| 1 | /* |
| 2 | * Elliptic curve arithmetic. |
| 3 | */ |
| 4 | |
| 5 | #include "bn-internal.h" |
| 6 | #include "ssh.h" |
| 7 | |
| 8 | /* Some background. |
| 9 | * |
| 10 | * These notes are intended to help programmers who are mathematically |
| 11 | * inclined, but not well versed in the theory of elliptic curves. If more |
| 12 | * detail is required, introductory texts of various levels of sophistication |
| 13 | * are readily available. I found |
| 14 | * |
| 15 | * L. Washington, `Elliptic Curves: Number Theory and Cryptography', CRC |
| 16 | * Press, 2003 |
| 17 | * |
| 18 | * useful. Further background on algebra and number theory can be found in |
| 19 | * |
| 20 | * V. Shoup, `A Computational Introduction to Number Theory and Algebra', |
| 21 | * version 2, Cambridge University Press, 2008; http://shoup.net/ntb/ |
| 22 | * [CC-ND-NC]. |
| 23 | * |
| 24 | * We work in a finite field k = GF(p^m). To make things easier, we'll only |
| 25 | * deal with the cases where p = 2 (`binary' fields) or m = 1 (`prime' |
| 26 | * fields). In fact, we don't currently implement binary fields either. |
| 27 | * |
| 28 | * We start in two-dimensional projective space over k, denoted P^2(k). This |
| 29 | * is the space of classes of triples of elements of k { (a x : a y : a z) | |
| 30 | * a in k^* } such that x y z /= 0. Projective points with z /= 0 are called |
| 31 | * `finite', and correspond to points in the more familiar two-dimensional |
| 32 | * affine space (x/z, y/z); points with z = 0 are called `infinite'. |
| 33 | * |
| 34 | * An elliptic curve is the set of projective points satisfying an equation |
| 35 | * of the form |
| 36 | * |
| 37 | * y^2 z + a_1 x y z + a_3 y z^2 = x^3 + a_2 x^2 z + a_4 x z^2 + a_6 z^3 |
| 38 | * |
| 39 | * (the `Weierstrass form'), such that the partial derivatives don't both |
| 40 | * vanish at any point. (Notice that this is a homogeneous equation -- all |
| 41 | * of the monomials have the same degree -- so this is a well-defined subset |
| 42 | * of our projective space. Don't worry about the numbering of the |
| 43 | * coefficients: maybe the later stuff about Jacobian coordinates will make |
| 44 | * them clear.) |
| 45 | * |
| 46 | * Let's consider, for a moment, a straight line r x + s y + t z = 0 which |
| 47 | * meets the curve at two points P and Q; then they must also meet again at |
| 48 | * some other point R. (Here, any or all of P, Q, and R might be equal; |
| 49 | * in this case, the simultaneous equations have repeated solutions.) Let's |
| 50 | * go out on a limb and write |
| 51 | * |
| 52 | * P + Q + R = 0 |
| 53 | * |
| 54 | * whenever three points P, Q, and R on the curve are collinear. Obviously |
| 55 | * this notion of `addition' is commutative: three points are collinear |
| 56 | * whichever order we write them. If we designate some point O as being an |
| 57 | * `identity' then we can introduce a notion of inverses. And then some |
| 58 | * magic occurs: this `addition' operation we've just invented is also |
| 59 | * associative -- so it's an abelian group operation! (This can be |
| 60 | * demonstrated by slogging through the algebra, but there's a fancy way |
| 61 | * involving the Weil pairing. Washington's book has both versions.) |
| 62 | * |
| 63 | * So we have an abelian group. It's actually quite a fun group, for |
| 64 | * complicated reasons I'm not going to do into here. For our purposes it's |
| 65 | * sufficient to note that, in most elliptic curve groups, the discrete |
| 66 | * logarithm problem -- finding x given x P -- seems really very hard. I |
| 67 | * mean, much harder than in multiplicative subgroups of finite fields, |
| 68 | * where index calculus techniques can give you the answer in subexponential |
| 69 | * time. The best algorithms known for solving this problem in elliptic |
| 70 | * curve groups are the ones which work in any old group G -- and they run in |
| 71 | * O(sqrt(#G)) time. |
| 72 | * |
| 73 | * Staring at the Weierstrass equation some more, we can determine that in |
| 74 | * fact there's exactly one infinite point on the curve: if z = 0 then we |
| 75 | * must have x = 0, so (0 : 1 : 0) is precisely that point. That's kind of |
| 76 | * handy. |
| 77 | * |
| 78 | * The Weierstrass form is pretty horrid, really. Fortunately, it's always |
| 79 | * possible to apply a linear transformation to make it less nasty. If p > 3 |
