--- /dev/null
+/*
+ * Elliptic curve arithmetic.
+ */
+
+#include "bn-internal.h"
+#include "ssh.h"
+
+/* Some background.
+ *
+ * These notes are intended to help programmers who are mathematically
+ * inclined, but not well versed in the theory of elliptic curves. If more
+ * detail is required, introductory texts of various levels of sophistication
+ * are readily available. I found
+ *
+ * L. Washington, `Elliptic Curves: Number Theory and Cryptography', CRC
+ * Press, 2003
+ *
+ * useful. Further background on algebra and number theory can be found in
+ *
+ * V. Shoup, `A Computational Introduction to Number Theory and Algebra',
+ * version 2, Cambridge University Press, 2008; http://shoup.net/ntb/
+ * [CC-ND-NC].
+ *
+ * We work in a finite field k = GF(p^m). To make things easier, we'll only
+ * deal with the cases where p = 2 (`binary' fields) or m = 1 (`prime'
+ * fields). In fact, we don't currently implement binary fields either.
+ *
+ * We start in two-dimensional projective space over k, denoted P^2(k). This
+ * is the space of classes of triples of elements of k { (a x : a y : a z) |
+ * a in k^* } such that x y z /= 0. Projective points with z /= 0 are called
+ * `finite', and correspond to points in the more familiar two-dimensional
+ * affine space (x/z, y/z); points with z = 0 are called `infinite'.
+ *
+ * An elliptic curve is the set of projective points satisfying an equation
+ * of the form
+ *
+ * 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
+ *
+ * (the `Weierstrass form'), such that the partial derivatives don't both
+ * vanish at any point. (Notice that this is a homogeneous equation -- all
+ * of the monomials have the same degree -- so this is a well-defined subset
+ * of our projective space. Don't worry about the numbering of the
+ * coefficients: maybe the later stuff about Jacobian coordinates will make
+ * them clear.)
+ *
+ * Let's consider, for a moment, a straight line r x + s y + t z = 0 which
+ * meets the curve at two points P and Q; then they must also meet again at
+ * some other point R. (Here, any or all of P, Q, and R might be equal;
+ * in this case, the simultaneous equations have repeated solutions.) Let's
+ * go out on a limb and write
+ *
+ * P + Q + R = 0
+ *
+ * whenever three points P, Q, and R on the curve are collinear. Obviously
+ * this notion of `addition' is commutative: three points are collinear
+ * whichever order we write them. If we designate some point O as being an
+ * `identity' then we can introduce a notion of inverses. And then some
+ * magic occurs: this `addition' operation we've just invented is also
+ * associative -- so it's an abelian group operation! (This can be
+ * demonstrated by slogging through the algebra, but there's a fancy way
+ * involving the Weil pairing. Washington's book has both versions.)
+ *
+ * So we have an abelian group. It's actually quite a fun group, for
+ * complicated reasons I'm not going to do into here. For our purposes it's
+ * sufficient to note that, in most elliptic curve groups, the discrete
+ * logarithm problem -- finding x given x P -- seems really very hard. I
+ * mean, much harder than in multiplicative subgroups of finite fields,
+ * where index calculus techniques can give you the answer in subexponential
+ * time. The best algorithms known for solving this problem in elliptic
+ * curve groups are the ones which work in any old group G -- and they run in
+ * O(sqrt(#G)) time.
+ *
+ * Staring at the Weierstrass equation some more, we can determine that in
+ * fact there's exactly one infinite point on the curve: if z = 0 then we
+ * must have x = 0, so (0 : 1 : 0) is precisely that point. That's kind of
+ * handy.
+ *
+ * The Weierstrass form is pretty horrid, really. Fortunately, it's always
+ * possible to apply a linear transformation to make it less nasty. If p > 3
+ * (which it will be for the prime fields we care about), we can simplify
+ * this to
+ *
+ * y^2 z = x^3 + a x z^2 + b z^3 (with 4 a^3 + 27 b^2 /= 0)
+ *
+ * and in the binary case we can simplify to
+ *
+ * y^2 z + x y z = x^3 + a x^2 z + b z^3 (with b /= 0)
+ *
+ * and it's these kinds of forms which we're actually going to work with.
