1 /* Copyright 2008, Google Inc.
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30 * curve25519: Curve25519 elliptic curve, public key function
32 * http://code.google.com/p/curve25519-donna/
34 * Adam Langley <agl@imperialviolet.org>
36 * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>
38 * More information about curve25519 can be found here
39 * http://cr.yp.to/ecdh.html
41 * djb's sample implementation of curve25519 is written in a special assembly
42 * language called qhasm and uses the floating point registers.
44 * This is, almost, a clean room reimplementation from the curve25519 paper. It
45 * uses many of the tricks described therein. Only the crecip function is taken
46 * from the sample implementation.
54 /* Sum two numbers: output += in */
55 static void fsum(felem *output, felem *in) {
57 for (i = 0; i < 10; i += 2) {
58 output[0+i] = (output[0+i] + in[0+i]);
59 output[1+i] = (output[1+i] + in[1+i]);
63 /* Find the difference of two numbers: output = in - output
64 * (note the order of the arguments!)
66 static void fdifference(felem *output, felem *in) {
68 for (i = 0; i < 10; ++i) {
69 output[i] = (in[i] - output[i]);
73 /* Multiply a number my a scalar: output = in * scalar */
74 static void fscalar_product(felem *output, felem *in, felem scalar) {
76 for (i = 0; i < 10; ++i) {
77 output[i] = in[i] * scalar;
81 /* Multiply two numbers: output = in2 * in
83 * output must be distinct to both inputs. The inputs are reduced coefficient
84 * form, the output is not.
86 static void fproduct(felem *output, felem *in2, felem *in) {
87 output[0] = in2[0] * in[0];
88 output[1] = in2[0] * in[1] +
90 output[2] = 2 * in2[1] * in[1] +
93 output[3] = in2[1] * in[2] +
97 output[4] = in2[2] * in[2] +
102 output[5] = in2[2] * in[3] +
108 output[6] = 2 * (in2[3] * in[3] +
115 output[7] = in2[3] * in[4] +
123 output[8] = in2[4] * in[4] +
124 2 * (in2[3] * in[5] +
132 output[9] = in2[4] * in[5] +
142 output[10] = 2 * (in2[5] * in[5] +
151 output[11] = in2[5] * in[6] +
159 output[12] = in2[6] * in[6] +
160 2 * (in2[5] * in[7] +
166 output[13] = in2[6] * in[7] +
172 output[14] = 2 * (in2[7] * in[7] +
177 output[15] = in2[7] * in[8] +
181 output[16] = in2[8] * in[8] +
182 2 * (in2[7] * in[9] +
184 output[17] = in2[8] * in[9] +
186 output[18] = 2 * in2[9] * in[9];
189 /* Reduce a long form to a short form by taking the input mod 2^255 - 19. */
190 static void freduce_degree(felem *output) {
191 output[8] += 19 * output[18];
192 output[7] += 19 * output[17];
193 output[6] += 19 * output[16];
194 output[5] += 19 * output[15];
195 output[4] += 19 * output[14];
196 output[3] += 19 * output[13];
197 output[2] += 19 * output[12];
198 output[1] += 19 * output[11];
199 output[0] += 19 * output[10];
202 /* Reduce all coefficients of the short form input to be -2**25 <= x <= 2**25
204 static void freduce_coefficients(felem *output) {
209 for (i = 0; i < 10; i += 2) {
210 felem over = output[i] / 0x2000000l;
211 felem over2 = (over + ((over >> 63) * 2) + 1) / 2;
212 output[i+1] += over2;
213 output[i] -= over2 * 0x4000000l;
215 over = output[i+1] / 0x2000000;
217 output[i+1] -= over * 0x2000000;
219 output[0] += 19 * output[10];
220 } while (output[10]);
223 /* A helpful wrapper around fproduct: output = in * in2.
225 * output must be distinct to both inputs. The output is reduced degree and
226 * reduced coefficient.