| 80 | * (which it will be for the prime fields we care about), we can simplify |
| 81 | * this to |
| 82 | * |
| 83 | * y^2 z = x^3 + a x z^2 + b z^3 (with 4 a^3 + 27 b^2 /= 0) |
| 84 | * |
| 85 | * and in the binary case we can simplify to |
| 86 | * |
| 87 | * y^2 z + x y z = x^3 + a x^2 z + b z^3 (with b /= 0) |
| 88 | * |
| 89 | * and it's these kinds of forms which we're actually going to work with. |
| 90 | * (There are other forms of elliptic curves which have interesting benefits, |
| 91 | * e.g., the Montgomery and Edwards forms, but they've not really caught on |
| 92 | * in standards.) Linear transformations don't mess up the geometry, so the |
| 93 | * group we constructed above still works -- and, indeed, the transformation |
| 94 | * is an isomorphism of groups. And, finally, the point at infinity we found |
| 95 | * above is still infinite. Let's call that point O, and declare it to be |
| 96 | * our additive identity. That leaves the finite points (x : y : 1), which |
| 97 | * correspond to the affine points |
| 98 | * |
| 99 | * y^2 = x^3 + a x + b or y^2 + x y = x^3 + a x^2 + b |
| 100 | * |
| 101 | * So, if we want to add two points P and Q, we draw the line between them; |
| 102 | * if P = Q then the line in question is the tangent to the curve at P. (The |
| 103 | * smoothness condition above is precisely what we need for this tangent to |
| 104 | * be well-defined. Unsmooth curves aren't worth worrying about, because |
| 105 | * they're cryptographically useless: they're isomorphic to some other group |
| 106 | * in which discrete logs are very easy.) In prime fields, if this line is |
| 107 | * vertical then it meets the curve again at O, and P + Q = O; otherwise it |
| 108 | * meets the curve at some other finite point R = (x, y), and the answer we |
| 109 | * want is P + Q = -R = (x, -y). In binary fields, the situation is similar |
| 110 | * but a little messier because of the x y term we couldn't eliminate. |
| 111 | * |
| 112 | * It's not very hard to work out where this point R is. You get a cubic |
| 113 | * equation out the far end which looks kind of daunting until you notice |
| 114 | * that the negation of the x^2 coefficient is exactly the sum of the roots, |
| 115 | * and you already know two of them. Unfortunately, the solution is still |
| 116 | * somewhat unpleasant and involves a bunch of division -- and in finite |
| 117 | * fields that involves computing a modular inverse, which is rather slow. |
| 118 | * |
| 119 | * Wouldn't it be really nice if we could somehow keep the denominators |
| 120 | * separate, so we could just do a few `big' divisions at the end of some |
| 121 | * calculation? Of course, we'd need to tweak the point-addition formulae so |
| 122 | * that we could cope with input points which have separate denominators. |
| 123 | * And so we enter the realm of fun coordinate systems. |
| 124 | * |
| 125 | * Perhaps an obvious choice of coordinates is the projective form above: |
| 126 | * rather than working out (x/m, y/n), we could just keep track of a |
| 127 | * projective point (x : y : m n). It turns out that there's a slightly |
| 128 | * different variant which is a little better: `Jacobian coordinates' are |
| 129 | * classes of triples (x :: y :: z), where x y z /= 0, and the point (x :: y |
| 130 | * :: z) is equivalent to (a^2 x :: a^3 y :: a z). |
| 131 | * |
| 132 | * The (simplified) Wieierstrass equation, using Jacobian projective |
| 133 | * coordinates, looks like this. |
| 134 | * |
| 135 | * y^2 = x^3 + a x z^2 + b z^6 |
| 136 | * |
| 137 | * or |
| 138 | * |
| 139 | * y^2 + x y z = x^3 + a x^2 z + b z^3 |
| 140 | * |
| 141 | * (Maybe this is a hint about the strange coefficient numbering in the |
| 142 | * general Weierstrass form.) Anyway, if a point is infinite then we have |
| 143 | * y^2 = x^3, which has the obvious solution x = y = 1. |
| 144 | * |
| 145 | * That's about all the theory we need for now. Let's get into the code. |
| 146 | */ |
| 147 | |
| 148 | /* Let's start by making an abstract interface for finite field arithmetic. |
| 149 | * If we need to do clever things later then this will be a convenient |
| 150 | * extension point (e.g., supporting binary fields, working in extension |
| 151 | * fields as we'd need if we wanted to implement Tate or Weil pairings, or |
| 152 | * just doing different kinds of optimizations for the prime case). |
| 153 | * |