+ * (There are other forms of elliptic curves which have interesting benefits,
+ * e.g., the Montgomery and Edwards forms, but they've not really caught on
+ * in standards.) Linear transformations don't mess up the geometry, so the
+ * group we constructed above still works -- and, indeed, the transformation
+ * is an isomorphism of groups. And, finally, the point at infinity we found
+ * above is still infinite. Let's call that point O, and declare it to be
+ * our additive identity. That leaves the finite points (x : y : 1), which
+ * correspond to the affine points
+ *
+ * y^2 = x^3 + a x + b or y^2 + x y = x^3 + a x^2 + b
+ *
+ * So, if we want to add two points P and Q, we draw the line between them;
+ * if P = Q then the line in question is the tangent to the curve at P. (The
+ * smoothness condition above is precisely what we need for this tangent to
+ * be well-defined. Unsmooth curves aren't worth worrying about, because
+ * they're cryptographically useless: they're isomorphic to some other group
+ * in which discrete logs are very easy.) In prime fields, if this line is
+ * vertical then it meets the curve again at O, and P + Q = O; otherwise it
+ * meets the curve at some other finite point R = (x, y), and the answer we
+ * want is P + Q = -R = (x, -y). In binary fields, the situation is similar
+ * but a little messier because of the x y term we couldn't eliminate.
+ *
+ * It's not very hard to work out where this point R is. You get a cubic
+ * equation out the far end which looks kind of daunting until you notice
+ * that the negation of the x^2 coefficient is exactly the sum of the roots,
+ * and you already know two of them. Unfortunately, the solution is still
+ * somewhat unpleasant and involves a bunch of division -- and in finite
+ * fields that involves computing a modular inverse, which is rather slow.
+ *
+ * Wouldn't it be really nice if we could somehow keep the denominators
+ * separate, so we could just do a few `big' divisions at the end of some
+ * calculation? Of course, we'd need to tweak the point-addition formulae so
+ * that we could cope with input points which have separate denominators.
+ * And so we enter the realm of fun coordinate systems.
+ *
+ * Perhaps an obvious choice of coordinates is the projective form above:
+ * rather than working out (x/m, y/n), we could just keep track of a
+ * projective point (x : y : m n). It turns out that there's a slightly
+ * different variant which is a little better: `Jacobian coordinates' are
+ * classes of triples (x :: y :: z), where x y z /= 0, and the point (x :: y
+ * :: z) is equivalent to (a^2 x :: a^3 y :: a z).
+ *
+ * The (simplified) Wieierstrass equation, using Jacobian projective
+ * coordinates, looks like this.
+ *
+ * y^2 = x^3 + a x z^2 + b z^6
+ *
+ * or
+ *
+ * y^2 + x y z = x^3 + a x^2 z + b z^3
+ *
+ * (Maybe this is a hint about the strange coefficient numbering in the
+ * general Weierstrass form.) Anyway, if a point is infinite then we have
+ * y^2 = x^3, which has the obvious solution x = y = 1.
+ *
+ * That's about all the theory we need for now. Let's get into the code.
+ */
+
+/* Let's start by making an abstract interface for finite field arithmetic.
+ * If we need to do clever things later then this will be a convenient
+ * extension point (e.g., supporting binary fields, working in extension
+ * fields as we'd need if we wanted to implement Tate or Weil pairings, or
+ * just doing different kinds of optimizations for the prime case).
+ *
+ * This is quite cheesy, but it's enough for now.
+ */
+
+union fieldelt {
+ Bignum n;
+};
+
+struct field {
+ const struct field_ops *ops; /* Field operations table. */
+ union fieldelt zero, one; /* Identity elements. */
+ Bignum q; /* The field size. */
+};
+
+struct field_ops {
+ void (*destroy)(struct field *f); /* Destroy the field object. */
+
+ void (*init)(struct field *f, union fieldelt *x);
+ /* Initialize x (to null). */
+
+ void (*free)(struct field *f, union fieldelt *x);
+ /* Free an element x, leaving it
+ * null.
+ */
+
+ void (*copy)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Make d be a copy of x. */
+
+ void (*frombn)(struct field *f, union fieldelt *d, Bignum n);
+ /* Convert n to an element. */
+
+ Bignum (*tobn)(struct field *f, union fieldelt *x);
+ /* Convert x to an integer. */
+
+ int (*zerop)(struct field *f, union fieldelt *x);
+ /* Return nonzero if x is zero. */
+
+ void (*dbl)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Store 2 x in d. */
+
+ void (*add)(struct field *f, union fieldelt *d,
+ union fieldelt *x, union fieldelt *y);
+ /* Store x + y in d. */
+
+ void (*sub)(struct field *f, union fieldelt *d,
+ union fieldelt *x, union fieldelt *y);
+ /* Store x - y in d. */
+
+ void (*mul)(struct field *f, union fieldelt *d,
+ union fieldelt *x, union fieldelt *y);
+ /* Store x y in d. */
+
+ void (*sqr)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Store x^2 in d. */
+
+ void (*inv)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Store 1/x in d; it is an error
+ * for x to be zero.