229 fmul(felem *output, felem *in, felem *in2) {
231 fproduct(t, in, in2);
233 freduce_coefficients(t);
234 memcpy(output, t, sizeof(felem) * 10);
237 static void fsquare_inner(felem *output, felem *in) {
239 output[0] = in[0] * in[0];
240 output[1] = 2 * in[0] * in[1];
241 output[2] = 2 * (in[1] * in[1] +
243 output[3] = 2 * (in[1] * in[2] +
245 output[4] = in[2] * in[2] +
248 output[5] = 2 * (in[2] * in[3] +
251 output[6] = 2 * (in[3] * in[3] +
255 output[7] = 2 * (in[3] * in[4] +
259 tmp = in[1] * in[7] + in[3] * in[5];
260 output[8] = in[4] * in[4] +
264 output[9] = 2 * (in[4] * in[5] +
269 tmp = in[3] * in[7] + in[1] * in[9];
270 output[10] = 2 * (in[5] * in[5] +
274 output[11] = 2 * (in[5] * in[6] +
278 output[12] = in[6] * in[6] +
282 output[13] = 2 * (in[6] * in[7] +
285 output[14] = 2 * (in[7] * in[7] +
288 output[15] = 2 * (in[7] * in[8] +
290 output[16] = in[8] * in[8] +
292 output[17] = 2 * in[8] * in[9];
293 output[18] = 2 * in[9] * in[9];
297 fsquare(felem *output, felem *in) {
299 fsquare_inner(t, in);
301 freduce_coefficients(t);
302 memcpy(output, t, sizeof(felem) * 10);
305 /* Take a little-endian, 32-byte number and expand it into polynomial form */
307 fexpand(felem *output, uchar *input) {
308 #define F(n,start,shift,mask) \
309 output[n] = ((((felem) input[start + 0]) | \
310 ((felem) input[start + 1]) << 8 | \
311 ((felem) input[start + 2]) << 16 | \
312 ((felem) input[start + 3]) << 24) >> shift) & mask;
313 F(0, 0, 0, 0x3ffffff);
314 F(1, 3, 2, 0x1ffffff);
315 F(2, 6, 3, 0x3ffffff);
316 F(3, 9, 5, 0x1ffffff);
317 F(4, 12, 6, 0x3ffffff);
318 F(5, 16, 0, 0x1ffffff);
319 F(6, 19, 1, 0x3ffffff);
320 F(7, 22, 3, 0x1ffffff);
321 F(8, 25, 4, 0x3ffffff);
322 F(9, 28, 6, 0x1ffffff);
326 /* Take a fully reduced polynomial form number and contract it into a
327 * little-endian, 32-byte array
330 fcontract(uchar *output, felem *input) {
334 for (i = 0; i < 9; ++i) {
336 while (input[i] < 0) {
337 input[i] += 0x2000000;
341 while (input[i] < 0) {
342 input[i] += 0x4000000;
347 while (input[9] < 0) {
348 input[9] += 0x2000000;
351 } while (input[0] < 0);
362 output[s+0] |= input[i] & 0xff; \
363 output[s+1] = (input[i] >> 8) & 0xff; \
364 output[s+2] = (input[i] >> 16) & 0xff; \
365 output[s+3] = (input[i] >> 24) & 0xff;
381 /* Input: Q, Q', Q-Q'
386 * x z: short form, destroyed
387 * xprime zprime: short form, destroyed
388 * qmqp: short form, preserved
390 static void fmonty(felem *x2, felem *z2, /* output 2Q */
391 felem *x3, felem *z3, /* output Q + Q' */
392 felem *x, felem *z, /* input Q */
393 felem *xprime, felem *zprime, /* input Q' */
394 felem *qmqp /* input Q - Q' */) {
395 felem origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19],
396 zzprime[19], zzzprime[19], xxxprime[19];
398 memcpy(origx, x, 10 * sizeof(felem));
400 fdifference(z, origx); // does x - z
402 memcpy(origxprime, xprime, sizeof(felem) * 10);
403 fsum(xprime, zprime);
404 fdifference(zprime, origxprime);
405 fproduct(xxprime, xprime, z);
406 fproduct(zzprime, x, zprime);
407 freduce_degree(xxprime);
408 freduce_coefficients(xxprime);
409 freduce_degree(zzprime);
410 freduce_coefficients(zzprime);
411 memcpy(origxprime, xxprime, sizeof(felem) * 10);
412 fsum(xxprime, zzprime);
413 fdifference(zzprime, origxprime);
414 fsquare(xxxprime, xxprime);
415 fsquare(zzzprime, zzprime);
416 fproduct(zzprime, zzzprime, qmqp);
417 freduce_degree(zzprime);
418 freduce_coefficients(zzprime);