| 154 | * This is quite cheesy, but it's enough for now. |
| 155 | */ |
| 156 | |
| 157 | union fieldelt { |
| 158 | Bignum n; |
| 159 | }; |
| 160 | |
| 161 | struct field { |
| 162 | const struct field_ops *ops; /* Field operations table. */ |
| 163 | union fieldelt zero, one; /* Identity elements. */ |
| 164 | Bignum q; /* The field size. */ |
| 165 | }; |
| 166 | |
| 167 | struct field_ops { |
| 168 | void (*destroy)(struct field *f); /* Destroy the field object. */ |
| 169 | |
| 170 | void (*init)(struct field *f, union fieldelt *x); |
| 171 | /* Initialize x (to null). */ |
| 172 | |
| 173 | void (*free)(struct field *f, union fieldelt *x); |
| 174 | /* Free an element x, leaving it |
| 175 | * null. |
| 176 | */ |
| 177 | |
| 178 | void (*copy)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 179 | /* Make d be a copy of x. */ |
| 180 | |
| 181 | void (*frombn)(struct field *f, union fieldelt *d, Bignum n); |
| 182 | /* Convert n to an element. */ |
| 183 | |
| 184 | Bignum (*tobn)(struct field *f, union fieldelt *x); |
| 185 | /* Convert x to an integer. */ |
| 186 | |
| 187 | int (*zerop)(struct field *f, union fieldelt *x); |
| 188 | /* Return nonzero if x is zero. */ |
| 189 | |
| 190 | void (*dbl)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 191 | /* Store 2 x in d. */ |
| 192 | |
| 193 | void (*add)(struct field *f, union fieldelt *d, |
| 194 | union fieldelt *x, union fieldelt *y); |
| 195 | /* Store x + y in d. */ |
| 196 | |
| 197 | void (*sub)(struct field *f, union fieldelt *d, |
| 198 | union fieldelt *x, union fieldelt *y); |
| 199 | /* Store x - y in d. */ |
| 200 | |
| 201 | void (*mul)(struct field *f, union fieldelt *d, |
| 202 | union fieldelt *x, union fieldelt *y); |
| 203 | /* Store x y in d. */ |
| 204 | |
| 205 | void (*sqr)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 206 | /* Store x^2 in d. */ |
| 207 | |
| 208 | void (*inv)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 209 | /* Store 1/x in d; it is an error |
| 210 | * for x to be zero. |
| 211 | */ |
| 212 | |
| 213 | int (*sqrt)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 214 | /* If x is a square, store a square |
| 215 | * root (chosen arbitrarily) in d |
| 216 | * and return nonzero; otherwise |
| 217 | * return zero leaving d unchanged. |
| 218 | */ |
| 219 | |
| 220 | /* For internal use only. The standard methods call on these for |
| 221 | * detailed field-specific behaviour. |
| 222 | */ |
| 223 | void (*_reduce)(struct field *f, union fieldelt *x); |
| 224 | /* Reduce an intermediate result so |
| 225 | * that it's within whatever bounds |
| 226 | * are appropriate. |
| 227 | */ |
| 228 | |
| 229 | /* These functions are only important for prime fields. */ |
| 230 | void (*dbl)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 231 | /* Store 2 x in d. */ |
| 232 | |
| 233 | void (*tpl)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 234 | /* Store 3 x in d. */ |
| 235 | |
| 236 | void (*dql)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 237 | /* Store 4 x in d. */ |
| 238 | |
| 239 | void (*hlv)(struct field *f, union fieldelt *d, union fieldelt *x); |
| 240 | /* Store x/2 in d. */ |
| 241 | }; |
| 242 | |
| 243 | /* Some standard methods for prime fields. */ |
| 244 | |
| 245 | static void gen_destroy(struct field *f) |
| 246 | { sfree(f); } |
| 247 | |
| 248 | static void prime_init(struct field *f, union fieldelt *x) |
| 249 | { x->n = 0; } |
| 250 | |
| 251 | static void prime_free(struct field *f, union fieldelt *x) |
| 252 | { if (x->n) { freebn(x->n); x->n = 0; } } |
| 253 | |
| 254 | static void prime_frombn(struct field *f, union fieldelt *d, Bignum n) |
| 255 | { f->ops->free(f, d); d->n = copybn(n); } |
| 256 | |
| 257 | static Bignum prime_tobn(struct field *f, union fieldelt *x) |
| 258 | { return copybn(x->n); } |
| 259 | |
| 260 | static Bignum prime_zerop(struct field *f, union fieldelt *x) |
| 261 | { return bignum_cmp(x->n, f->zero.n) == 0; } |
| 262 | |
| 263 | static void prime_add(struct field *f, union fieldelt *d, |
| 264 | union fieldelt *x, union fieldelt *y) |
| 265 | { |
| 266 | Bignum sum = bigadd(x, y); |
| 267 | Bignum red = bigsub(sum, f->q); |
| 268 | |
| 269 | f->ops->free(f, d); |
| 270 | if (!red) |
| 271 | d->n = sum; |
| 272 | else { |
| 273 | freebn(sum); |
| 274 | d->n = red; |
| 275 | } |
| 276 | } |
| 277 | |
| 278 | static void prime_sub(struct field *f, union fieldelt *d, |