+ */
+
+ int (*sqrt)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* If x is a square, store a square
+ * root (chosen arbitrarily) in d
+ * and return nonzero; otherwise
+ * return zero leaving d unchanged.
+ */
+
+ /* For internal use only. The standard methods call on these for
+ * detailed field-specific behaviour.
+ */
+ void (*_reduce)(struct field *f, union fieldelt *x);
+ /* Reduce an intermediate result so
+ * that it's within whatever bounds
+ * are appropriate.
+ */
+
+ /* These functions are only important for prime fields. */
+ void (*dbl)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Store 2 x in d. */
+
+ void (*tpl)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Store 3 x in d. */
+
+ void (*dql)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Store 4 x in d. */
+
+ void (*hlv)(struct field *f, union fieldelt *d, union fieldelt *x);
+ /* Store x/2 in d. */
+};
+
+/* Some standard methods for prime fields. */
+
+static void gen_destroy(struct field *f)
+ { sfree(f); }
+
+static void prime_init(struct field *f, union fieldelt *x)
+ { x->n = 0; }
+
+static void prime_free(struct field *f, union fieldelt *x)
+ { if (x->n) { freebn(x->n); x->n = 0; } }
+
+static void prime_frombn(struct field *f, union fieldelt *d, Bignum n)
+ { f->ops->free(f, d); d->n = copybn(n); }
+
+static Bignum prime_tobn(struct field *f, union fieldelt *x)
+ { return copybn(x->n); }
+
+static Bignum prime_zerop(struct field *f, union fieldelt *x)
+ { return bignum_cmp(x->n, f->zero.n) == 0; }
+
+static void prime_add(struct field *f, union fieldelt *d,
+ union fieldelt *x, union fieldelt *y)
+{
+ Bignum sum = bigadd(x, y);
+ Bignum red = bigsub(sum, f->q);
+
+ f->ops->free(f, d);
+ if (!red)
+ d->n = sum;
+ else {
+ freebn(sum);
+ d->n = red;
+ }
+}
+
+static void prime_sub(struct field *f, union fieldelt *d,
+ union fieldelt *x, union fieldelt *y)
+{
+ Bignum raise = bigadd(x, f->q);
+ Bignum diff = bigsub(raise, y);
+ Bignum drop = bigsub(diff, f->q);
+
+ f->ops->free(f, d); freebn(raise);
+ if (!drop)
+ d->n = diff;
+ else {
+ freebn(diff);
+ d->n = drop;
+ }
+}
+
+static void prime_mul(struct field *f, union fieldelt *d,
+ union fieldelt *x, union fieldelt *y)
+{
+ Bignum prod = bigmul(x, y);
+
+ f->ops->free(f, d);
+ d->n = prod;
+ f->ops->_reduce(f, d);
+}
+
+static void gen_sqr(struct field *f, union fieldelt *d, union fieldelt *x)
+ { f->ops->mul(f->ops, d, x, x); }
+
+static void prime_inv(struct field *f, union fieldelt *d, union fieldelt *x)
+{
+ Bignum inv = modinv(x->n, f->q);
+
+ f->ops->free(f, d);
+ d->n = inv;
+}
+
+static int prime_sqrt(struct field *f, union fieldelt *d, union fieldelt *x)
+{
+ Bignum sqrt = modsqrt(x->n, f->q);
+
+ if (!sqrt) return 0;
+ f->ops->free(f, d);
+ d->n = sqrt;
+ return 1;
+}
+
+static void prime_dbl(struct field *f, union_fieldelt *d, union fieldelt *x)
+ { f->ops->add(f, d, x, x); }
+
+static void prime_tpl(struct field *f, union_fieldelt *d, union fieldelt *x)
+{
+ union_fieldelt t;
+
+ if (d != x) {
+ f->ops->add(f, d, x, x);
+ f->ops->add(f, d, d, x);
+ } else {
+ f->ops->init(f, &t);
+ f->ops->copy(f, d, x);
+ f->ops->add(f, d, &t, &t);
+ f->ops->add(f, d, d, &t);
+ f->ops->free(f, &t);
+ }
+}
+
+static void prime_qdl(struct field *f, union fieldelt *d,
+ union fieldelt *x)
+ { f->ops->add(f, d, x, x); f->ops->add(f, d, d, d); }
+
+static void prime_hlv(struct field *f, union fieldelt *d, union fieldelt *x)
+{
+ Bignum t, u;
+
+ if (!x->n[0])
+ f->ops->copy(f, d, x);
+ else {
+ /* The tedious answer is to multiply by the inverse of 2 in this
+ * field. But there's a better way: either x or q - x is actually
+ * divisible by 2, so the answer we want is either x >> 1 or p -
+ * ((p - x) >> 1) depending on whether x is even.