419 memcpy(x3, xxxprime, sizeof(felem) * 10);
420 memcpy(z3, zzprime, sizeof(felem) * 10);
424 fproduct(x2, xx, zz);
426 freduce_coefficients(x2);
427 fdifference(zz, xx); // does zz = xx - zz
428 memset(zzz + 10, 0, sizeof(felem) * 9);
429 fscalar_product(zzz, zz, 121665);
431 freduce_coefficients(zzz);
433 fproduct(z2, zz, zzz);
435 freduce_coefficients(z2);
438 /* Calculates nQ where Q is the x-coordinate of a point on the curve
440 * resultx/resultz: the x coordinate of the resulting curve point (short form)
441 * n: a little endian, 32-byte number
442 * q: a point of the curve (short form)
445 cmult(felem *resultx, felem *resultz, uchar *n, felem *q) {
446 felem a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
447 felem *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
448 felem e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1};
449 felem *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
453 memcpy(nqpqx, q, sizeof(felem) * 10);
455 for (i = 0; i < 32; ++i) {
456 uchar byte = n[31 - i];
457 for (j = 0; j < 8; ++j) {
459 fmonty(nqpqx2, nqpqz2,
489 memcpy(resultx, nqx, sizeof(felem) * 10);
490 memcpy(resultz, nqz, sizeof(felem) * 10);
493 // -----------------------------------------------------------------------------
494 // Shamelessly copied from djb's code
495 // -----------------------------------------------------------------------------
497 crecip(felem *out, felem *z) {
510 /* 2 */ fsquare(z2,z);
511 /* 4 */ fsquare(t1,z2);
512 /* 8 */ fsquare(t0,t1);
513 /* 9 */ fmul(z9,t0,z);
514 /* 11 */ fmul(z11,z9,z2);
515 /* 22 */ fsquare(t0,z11);
516 /* 2^5 - 2^0 = 31 */ fmul(z2_5_0,t0,z9);
518 /* 2^6 - 2^1 */ fsquare(t0,z2_5_0);
519 /* 2^7 - 2^2 */ fsquare(t1,t0);
520 /* 2^8 - 2^3 */ fsquare(t0,t1);
521 /* 2^9 - 2^4 */ fsquare(t1,t0);
522 /* 2^10 - 2^5 */ fsquare(t0,t1);
523 /* 2^10 - 2^0 */ fmul(z2_10_0,t0,z2_5_0);
525 /* 2^11 - 2^1 */ fsquare(t0,z2_10_0);
526 /* 2^12 - 2^2 */ fsquare(t1,t0);
527 /* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
528 /* 2^20 - 2^0 */ fmul(z2_20_0,t1,z2_10_0);
530 /* 2^21 - 2^1 */ fsquare(t0,z2_20_0);
531 /* 2^22 - 2^2 */ fsquare(t1,t0);
532 /* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
533 /* 2^40 - 2^0 */ fmul(t0,t1,z2_20_0);
535 /* 2^41 - 2^1 */ fsquare(t1,t0);
536 /* 2^42 - 2^2 */ fsquare(t0,t1);
537 /* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
538 /* 2^50 - 2^0 */ fmul(z2_50_0,t0,z2_10_0);
540 /* 2^51 - 2^1 */ fsquare(t0,z2_50_0);
541 /* 2^52 - 2^2 */ fsquare(t1,t0);
542 /* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
543 /* 2^100 - 2^0 */ fmul(z2_100_0,t1,z2_50_0);
545 /* 2^101 - 2^1 */ fsquare(t1,z2_100_0);
546 /* 2^102 - 2^2 */ fsquare(t0,t1);
547 /* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
548 /* 2^200 - 2^0 */ fmul(t1,t0,z2_100_0);
550 /* 2^201 - 2^1 */ fsquare(t0,t1);
551 /* 2^202 - 2^2 */ fsquare(t1,t0);
552 /* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
553 /* 2^250 - 2^0 */ fmul(t0,t1,z2_50_0);
555 /* 2^251 - 2^1 */ fsquare(t1,t0);
556 /* 2^252 - 2^2 */ fsquare(t0,t1);
557 /* 2^253 - 2^3 */ fsquare(t1,t0);
558 /* 2^254 - 2^4 */ fsquare(t0,t1);
559 /* 2^255 - 2^5 */ fsquare(t1,t0);
560 /* 2^255 - 21 */ fmul(out,t1,z11);
564 curve25519(uchar mypublic[32], uchar secret[32], uchar basepoint[32]) {
565 felem bp[10], x[10], z[10], zmone[10];
566 fexpand(bp, basepoint);
567 cmult(x, z, secret, bp);
570 fcontract(mypublic, z);