| 279 | union fieldelt *x, union fieldelt *y) |
| 280 | { |
| 281 | Bignum raise = bigadd(x, f->q); |
| 282 | Bignum diff = bigsub(raise, y); |
| 283 | Bignum drop = bigsub(diff, f->q); |
| 284 | |
| 285 | f->ops->free(f, d); freebn(raise); |
| 286 | if (!drop) |
| 287 | d->n = diff; |
| 288 | else { |
| 289 | freebn(diff); |
| 290 | d->n = drop; |
| 291 | } |
| 292 | } |
| 293 | |
| 294 | static void prime_mul(struct field *f, union fieldelt *d, |
| 295 | union fieldelt *x, union fieldelt *y) |
| 296 | { |
| 297 | Bignum prod = bigmul(x, y); |
| 298 | |
| 299 | f->ops->free(f, d); |
| 300 | d->n = prod; |
| 301 | f->ops->_reduce(f, d); |
| 302 | } |
| 303 | |
| 304 | static void gen_sqr(struct field *f, union fieldelt *d, union fieldelt *x) |
| 305 | { f->ops->mul(f->ops, d, x, x); } |
| 306 | |
| 307 | static void prime_inv(struct field *f, union fieldelt *d, union fieldelt *x) |
| 308 | { |
| 309 | Bignum inv = modinv(x->n, f->q); |
| 310 | |
| 311 | f->ops->free(f, d); |
| 312 | d->n = inv; |
| 313 | } |
| 314 | |
| 315 | static int prime_sqrt(struct field *f, union fieldelt *d, union fieldelt *x) |
| 316 | { |
| 317 | Bignum sqrt = modsqrt(x->n, f->q); |
| 318 | |
| 319 | if (!sqrt) return 0; |
| 320 | f->ops->free(f, d); |
| 321 | d->n = sqrt; |
| 322 | return 1; |
| 323 | } |
| 324 | |
| 325 | static void prime_dbl(struct field *f, union_fieldelt *d, union fieldelt *x) |
| 326 | { f->ops->add(f, d, x, x); } |
| 327 | |
| 328 | static void prime_tpl(struct field *f, union_fieldelt *d, union fieldelt *x) |
| 329 | { |
| 330 | union_fieldelt t; |
| 331 | |
| 332 | if (d != x) { |
| 333 | f->ops->add(f, d, x, x); |
| 334 | f->ops->add(f, d, d, x); |
| 335 | } else { |
| 336 | f->ops->init(f, &t); |
| 337 | f->ops->copy(f, d, x); |
| 338 | f->ops->add(f, d, &t, &t); |
| 339 | f->ops->add(f, d, d, &t); |
| 340 | f->ops->free(f, &t); |
| 341 | } |
| 342 | } |
| 343 | |
| 344 | static void prime_qdl(struct field *f, union fieldelt *d, |
| 345 | union fieldelt *x) |
| 346 | { f->ops->add(f, d, x, x); f->ops->add(f, d, d, d); } |
| 347 | |
| 348 | static void prime_hlv(struct field *f, union fieldelt *d, union fieldelt *x) |
| 349 | { |
| 350 | Bignum t, u; |
| 351 | |
| 352 | if (!x->n[0]) |
| 353 | f->ops->copy(f, d, x); |
| 354 | else { |
| 355 | /* The tedious answer is to multiply by the inverse of 2 in this |
| 356 | * field. But there's a better way: either x or q - x is actually |
| 357 | * divisible by 2, so the answer we want is either x >> 1 or p - |
| 358 | * ((p - x) >> 1) depending on whether x is even. |
| 359 | */ |
| 360 | if (!(x->n[1] & 1)) { |
| 361 | t = copybn(x->n); |
| 362 | shift_right(t + 1, t[0], 1); |
| 363 | } else { |
| 364 | u = bigsub(f->q, x); |
| 365 | shift_right(u + 1, u[0], 1); |
| 366 | t = bigsub(f->q, u); |
| 367 | freebn(u); |
| 368 | } |
| 369 | f->ops->free(f, d); |
| 370 | d->n = t; |
| 371 | } |
| 372 | |
| 373 | /* Now some utilities for modular reduction. All of the primes we're |
| 374 | * concerned about are of the form p = 2^n - d for `convenient' values of d |
| 375 | * -- such numbers are called `pseudo-Mersenne primes'. Suppose we're given |
| 376 | * a value x = x_0 2^n + x_1: then x - x_0 p = x_1 + x_0 d is clearly less |
| 377 | * than x and congruent to it mod p. So we can reduce x below 2^n by |
| 378 | * iterating this trick; if we're unlucky enough to have p <= x < 2^n then we |
| 379 | * can just subtract p. |
| 380 | * |
| 381 | * The values of d we'll be dealing with are specifically convenient because |
| 382 | * they satisfy the following properties. |
| 383 | * |
| 384 | * * We can write d = SUM_i d_i 2^i, with d_i in { -1, 0, +1 }. (This is |
| 385 | * no surprise.) |
| 386 | * |
| 387 | * * d > 0. |
| 388 | * |
| 389 | * * Very few d_i /= 0. With one small exception, if d_i /= 0 then i is |
| 390 | * divisible by 32. |
| 391 | * |
| 392 | * The following functions will therefore come in very handy. |
| 393 | */ |
| 394 | |
| 395 | static void add_imm_lsl(BignumInt *x, BignumInt y, unsigned shift) |
| 396 | { |
| 397 | BignumDbl c = y << (shift % BIGNUM_INT_BITS); |
| 398 | |
| 399 | for (x += shift/BIGNUM_INT_BITS; c; x++) { |
| 400 | c += *x; |
| 401 | *x = c & BIGNUM_INT_MASK; |
| 402 | c >>= BIGNUM_INT_BITS; |
| 403 | } |
| 404 | } |
| 405 | |
| 406 | static void sub_imm_lsl(BignumInt *x, BignumInt y, unsigned shift) |
| 407 | { |