+ */
+ if (!(x->n[1] & 1)) {
+ t = copybn(x->n);
+ shift_right(t + 1, t[0], 1);
+ } else {
+ u = bigsub(f->q, x);
+ shift_right(u + 1, u[0], 1);
+ t = bigsub(f->q, u);
+ freebn(u);
+ }
+ f->ops->free(f, d);
+ d->n = t;
+}
+
+/* Now some utilities for modular reduction. All of the primes we're
+ * concerned about are of the form p = 2^n - d for `convenient' values of d
+ * -- such numbers are called `pseudo-Mersenne primes'. Suppose we're given
+ * a value x = x_0 2^n + x_1: then x - x_0 p = x_1 + x_0 d is clearly less
+ * than x and congruent to it mod p. So we can reduce x below 2^n by
+ * iterating this trick; if we're unlucky enough to have p <= x < 2^n then we
+ * can just subtract p.
+ *
+ * The values of d we'll be dealing with are specifically convenient because
+ * they satisfy the following properties.
+ *
+ * * We can write d = SUM_i d_i 2^i, with d_i in { -1, 0, +1 }. (This is
+ * no surprise.)
+ *
+ * * d > 0.
+ *
+ * * Very few d_i /= 0. With one small exception, if d_i /= 0 then i is
+ * divisible by 32.
+ *
+ * The following functions will therefore come in very handy.
+ */
+
+static void add_imm_lsl(BignumInt *x, BignumInt y, unsigned shift)
+{
+ BignumDbl c = y << (shift % BIGNUM_INT_BITS);
+
+ for (x += shift/BIGNUM_INT_BITS; c; x++) {
+ c += *x;
+ *x = c & BIGNUM_INT_MASK;
+ c >>= BIGNUM_INT_BITS;
+ }
+}
+
+static void sub_imm_lsl(BignumInt *x, BignumInt y, unsigned shift)
+{
+ BignumDbl c = y << (shift % BIGNUM_INT_BITS);
+ BignumDbl t;
+
+ c--;
+ for (x += shift/BIGNUM_INT_BITS; c; x++) {
+ c ^= BIGNUM_INT_MASK;
+ c += *x;
+ *x = c & BIGNUM_INT_MASK;
+ c >>= BIGNUM_INT_BITS;
+ }
+}
+
+static void pmp_reduce(struct field *f, union fieldelt *xx,
+ void (*add_mul_d_lsl)(BignumInt *x,
+ BignumInt y,
+ unsigned shift))
+{
+ /* Notation: write w = BIGNUM_INT_WORDS, and B = 2^w is the base we're
+ * working in. We assume that p = 2^n - d for some d. The job of
+ * add_mul_d_lsl is to add to its argument x the value y d 2^shift.
+ */
+
+ BignumInt top, *p, *x;
+ unsigned plen, xlen, i;
+ unsigned topbits, topclear;
+ Bignum t;
+
+ /* Initialization: if x is shorter than p then there's nothing to do.
+ * Otherwise, ensure that it's at least one word longer, since this is
+ * required by the utility functions which add_mul_d_lsl will call.
+ */
+ p = f->q + 1; plen = f->q[0];
+ x = xx->n + 1; xlen = xx->n[0];
+ if (xlen < plen) return;
+ else if (xlen == plen) {
+ t = newbn(plen + 1);
+ for (i = 0; i < plen; i++) t[i + 1] = x[i];
+ t[plen] = 0;
+ freebn(xx->n); xx->n = t;
+ x = t + 1; xlen = plen + 1;
+ }
+
+ /* Preparation: Work out the bit length of p. At the end of this, we'll
+ * have topbits such that n = w (plen - 1) + topbits, and topclear = w -
+ * topbits, so that n = w plen - topclear.
