0x2000000; input[0] -= 19; } } while (input[0] < 0); input[1] <<= 2; input[2] <<= 3; input[3] <<= 5; input[4] <<= 6; input[6] <<= 1; input[7] <<= 3; input[8] <<= 4; input[9] <<= 6; #define F(i, s) \ output[s+0] |= input[i] & 0xff; \ output[s+1] = (input[i] >> 8) & 0xff; \ output[s+2] = (input[i] >> 16) & 0xff; \ output[s+3] = (input[i] >> 24) & 0xff; output[0] = 0; output[16] = 0; F(0,0); F(1,3); F(2,6); F(3,9); F(4,12); F(5,16); F(6,19); F(7,22); F(8,25); F(9,28); #undef F } /* 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(felem *x2, felem *z2, /* output 2Q */ felem *x3, felem *z3, /* output Q + Q' */ felem *x, felem *z, /* input Q */ felem *xprime, felem *zprime, /* input Q' */ felem *qmqp /* input Q - Q' */) { felem origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19], zzprime[19], zzzprime[19], xxxprime[19]; memcpy(origx, x, 10 * sizeof(felem)); fsum(x, z); fdifference(z, origx); // does x - z memcpy(origxprime, xprime, sizeof(felem) * 10); fsum(xprime, zprime); fdifference(zprime, origxprime); fproduct(xxprime, xprime, z); fproduct(zzprime, x, zprime); freduce_degree(xxprime); freduce_coefficients(xxprime); freduce_degree(zzprime); freduce_coefficients(zzprime); memcpy(origxprime, xxprime, sizeof(felem) * 10); fsum(xxprime, zzprime); fdifference(zzprime, origxprime); fsquare(xxxprime, xxprime); fsquare(zzzprime, zzprime); fproduct(zzprime, zzzprime, qmqp); freduce_degree(zzprime); freduce_coefficients(zzprime); memcpy(x3, xxxprime, sizeof(felem) * 10); memcpy(z3, zzprime, sizeof(felem) * 10); fsquare(xx, x); fsquare(zz, z); fproduct(x2, xx, zz); freduce_degree(x2); freduce_coefficients(x2); fdifference(zz, xx); // does zz = xx - zz memset(zzz + 10, 0, sizeof(felem) * 9); fscalar_product(zzz, zz, 121665); freduce_degree(zzz); freduce_coefficients(zzz); fsum(zzz, xx); fproduct(z2, zz, zzz); freduce_degree(z2); freduce_coefficients(z2); } /* 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(felem *resultx, felem *resultz, uchar *n, felem *q) { felem a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0}; felem *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t; felem e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1}; felem *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h; unsigned i, j; memcpy(nqpqx, q, sizeof(felem) * 10); for (i = 0; i < 32; ++i) { uchar byte = n[31 - i]; for (j = 0; j < 8; ++j) { if (byte & 0x80) { fmonty(nqpqx2, nqpqz2, nqx2, nqz2, nqpqx, nqpqz, nqx, nqz, q); } else { fmonty(nqx2, nqz2, nqpqx2, nqpqz2, nqx, nqz, nqpqx, nqpqz, q); } 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(felem) * 10); memcpy(resultz, nqz, sizeof(felem) * 10); } // ----------------------------------------------------------------------------- // Shamelessly copied from djb's code // ----------------------------------------------------------------------------- static void crecip(felem *out, felem *z) { felem z2[10]; felem z9[10]; felem z11[10]; felem z2_5_0[10]; felem z2_10_0[10]; felem z2_20_0[10]; felem z2_50_0[10]; felem z2_100_0[10]; felem t0[10]; felem t1[10]; int i; /* 2 */ fsquare(z2,z); /* 4 */ fsquare(t1,z2); /* 8 */ fsquare(t0,t1); /* 9 */ fmul(z9,t0,z); /* 11 */ fmul(z11,z9,z2); /* 22 */ fsquare(t0,z11); /* 2^5 - 2^0 = 31 */ fmul(z2_5_0,t0,z9); /* 2^6 - 2^1 */ fsquare(t0,z2_5_0); /* 2^7 - 2^2 */ fsquare(t1,t0); /* 2^8 - 2^3 */ fsquare(t0,t1); /* 2^9 - 2^4 */ fsquare(t1,t0); /* 2^10 - 2^5 */ fsquare(t0,t1); /* 2^10 - 2^0 */ fmul(z2_10_0,t0,z2_5_0); /* 2^11 - 2^1 */ fsquare(t0,z2_10_0); /* 2^12 - 2^2 */ fsquare(t1,t0); /* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } /* 2^20 - 2^0 */ fmul(z2_20_0,t1,z2_10_0); /* 2^21 - 2^1 */ fsquare(t0,z2_20_0); /* 2^22 - 2^2 */ fsquare(t1,t0); /* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } /* 2^40 - 2^0 */ fmul(t0,t1,z2_20_0); /* 2^41 - 2^1 */ fsquare(t1,t0); /* 2^42 - 2^2 */ fsquare(t0,t1); /* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t1,t0); fsquare(t0,t1); } /* 2^50 - 2^0 */ fmul(z2_50_0,t0,z2_10_0); /* 2^51 - 2^1 */ fsquare(t0,z2_50_0); /* 2^52 - 2^2 */ fsquare(t1,t0); /* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } /* 2^100 - 2^0 */ fmul(z2_100_0,t1,z2_50_0); /* 2^101 - 2^1 */ fsquare(t1,z2_100_0); /* 2^102 - 2^2 */ fsquare(t0,t1); /* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fsquare(t1,t0); fsquare(t0,t1); } /* 2^200 - 2^0 */ fmul(t1,t0,z2_100_0); /* 2^201 - 2^1 */ fsquare(t0,t1); /* 2^202 - 2^2 */ fsquare(t1,t0); /* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } /* 2^250 - 2^0 */ fmul(t0,t1,z2_50_0); /* 2^251 - 2^1 */ fsquare(t1,t0); /* 2^252 - 2^2 */ fsquare(t0,t1); /* 2^253 - 2^3 */ fsquare(t1,t0); /* 2^254 - 2^4 */ fsquare(t0,t1); /* 2^255 - 2^5 */ fsquare(t1,t0); /* 2^255 - 21 */ fmul(out,t1,z11); } void curve25519(uchar mypublic[32], uchar secret[32], uchar basepoint[32]) { felem bp[10], x[10], z[10], zmone[10]; fexpand(bp, basepoint); cmult(x, z, secret, bp); crecip(zmone, z); fmul(z, x, zmone); fcontract(mypublic, z); }