| 408 | BignumDbl c = y << (shift % BIGNUM_INT_BITS); |
| 409 | BignumDbl t; |
| 410 | |
| 411 | c--; |
| 412 | for (x += shift/BIGNUM_INT_BITS; c; x++) { |
| 413 | c ^= BIGNUM_INT_MASK; |
| 414 | c += *x; |
| 415 | *x = c & BIGNUM_INT_MASK; |
| 416 | c >>= BIGNUM_INT_BITS; |
| 417 | } |
| 418 | } |
| 419 | |
| 420 | static void pmp_reduce(struct field *f, union fieldelt *xx, |
| 421 | void (*add_mul_d_lsl)(BignumInt *x, |
| 422 | BignumInt y, |
| 423 | unsigned shift)) |
| 424 | { |
| 425 | /* Notation: write w = BIGNUM_INT_WORDS, and B = 2^w is the base we're |
| 426 | * working in. We assume that p = 2^n - d for some d. The job of |
| 427 | * add_mul_d_lsl is to add to its argument x the value y d 2^shift. |
| 428 | */ |
| 429 | |
| 430 | BignumInt top, *p, *x; |
| 431 | unsigned plen, xlen, i; |
| 432 | unsigned topbits, topclear; |
| 433 | Bignum t; |
| 434 | |
| 435 | /* Initialization: if x is shorter than p then there's nothing to do. |
| 436 | * Otherwise, ensure that it's at least one word longer, since this is |
| 437 | * required by the utility functions which add_mul_d_lsl will call. |
| 438 | */ |
| 439 | p = f->q + 1; plen = f->q[0]; |
| 440 | x = xx->n + 1; xlen = xx->n[0]; |
| 441 | if (xlen < plen) return; |
| 442 | else if (xlen == plen) { |
| 443 | t = newbn(plen + 1); |
| 444 | for (i = 0; i < plen; i++) t[i + 1] = x[i]; |
| 445 | t[plen] = 0; |
| 446 | freebn(xx->n); xx->n = t; |
| 447 | x = t + 1; xlen = plen + 1; |
| 448 | } |
| 449 | |
| 450 | /* Preparation: Work out the bit length of p. At the end of this, we'll |
| 451 | * have topbits such that n = w (plen - 1) + topbits, and topclear = w - |
| 452 | * topbits, so that n = w plen - topclear. |
| 453 | */ |
| 454 | while (plen > 0 && !p[plen - 1]) plen--; |
| 455 | top = p[plen - 1]; |
| 456 | assert(top); |
| 457 | if (top & BIGNUM_TOP_BIT) topbits = BIGNUM_INT_BITS; |
| 458 | else for (topbits = 0; top >> topbits; topbits++); |
| 459 | topclear = BIGNUM_INT_BITS - topbits; |
| 460 | |
| 461 | /* Step 1: Trim x down to the right number of words. */ |
| 462 | while (xlen > plen) { |
| 463 | |
| 464 | /* Pick out the topmost word; call it y for now. */ |
| 465 | y = x[xlen - 1]; |
| 466 | if (!y) { |
| 467 | xlen--; |
| 468 | continue; |
| 469 | } |
| 470 | |
| 471 | /* Observe that d == 2^n (mod p). Clear that top word. This |
| 472 | * effectively subtracts y B^{xlen-1} from x, so we must add back d |
| 473 | * 2^{w(xlen-1)-n} in order to preserve congruence modulo p. Since d |
| 474 | * is smaller than 2^n, this will reduce the absolute value of x, so |
| 475 | * we make (at least some) progress. In practice, we expect that d |
| 476 | * is a lot smaller. |
| 477 | * |
| 478 | * Note that w (xlen - 1) - n = w xlen - w - n = w (xlen - plen - 1) |
| 479 | * + topclear. |
| 480 | */ |
| 481 | x[xlen - 1] = 0; |
| 482 | add_mul_d_lsl(x + xlen - plen - 1, y, topclear); |
| 483 | } |
| 484 | |
| 485 | /* Step 2: Trim off the high bits of x, beyond the top bit of p. In more |
| 486 | * detail: if x >= 2^n > p, then write x = x_0 + 2^n y; then we can clear |
| 487 | * the top topclear bits of x, and add y d to preserve congruence mod p. |
| 488 | * |
| 489 | * If topclear is zero then there's nothing to do here: step 1 will |
| 490 | * already have arranged that x < 2^n. |
| 491 | */ |
| 492 | if (topclear) { |
| 493 | for (;;) { |
| 494 | y = x[xlen - 1] >> topbits; |
| 495 | if (!y) break; |
| 496 | x[xlen - 1] &= (1 << topbits) - 1; |
| 497 | add_mul_d_lsl(x, y, 0); |
| 498 | } |
| 499 | } |
| 500 | |
| 501 | /* Step 3: If x >= p then subtract p from x. */ |
| 502 | for (i = plen; i-- && x[i] == p[i]; ); |
| 503 | if (i >= plen || x[i] > p[i]) internal_sub(x, p, plen); |
| 504 | |
| 505 | /* We're done now. */ |
| 506 | d->n[0] = xlen; |
| 507 | bn_restore_invariant(d->n); |
| 508 | } |
| 509 | |
| 510 | /* Putting together a field-ops structure for a pseudo-Mersenne prime field. |
| 511 | */ |
| 512 | |
| 513 | static void pmp_destroy(struct field *f) |
| 514 | { freebn(f->q); sfree(f); } |
| 515 | |
| 516 | #define DEFINE_PMP_FIELD(fname) \ |
| 517 | static const struct field_ops fname##_fieldops { \ |
| 518 | pmp_destroy, \ |
| 519 | prime_init, prime_free, prime_copy, \ |
| 520 | prime_frombn, prime_tobn, \ |
| 521 | prime_zerop, \ |
| 522 | prime_add, prime_sub, prime_mul, gen_sqr, prime_inv, prime_sqrt, \ |
| 523 | fname_reduce, \ |
| 524 | gen_dbl, gen_tpl, gen_qdl, \ |
| 525 | prime_hlv \ |