+ */
+ while (plen > 0 && !p[plen - 1]) plen--;
+ top = p[plen - 1];
+ assert(top);
+ if (top & BIGNUM_TOP_BIT) topbits = BIGNUM_INT_BITS;
+ else for (topbits = 0; top >> topbits; topbits++);
+ topclear = BIGNUM_INT_BITS - topbits;
+
+ /* Step 1: Trim x down to the right number of words. */
+ while (xlen > plen) {
+
+ /* Pick out the topmost word; call it y for now. */
+ y = x[xlen - 1];
+ if (!y) {
+ xlen--;
+ continue;
+ }
+
+ /* Observe that d == 2^n (mod p). Clear that top word. This
+ * effectively subtracts y B^{xlen-1} from x, so we must add back d
+ * 2^{w(xlen-1)-n} in order to preserve congruence modulo p. Since d
+ * is smaller than 2^n, this will reduce the absolute value of x, so
+ * we make (at least some) progress. In practice, we expect that d
+ * is a lot smaller.
+ *
+ * Note that w (xlen - 1) - n = w xlen - w - n = w (xlen - plen - 1)
+ * + topclear.
+ */
+ x[xlen - 1] = 0;
+ add_mul_d_lsl(x + xlen - plen - 1, y, topclear);
+ }
+
+ /* Step 2: Trim off the high bits of x, beyond the top bit of p. In more
+ * detail: if x >= 2^n > p, then write x = x_0 + 2^n y; then we can clear
+ * the top topclear bits of x, and add y d to preserve congruence mod p.
+ *
+ * If topclear is zero then there's nothing to do here: step 1 will
+ * already have arranged that x < 2^n.
+ */
+ if (topclear) {
+ for (;;) {
+ y = x[xlen - 1] >> topbits;
+ if (!y) break;
+ x[xlen - 1] &= (1 << topbits) - 1;
+ add_mul_d_lsl(x, y, 0);
+ }
+ }
+
+ /* Step 3: If x >= p then subtract p from x. */
+ for (i = plen; i-- && x[i] == p[i]; );
+ if (i >= plen || x[i] > p[i]) internal_sub(x, p, plen);
+
+ /* We're done now. */
+ d->n[0] = xlen;
+ bn_restore_invariant(d->n);
+}
+
+/* Putting together a field-ops structure for a pseudo-Mersenne prime field.
+ */
+
+static void pmp_destroy(struct field *f)
+ { freebn(f->q); sfree(f); }
+
+#define DEFINE_PMP_FIELD(fname) \
+static const struct field_ops fname##_fieldops { \
+ pmp_destroy, \
+ prime_init, prime_free, prime_copy, \
+ prime_frombn, prime_tobn, \
+ prime_zerop, \
+ prime_add, prime_sub, prime_mul, gen_sqr, prime_inv, prime_sqrt, \
+ fname_reduce, \
+ gen_dbl, gen_tpl, gen_qdl, \
+ prime_hlv \
+}; \
+ \
+static struct field *make_##fname##_field(void) \
+{ \
+ struct field *f = snew(struct field); \
+ f->ops = &fname##_fieldops; \
+ f->zero.n = Zero; \
+ f->one.n = One; \
+ f->q = bigunm_from_bytes(fname##_p, sizeof(fname##_p)); \
+ return f; \
+}
+
+/* Some actual field definitions. */
+
+static const unsigned char p256_p[] = {
+ 0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,
+ 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff
+};
+
+static void p256_add_mul_d_lsl(BignumInt *x, BignumInt y, unsigned shift)
+{
+ /* p_{256} = 2^{256} - 2^{224} + 2^{192} + 2^{96} - 1 */
+ add_imm_lsl(x, y, shift + 224);
+ sub_imm_lsl(x, y, shift + 192);
+ sub_imm_lsl(x, y, shift + 96);
+ add_imm_lsl(x, y, shift + 0);
+}
+
+static void p256_reduce(struct field *f, union fieldelt *d)
+ { pmp_reduce(f, d, p256_add_mul_d_lsl); }
+
+DEFINE_PMP_FIELD(p256)
+
+static const unsigned char p384_p[] = {
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe,
+ 0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x00,
+ 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff
+};
+
+static void p384_add_mul_d_lsl(BignumInt *x, BignumInt y, unsigned shift)
+{
+ /* p_{384} = 2^{384} - 2^{128} - 2^{96} + 2^{32} - 1 */
+ add_imm_lsl(x, y, shift + 128);
+ add_imm_lsl(x, y, shift + 96);
+ sub_imm_lsl(x, y, shift + 32);
+ add_imm_lsl(x, y, shift + 0);
+}
+
+static void p384_reduce(struct field *f, union fieldelt *d)
+ { pmp_reduce(f, d, p384_add_mul_d_lsl); }
+
+DEFINE_PMP_FIELD(p384)
+
+static const unsigned char p521_p[] = {
+ 0x01, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
+ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff
+};
+
+static void p521_add_mul_d_lsl(BignumInt *x, BignumInt y, unsigned shift)
+ { add_imm_lsl(x, y, shift); } /* p_{521} = 2^{521} - 1 */
+
+static void p521_reduce(struct field *f, union fieldelt *d)
+ { pmp_reduce(f, d, p521_add_mul_d_lsl); }
+
+DEFINE_PMP_FIELD(p521)
+
+/* And now, some actual elliptic curves.