| 526 | }; \ |
| 527 | \ |
| 528 | static struct field *make_##fname##_field(void) \ |
| 529 | { \ |
| 530 | struct field *f = snew(struct field); \ |
| 531 | f->ops = &fname##_fieldops; \ |
| 532 | f->zero.n = Zero; \ |
| 533 | f->one.n = One; \ |
| 534 | f->q = bigunm_from_bytes(fname##_p, sizeof(fname##_p)); \ |
| 535 | return f; \ |
| 536 | } |
| 537 | |
| 538 | /* Some actual field definitions. */ |
| 539 | |
| 540 | static const unsigned char p256_p[] = { |
| 541 | 0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, |
| 542 | 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, |
| 543 | 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff, |
| 544 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff |
| 545 | }; |
| 546 | |
| 547 | static void p256_add_mul_d_lsl(BignumInt *x, BignumInt y, unsigned shift) |
| 548 | { |
| 549 | /* p_{256} = 2^{256} - 2^{224} + 2^{192} + 2^{96} - 1 */ |
| 550 | add_imm_lsl(x, y, shift + 224); |
| 551 | sub_imm_lsl(x, y, shift + 192); |
| 552 | sub_imm_lsl(x, y, shift + 96); |
| 553 | add_imm_lsl(x, y, shift + 0); |
| 554 | } |
| 555 | |
| 556 | static void p256_reduce(struct field *f, union fieldelt *d) |
| 557 | { pmp_reduce(f, d, p256_add_mul_d_lsl); } |
| 558 | |
| 559 | DEFINE_PMP_FIELD(p256) |
| 560 | |
| 561 | static const unsigned char p384_p[] = { |
| 562 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 563 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 564 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 565 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe, |
| 566 | 0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x00, |
| 567 | 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff |
| 568 | }; |
| 569 | |
| 570 | static void p384_add_mul_d_lsl(BignumInt *x, BignumInt y, unsigned shift) |
| 571 | { |
| 572 | /* p_{384} = 2^{384} - 2^{128} - 2^{96} + 2^{32} - 1 */ |
| 573 | add_imm_lsl(x, y, shift + 128); |
| 574 | add_imm_lsl(x, y, shift + 96); |
| 575 | sub_imm_lsl(x, y, shift + 32); |
| 576 | add_imm_lsl(x, y, shift + 0); |
| 577 | } |
| 578 | |
| 579 | static void p384_reduce(struct field *f, union fieldelt *d) |
| 580 | { pmp_reduce(f, d, p384_add_mul_d_lsl); } |
| 581 | |
| 582 | DEFINE_PMP_FIELD(p384) |
| 583 | |
| 584 | static const unsigned char p521_p[] = { |
| 585 | 0x01, 0xff, |
| 586 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 587 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 588 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 589 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 590 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 591 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 592 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
| 593 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff |
| 594 | }; |
| 595 | |
| 596 | static void p521_add_mul_d_lsl(BignumInt *x, BignumInt y, unsigned shift) |
| 597 | { add_imm_lsl(x, y, shift); } /* p_{521} = 2^{521} - 1 */ |
| 598 | |
| 599 | static void p521_reduce(struct field *f, union fieldelt *d) |
| 600 | { pmp_reduce(f, d, p521_add_mul_d_lsl); } |
| 601 | |
| 602 | DEFINE_PMP_FIELD(p521) |
| 603 | |
| 604 | /* And now, some actual elliptic curves. |
| 605 | * |
| 606 | * As with fields, we start by defining an abstract interface which may have |
| 607 | * various implementations (e.g., different algorithms with different |
| 608 | * performance properties, or based on different kinds of underlying fields. |
| 609 | */ |
| 610 | |
| 611 | /* An external-form elliptic curve point. */ |
| 612 | struct ecpt { |
| 613 | unsigned f; /* Flags */ |
| 614 | #define ECPTF_INF 1u /* Point is at infinity */ |
| 615 | union fieldelt x, y; /* x- and y-coordinates */ |
| 616 | }; |
| 617 | |
| 618 | /* An internal-form elliptic curve point. This is where most of the action |
| 619 | * happens. |
| 620 | */ |
| 621 | union iecpt { |
| 622 | struct { |
| 623 | union fieldelt x, y, z; /* Projective-form triple. */ |
| 624 | } jac; /* Jacobian projective coords. */ |
| 625 | }; |
| 626 | |
| 627 | struct ec { |
| 628 | const struct ec_ops *ops; /* Curve operations table. */ |
| 629 | struct field *f; /* The underlying finite field. */ |
| 630 | struct iecpt p; /* The well-known generator point */ |
| 631 | Bignum r; /* The size of the generated group */ |
| 632 | union { |
| 633 | struct { union fieldelt a, b; } w; /* Simple Weierstrass coeffs. */ |
| 634 | } u; |