+ *
+ * As with fields, we start by defining an abstract interface which may have
+ * various implementations (e.g., different algorithms with different
+ * performance properties, or based on different kinds of underlying fields.
+ */
+
+/* An external-form elliptic curve point. */
+struct ecpt {
+ unsigned f; /* Flags */
+#define ECPTF_INF 1u /* Point is at infinity */
+ union fieldelt x, y; /* x- and y-coordinates */
+};
+
+/* An internal-form elliptic curve point. This is where most of the action
+ * happens.
+ */
+union iecpt {
+ struct {
+ union fieldelt x, y, z; /* Projective-form triple. */
+ } jac; /* Jacobian projective coords. */
+};
+
+struct ec {
+ const struct ec_ops *ops; /* Curve operations table. */
+ struct field *f; /* The underlying finite field. */
+ struct iecpt p; /* The well-known generator point */
+ Bignum r; /* The size of the generated group */
+ union {
+ struct { union fieldelt a, b; } w; /* Simple Weierstrass coeffs. */
+ } u;
+};
+
+struct ec_ops {
+ void (*destroy)(struct ec *ec); /* Destroy the curve itself. */
+
+ void (*init)(struct ec *ec, union iecpt *p);
+ /* Initialize p (to null). */
+
+ void (*free)(struct ec *ec, union iecpt *p);
+ /* Free the point p. */
+
+ void (*setinf)(struct ec *ec, union iecpt *p);
+ /* Set p to be the point at infinity.
+ */
+
+ int (*infp)(struct ec *ec, union iecpt *p);
+ /* Answer whether p is infinite. */
+
+ void (*copy)(struct ec *ec, union iecpt *d, union iecpt *d);
+ /* Make p be a copy of d. */
+
+ void (*in)(struct ec *ec, union iecpt *d, struct ecpt *p);
+ /* Convert external-format p to
+ * internal-format d.
+ */
+
+ void (*out)(struct ec *ec, struct ecpt *d, union iecpt *p);
+ /* Convert internal-format p to
+ * external-format d.
+ */
+
+ void (*add)(struct ec *ec, union iecpt *d,
+ union iecpt *p, union iecpt *q);
+ /* Store in d the sum of the points p
+ * and q.
+ */
+
+ void (*sub)(struct ec *ec, union iecpt *d,
+ union iecpt *p, union iecpt *q);
+ /* Store in d the result of
+ * subtracting q from p.
+ */
+
+ void (*neg)(struct ec *ec, union iecpt *d, union iecpt *p);
+ /* Store in d the negation of p. */
+};
+
+/* And now some common utilities. */
+
+static void weier_destroy(struct ec *ec)
+{
+ ec->f->ops(ec->f, &ec->u.w.a);
+ ec->f->ops(ec->f, &ec->u.w.b);
+ freebn(ec->r);
+ f->ops->destroy(ec->f);
+ sfree(ec);
+}
+
+static void jac_init(struct ec *ec, union iecpt *p)
+{
+ ec->f->ops->init(ec->f, &p->proj.x);
+ ec->f->ops->init(ec->f, &p->proj.y);
+ ec->f->ops->init(ec->f, &p->proj.z);
+}
+
+static void jac_free(struct ec *ec, union iecpt *p)
+{
+ ec->f->ops->free(ec->f, &p->proj.x);
+ ec->f->ops->free(ec->f, &p->proj.y);
+ ec->f->ops->free(ec->f, &p->proj.z);
+}
+
+static void jac_setinf(struct ec *ec, union iecpt *d)
+{
+ ec->f->ops->copy(ec->f, &d->proj.x, &ec->f->one);
+ ec->f->ops->copy(ec->f, &d->proj.y, &ec->f->one);
+ ec->f->ops->copy(ec->f, &d->proj.z, &ec->f->zero);
+}
+
+static int jac_infp(struct ec *ec, union iecpt *d)
+ { return ec->f->ops->zerop(ec->f, &d->proj.z); }
+
+static void jac_copy(struct ec *ec, union iecpt *d, union iecpt *d)
+{
+ ec->f->ops->copy(ec->f, &d->proj.x, &p->proj.x);
+ ec->f->ops->copy(ec->f, &d->proj.y, &p->proj.y);
+ ec->f->ops->copy(ec->f, &d->proj.z, &p->proj.z);
+}
+
+static void jac_in(struct ec *ec, union iecpt *d, struct ecpt *p)
+{
+ ec->f->ops->copy(ec->f, &d->x, &p->proj.x);
+ ec->f->ops->copy(ec->f, &d->y, &p->proj.y);
+ ec->f->ops->copy(ec->f, &d->z, &ec->f->one);
+}
+
+static void jac_out(struct ec *ec, struct ecpt *d, union iecpt *p)
+{
+ union fieldelt iz, iz2, iz3;
+
+ if (ec->f->ops->zerop(ec->f, &p->proj.z)) {
+ ec->f->ops->free(ec->f, d->x);
+ ec->f->ops->free(ec->f, d->y);
+ d->f = ECPTF_INF;
+ } else {
+ ec->f->ops->init(ec->f, &iz2);
+ ec->f->ops->init(ec->f, &iz3);
+
+ ec->f->ops->inv(ec->f, &iz3, &p->proj.z);
+ ec->f->ops->sqr(ec->f, &iz2, &iz3);
+ ec->f->ops->mul(ec->f, &iz3, &iz3, &iz2);
+
+ ec->f->ops->mul(ec->f, &d->x, &p->proj.x, &iz2);
+ ec->f->ops->mul(ec->f, &d->y, &p->proj.x, &iz3);
+ d->f = 0;
+
+ ec->f->ops->free(ec->f, &iz2);
+ ec->f->ops->free(ec->f, &iz3);
+ }
+}
+
+static void gen_sub(struct ec *ec, union iecpt *d,
+ union iecpt *p, union iecpt *q)
+{
+ union iecpt t;
+
+ ec->ops->init(ec, &t);
+ ec->ops->neg(ec, &t, q);
+ ec->ops->add(ec, d, p, &t);
+ ec->ops->free(ec, &t);
+}
+
+/* Finally, let's define some actual curve arithmetic functions. */
+
+static void pwjac_dbl(struct ec *ec, union iecpt *d, union iecpt *p)
+{
+ struct field *f = ec->f;
+ union fieldelt m, s, t, u;
+
+ if (ec->ops->infp(ec, p)) {
+ ec->setinf(ec, d);
+ return;
+ }
+
+ f->ops->init(f, &m);
+ f->ops->init(f, &s);
+ f->ops->init(f, &t);
+ f->ops->init(f, &u);
+
+ f->ops->sqr(f, &u, &p->proj.z); /* z^2 */
+ f->ops->sqr(f, &t, &u); /* z^4 */
+ f->ops->mul(f, &u, &t, &ec->u.w.a); /* a z^4 */
+ f->ops->sqr(f, &m, &p->proj.x); /* x^2 */
+ f->ops->tpl(f, &m, &m); /* 3 x^2 */
+ f->ops->add(f, &m, &m, &u); /* m = 3 x^2 + a z^4 */