| 635 | }; |
| 636 | |
| 637 | struct ec_ops { |
| 638 | void (*destroy)(struct ec *ec); /* Destroy the curve itself. */ |
| 639 | |
| 640 | void (*init)(struct ec *ec, union iecpt *p); |
| 641 | /* Initialize p (to null). */ |
| 642 | |
| 643 | void (*free)(struct ec *ec, union iecpt *p); |
| 644 | /* Free the point p. */ |
| 645 | |
| 646 | void (*setinf)(struct ec *ec, union iecpt *p); |
| 647 | /* Set p to be the point at infinity. |
| 648 | */ |
| 649 | |
| 650 | int (*infp)(struct ec *ec, union iecpt *p); |
| 651 | /* Answer whether p is infinite. */ |
| 652 | |
| 653 | void (*copy)(struct ec *ec, union iecpt *d, union iecpt *d); |
| 654 | /* Make p be a copy of d. */ |
| 655 | |
| 656 | void (*in)(struct ec *ec, union iecpt *d, struct ecpt *p); |
| 657 | /* Convert external-format p to |
| 658 | * internal-format d. |
| 659 | */ |
| 660 | |
| 661 | void (*out)(struct ec *ec, struct ecpt *d, union iecpt *p); |
| 662 | /* Convert internal-format p to |
| 663 | * external-format d. |
| 664 | */ |
| 665 | |
| 666 | void (*add)(struct ec *ec, union iecpt *d, |
| 667 | union iecpt *p, union iecpt *q); |
| 668 | /* Store in d the sum of the points p |
| 669 | * and q. |
| 670 | */ |
| 671 | |
| 672 | void (*sub)(struct ec *ec, union iecpt *d, |
| 673 | union iecpt *p, union iecpt *q); |
| 674 | /* Store in d the result of |
| 675 | * subtracting q from p. |
| 676 | */ |
| 677 | |
| 678 | void (*neg)(struct ec *ec, union iecpt *d, union iecpt *p); |
| 679 | /* Store in d the negation of p. */ |
| 680 | }; |
| 681 | |
| 682 | /* And now some common utilities. */ |
| 683 | |
| 684 | static void weier_destroy(struct ec *ec) |
| 685 | { |
| 686 | ec->f->ops(ec->f, &ec->u.w.a); |
| 687 | ec->f->ops(ec->f, &ec->u.w.b); |
| 688 | freebn(ec->r); |
| 689 | f->ops->destroy(ec->f); |
| 690 | sfree(ec); |
| 691 | } |
| 692 | |
| 693 | static void jac_init(struct ec *ec, union iecpt *p) |
| 694 | { |
| 695 | ec->f->ops->init(ec->f, &p->proj.x); |
| 696 | ec->f->ops->init(ec->f, &p->proj.y); |
| 697 | ec->f->ops->init(ec->f, &p->proj.z); |
| 698 | } |
| 699 | |
| 700 | static void jac_free(struct ec *ec, union iecpt *p) |
| 701 | { |
| 702 | ec->f->ops->free(ec->f, &p->proj.x); |
| 703 | ec->f->ops->free(ec->f, &p->proj.y); |
| 704 | ec->f->ops->free(ec->f, &p->proj.z); |
| 705 | } |
| 706 | |
| 707 | static void jac_setinf(struct ec *ec, union iecpt *d) |
| 708 | { |
| 709 | ec->f->ops->copy(ec->f, &d->proj.x, &ec->f->one); |
| 710 | ec->f->ops->copy(ec->f, &d->proj.y, &ec->f->one); |
| 711 | ec->f->ops->copy(ec->f, &d->proj.z, &ec->f->zero); |
| 712 | } |
| 713 | |
| 714 | static int jac_infp(struct ec *ec, union iecpt *d) |
| 715 | { return ec->f->ops->zerop(ec->f, &d->proj.z); } |
| 716 | |
| 717 | static void jac_copy(struct ec *ec, union iecpt *d, union iecpt *d) |
| 718 | { |
| 719 | ec->f->ops->copy(ec->f, &d->proj.x, &p->proj.x); |
| 720 | ec->f->ops->copy(ec->f, &d->proj.y, &p->proj.y); |
| 721 | ec->f->ops->copy(ec->f, &d->proj.z, &p->proj.z); |
| 722 | } |
| 723 | |
| 724 | static void jac_in(struct ec *ec, union iecpt *d, struct ecpt *p) |
| 725 | { |
| 726 | ec->f->ops->copy(ec->f, &d->x, &p->proj.x); |
| 727 | ec->f->ops->copy(ec->f, &d->y, &p->proj.y); |
| 728 | ec->f->ops->copy(ec->f, &d->z, &ec->f->one); |
| 729 | } |
| 730 | |
| 731 | static void jac_out(struct ec *ec, struct ecpt *d, union iecpt *p) |
| 732 | { |
| 733 | union fieldelt iz, iz2, iz3; |
| 734 | |
| 735 | if (ec->f->ops->zerop(ec->f, &p->proj.z)) { |
| 736 | ec->f->ops->free(ec->f, d->x); |
| 737 | ec->f->ops->free(ec->f, d->y); |
| 738 | d->f = ECPTF_INF; |
| 739 | } else { |
| 740 | ec->f->ops->init(ec->f, &iz2); |
| 741 | ec->f->ops->init(ec->f, &iz3); |
| 742 | |
| 743 | ec->f->ops->inv(ec->f, &iz3, &p->proj.z); |
| 744 | ec->f->ops->sqr(ec->f, &iz2, &iz3); |
| 745 | ec->f->ops->mul(ec->f, &iz3, &iz3, &iz2); |
| 746 | |
| 747 | ec->f->ops->mul(ec->f, &d->x, &p->proj.x, &iz2); |
| 748 | ec->f->ops->mul(ec->f, &d->y, &p->proj.x, &iz3); |
| 749 | d->f = 0; |
| 750 | |
| 751 | ec->f->ops->free(ec->f, &iz2); |
| 752 | ec->f->ops->free(ec->f, &iz3); |
| 753 | } |
| 754 | } |
| 755 | |
| 756 | static void gen_sub(struct ec *ec, union iecpt *d, |
| 757 | union iecpt *p, union iecpt *q) |
| 758 | { |
| 759 | union iecpt t; |
| 760 | |
| 761 | ec->ops->init(ec, &t); |
| 762 | ec->ops->neg(ec, &t, q); |
| 763 | ec->ops->add(ec, d, p, &t); |
| 764 | ec->ops->free(ec, &t); |