+
+ f->ops->dbl(f, &t, &p->proj.y); /* 2 y */
+ f->ops->mul(f, &d->proj.z, &t, &p->proj.z); /* z' = 2 y z */
+
+ f->ops->sqr(f, &u, &t); /* 4 y^2 */
+ f->ops->mul(f, &s, &u, &p->proj.x); /* s = 4 x y^2 */
+ f->ops->sqr(f, &t, &u); /* 16 y^4 */
+ f->ops->hlv(f, &t, &t); /* t = 8 y^4 */
+
+ f->ops->dbl(f, &u, &s); /* 2 s */
+ f->ops->sqr(f, &d->proj.x, &m); /* m^2 */
+ f->ops->sub(f, &d->proj.x, &d->proj.x, &u); /* x' = m^2 - 2 s */
+
+ f->ops->sub(f, &u, &s, &d->proj.x); /* s - x' */
+ f->ops->mul(f, &d->proj.y, &m, &u); /* m (s - x') */
+ f->ops->sub(f, &d->proj.y, &d->proj.y, &t); /* y' = m (s - x') - t */
+
+ f->ops->free(f, &m);
+ f->ops->free(f, &s);
+ f->ops->free(f, &t);
+ f->ops->free(f, &u);
+}
+
+static void pwjac_add(struct ec *ec, union iecpt *d,
+ union iecpt *p, union iecpt *q)
+{
+ struct field *f = ec->f;
+ union fieldelt m, r, s, ss, t, u, uu, w;
+
+ if (a == b) { pwjac_dbl(ec, d, p); return; }
+ else if (ec->ops->infp(ec, p)) { ec->ops->copy(ec, d, q); return; }
+ else if (ec->ops->infp(ec, q)) { ec->ops->copy(ec, d, p); return; }
+
+ f->ops->init(f, &m);
+ f->ops->init(f, &r);
+ f->ops->init(f, &s);
+ f->ops->init(f, &ss);
+ f->ops->init(f, &t);
+ f->ops->init(f, &u);
+ f->ops->init(f, &uu);
+ f->ops->init(f, &w);
+
+ f->ops->sqr(f, &s, &p->proj.z); /* z_0^2 */
+ f->ops->mul(f, &u, &s, &q->proj.x); /* u = x_1 z_0^2 */
+ f->ops->mul(f, &s, &s, &p->proj.z); /* z_0^3 */
+ f->ops->mul(f, &s, &s, &q->proj.y); /* s = y_1 z_0^3 */
+
+ f->ops->sqr(f, &ss, &q->proj.z); /* z_1^2 */
+ f->ops->mul(f, &uu, &ss, &p->proj.x); /* uu = x_0 z_1^2 */
+ f->ops->mul(f, &ss, &ss, &q->proj.z); /* z_1^3 */
+ f->ops->mul(f, &ss, &ss, &p->proj.y); /* ss = y_0 z_1^3 */
+
+ f->ops->sub(f, &w, &uu, &u); /* w = uu - u */
+ f->ops->sub(f, &r, &ss, &s); /* r = ss - s */
+ if (f->ops->zerop(f, w)) {
+ if (f->ops->zerop(f, r)) ec->ops->dbl(ec, d, p);
+ else ec->ops->setinf(ec, d);
+ goto cleanup;
+ }
+
+ f->ops->add(f, &t, &u, &uu); /* t = uu + u */
+ f->ops->add(f, &m, &s, &ss); /* m = ss + s */
+
+ f->ops->mul(f, &uu, &p->proj.z, &w); /* z_0 w */
+ f->ops->mul(f, &d->proj.z, &uu, &q->proj.z); /* z' = z_0 z_1 w */
+
+ f->ops->sqr(f, &s, &w); /* w^2 */
+ f->ops->mul(f, &u, &s, &t); /* t w^2 */
+ f->ops->mul(f, &s, &s, &w); /* w^3 */
+ f->ops->mul(f, &s, &s, &m); /* m w^3 */
+
+ f->ops->sqr(f, &m, &r); /* r^2 */
+ f->ops->sub(f, &d->proj.x, &m, &u); /* x' = r^2 - t w^2 */
+
+ f->ops->dbl(f, &m, &d->proj.x); /* 2 x' */
+ f->ops->sub(f, &u, &u, &m); /* v = t w^2 - 2 x' */
+ f->ops->mul(f, &u, &u, &r); /* v r */
+ f->ops->sub(f, &u, &u, &s); /* v r - m w^3 */
+ f->ops->hlv(f, &d->proj.y, &m); /* y' = (v r - m w^3)/2 */
+
+cleanup:
+ f->ops->free(f, &m);
+ f->ops->free(f, &r);
+ f->ops->free(f, &s);
+ f->ops->free(f, &ss);
+ f->ops->free(f, &t);
+ f->ops->free(f, &u);
+ f->ops->free(f, &uu);
+ f->ops->free(f, &w);
+}
+
projects / u / mdw / putty / commitdiff