| 765 | } |
| 766 | |
| 767 | /* Finally, let's define some actual curve arithmetic functions. */ |
| 768 | |
| 769 | static void pwjac_dbl(struct ec *ec, union iecpt *d, union iecpt *p) |
| 770 | { |
| 771 | struct field *f = ec->f; |
| 772 | union fieldelt m, s, t, u; |
| 773 | |
| 774 | if (ec->ops->infp(ec, p)) { |
| 775 | ec->setinf(ec, d); |
| 776 | return; |
| 777 | } |
| 778 | |
| 779 | f->ops->init(f, &m); |
| 780 | f->ops->init(f, &s); |
| 781 | f->ops->init(f, &t); |
| 782 | f->ops->init(f, &u); |
| 783 | |
| 784 | f->ops->sqr(f, &u, &p->proj.z); /* z^2 */ |
| 785 | f->ops->sqr(f, &t, &u); /* z^4 */ |
| 786 | f->ops->mul(f, &u, &t, &ec->u.w.a); /* a z^4 */ |
| 787 | f->ops->sqr(f, &m, &p->proj.x); /* x^2 */ |
| 788 | f->ops->tpl(f, &m, &m); /* 3 x^2 */ |
| 789 | f->ops->add(f, &m, &m, &u); /* m = 3 x^2 + a z^4 */ |
| 790 | |
| 791 | f->ops->dbl(f, &t, &p->proj.y); /* 2 y */ |
| 792 | f->ops->mul(f, &d->proj.z, &t, &p->proj.z); /* z' = 2 y z */ |
| 793 | |
| 794 | f->ops->sqr(f, &u, &t); /* 4 y^2 */ |
| 795 | f->ops->mul(f, &s, &u, &p->proj.x); /* s = 4 x y^2 */ |
| 796 | f->ops->sqr(f, &t, &u); /* 16 y^4 */ |
| 797 | f->ops->hlv(f, &t, &t); /* t = 8 y^4 */ |
| 798 | |
| 799 | f->ops->dbl(f, &u, &s); /* 2 s */ |
| 800 | f->ops->sqr(f, &d->proj.x, &m); /* m^2 */ |
| 801 | f->ops->sub(f, &d->proj.x, &d->proj.x, &u); /* x' = m^2 - 2 s */ |
| 802 | |
| 803 | f->ops->sub(f, &u, &s, &d->proj.x); /* s - x' */ |
| 804 | f->ops->mul(f, &d->proj.y, &m, &u); /* m (s - x') */ |
| 805 | f->ops->sub(f, &d->proj.y, &d->proj.y, &t); /* y' = m (s - x') - t */ |
| 806 | |
| 807 | f->ops->free(f, &m); |
| 808 | f->ops->free(f, &s); |
| 809 | f->ops->free(f, &t); |
| 810 | f->ops->free(f, &u); |
| 811 | } |
| 812 | |
| 813 | static void pwjac_add(struct ec *ec, union iecpt *d, |
| 814 | union iecpt *p, union iecpt *q) |
| 815 | { |
| 816 | struct field *f = ec->f; |
| 817 | union fieldelt m, r, s, ss, t, u, uu, w; |
| 818 | |
| 819 | if (a == b) { pwjac_dbl(ec, d, p); return; } |
| 820 | else if (ec->ops->infp(ec, p)) { ec->ops->copy(ec, d, q); return; } |
| 821 | else if (ec->ops->infp(ec, q)) { ec->ops->copy(ec, d, p); return; } |
| 822 | |
| 823 | f->ops->init(f, &m); |
| 824 | f->ops->init(f, &r); |
| 825 | f->ops->init(f, &s); |
| 826 | f->ops->init(f, &ss); |
| 827 | f->ops->init(f, &t); |
| 828 | f->ops->init(f, &u); |
| 829 | f->ops->init(f, &uu); |
| 830 | f->ops->init(f, &w); |
| 831 | |
| 832 | f->ops->sqr(f, &s, &p->proj.z); /* z_0^2 */ |
| 833 | f->ops->mul(f, &u, &s, &q->proj.x); /* u = x_1 z_0^2 */ |
| 834 | f->ops->mul(f, &s, &s, &p->proj.z); /* z_0^3 */ |
| 835 | f->ops->mul(f, &s, &s, &q->proj.y); /* s = y_1 z_0^3 */ |
| 836 | |
| 837 | f->ops->sqr(f, &ss, &q->proj.z); /* z_1^2 */ |
| 838 | f->ops->mul(f, &uu, &ss, &p->proj.x); /* uu = x_0 z_1^2 */ |
| 839 | f->ops->mul(f, &ss, &ss, &q->proj.z); /* z_1^3 */ |
| 840 | f->ops->mul(f, &ss, &ss, &p->proj.y); /* ss = y_0 z_1^3 */ |
| 841 | |
| 842 | f->ops->sub(f, &w, &uu, &u); /* w = uu - u */ |
| 843 | f->ops->sub(f, &r, &ss, &s); /* r = ss - s */ |
| 844 | if (f->ops->zerop(f, w)) { |
| 845 | if (f->ops->zerop(f, r)) ec->ops->dbl(ec, d, p); |
| 846 | else ec->ops->setinf(ec, d); |
| 847 | goto cleanup; |
| 848 | } |
| 849 | |
| 850 | f->ops->add(f, &t, &u, &uu); /* t = uu + u */ |
| 851 | f->ops->add(f, &m, &s, &ss); /* m = ss + s */ |
| 852 | |
| 853 | f->ops->mul(f, &uu, &p->proj.z, &w); /* z_0 w */ |
| 854 | f->ops->mul(f, &d->proj.z, &uu, &q->proj.z); /* z' = z_0 z_1 w */ |
| 855 | |
| 856 | f->ops->sqr(f, &s, &w); /* w^2 */ |
| 857 | f->ops->mul(f, &u, &s, &t); /* t w^2 */ |
| 858 | f->ops->mul(f, &s, &s, &w); /* w^3 */ |
| 859 | f->ops->mul(f, &s, &s, &m); /* m w^3 */ |
| 860 | |
| 861 | f->ops->sqr(f, &m, &r); /* r^2 */ |
| 862 | f->ops->sub(f, &d->proj.x, &m, &u); /* x' = r^2 - t w^2 */ |
| 863 | |
| 864 | f->ops->dbl(f, &m, &d->proj.x); /* 2 x' */ |
| 865 | f->ops->sub(f, &u, &u, &m); /* v = t w^2 - 2 x' */ |
| 866 | f->ops->mul(f, &u, &u, &r); /* v r */ |
| 867 | f->ops->sub(f, &u, &u, &s); /* v r - m w^3 */ |
| 868 | f->ops->hlv(f, &d->proj.y, &m); /* y' = (v r - m w^3)/2 */ |
| 869 | |
| 870 | cleanup: |
| 871 | f->ops->free(f, &m); |
| 872 | f->ops->free(f, &r); |
| 873 | f->ops->free(f, &s); |
| 874 | f->ops->free(f, &ss); |
| 875 | f->ops->free(f, &t); |
| 876 | f->ops->free(f, &u); |
| 877 | f->ops->free(f, &uu); |
| 878 | f->ops->free(f, &w); |
| 879 | } |
| 880 | |