/*----- Header files ------------------------------------------------------*/
+#include <mLib/darray.h>
+
#include "ec.h"
#include "ectab.h"
#include "gf.h"
+#include "keysz.h"
+#include "mpbarrett.h"
#include "pgen.h"
+#include "primeiter.h"
#include "mprand.h"
#include "mpint.h"
#include "rabin.h"
+/*----- Embedding degree checking -----------------------------------------*
+ *
+ * Let %$q = p^m$% be a prime power, and let %$E$% be an elliptic curve over
+ * %$\gf{q}$% with %$n = \#E(\gf{q}) = r h$% where %$r$% is prime. Then the
+ * Weil and Tate pairings can be used to map %$r$%-torsion points on
+ * %$E(\gf{q})$% onto the %$r$%-th roots of unity (i.e., the order-%$r$%
+ * subgroup) in an extension field %$\gf{p^k}$% of %$\gf{p}$% (%%\emph{not}%%
+ * of %$\gf{q}$% -- see [Hitt]). We call the smallest such %$k$% the
+ * %%\emph{embedding degree}%% of the curve %$E$%. The
+ * Menezes-Okamoto-Vanstone (MOV) attack solves the discrete log problem in
+ * %$E(\gf{q})$% by using the pairing and then applying index calculus to
+ * extract a discrete log in %$\gf{p^k}$%; obviously this only works if %$k$%
+ * is small enough.
+ *
+ * The usual check, suggested in, e.g., [P1363] or [SEC1], only covers
+ * extension fields %$\gf{q^\ell}$% of %$\gf{q}$%, which is fine when %$q$%
+ * is prime, but when we're dealing with binary fields it works less well.
+ * Indeed, as [Hitt] demonstrates, the embedding field can actually be
+ * %%\emph{smaller}%% than %$\gf{q}$%, and choosing %$m$% prime doesn't help
+ * (even though I previously thought it did).
+ *
+ * Define the %%\emph{embedding degree bound}%% %$B$% to be the smallest
+ * %$i$% such that discrete logs in %$\gf{p^i}$% are about as hard as in
+ * %$E(\gf{q})$%.
+ *
+ * The embedding group is a subgroup of the multiplicative group
+ * %$\gf{p^k}^*$% which contains %$p^k - 1$% elements; therefore we must have
+ * %$r \mid p^k - 1$%, or, equivalently, %$p^k \equiv 1 \pmod{r}$%.
+ *
+ * The recommended checking procedure, e.g., in [P1363], is just to check
+ * %$q^i \not\equiv 1 \pmod{r}$% for each %$0 < i < B$%. This is fast when
+ * you only consider extension fields of %$\gf{q}$%, since %$B$% is at most
+ * about 27. However, as noted above, this is inadequate when %$q$% is a
+ * prime power, and we must check all the extension fields of %$p$%. Now
+ * %$B$% can be about 15000, which is rather scarier -- we need a better
+ * algorithm.
+ *
+ * As noted, we must have %$p^k \equiv 1 \pmod{r}$%; but by minimality of
+ * %$k$%, we must have %$p^i \not\equiv 1 \pmod{r}$% for %$0 < i < k$%.
+ * Therefore %$p$% generates an order-%$k$% subgroup in %$\gf{r}^*$%, so we
+ * must have %$k \mid r - 1$%.
+ *
+ * Of course, factoring %$r - 1$% is a mug's game; but we're not interested
+ * in the complete factorization -- just the %$B$%-smooth portion. An
+ * algorithm suggests itself:
+ *
+ * 1. Extract the factors of %$r - 1$% which are less than %$B$%.
+ *
+ * 2. For each divisor %$d$% of %$r - 1$% less than %$B$% (which we can
+ * construct using this factorization), make sure that
+ * %$p^d \not\equiv 1 \pmod{r}$%.
+ *
+ * This takes a little while but not ever-so long.
+ *
+ * This is enough for cryptosystems based on the computational Diffie-
+ * Hellman problem to be secure. However, it's %%\emph{not}%% enough for the
+ * %%\emph{decisional}%% Diffie-Hellman problem to be hard; it appears we
+ * also need to hope that there aren't any suitable distortion maps with
+ * which one can solve the DDH problem. I don't know how to check for those
+ * at the moment.
+ *
+ * We'll take the subgroup order as indicative of the security level actually
+ * wanted. Then, to ensure security against the MOV attack, we must ensure
+ * that the embedding degree is sufficiently large that discrete logs in
+ * %$\gf{q^m}$% are at least as hard as discrete logs over the curve.
+ *
+ * We actually allow a small amount of slop in the conversions, in order to
+ * let people pick nice round numbers for their key lengths.
+ *
+ * References:
+ *
+ * [Hitt] L. Hitt, On an improved definition of embedding degree;
+ * http://eprint.iacr.org/2006/415
+ *
+ * [P1363] IEEE 1363-2000: Standard Specifications for Public Key
+ * Cryptography; http://grouper.ieee.org/groups/1363/P1363/index.html
+ *
+ * [SEC1] SEC 1: Elliptic Curve Cryptography;
+ * http://www.secg.org/download/aid-385/sec1_final.pdf
+ */
+
+/* --- @movcheck@ --- *
+ *
+ * Arguments: @mp *r@ = curve subgroup order
+ * @mp *p@ = field characteristic
+ * @unsigned long B@ = embedding degree bound
+ *
+ * Returns: Zero if OK, nonzero if an embedding was found.
+ *
+ * Use: Checks a curve for embeddings with degree less than the
+ * stated bound %$B$%. See above for explanation and a
+ * description of the algorithm.
+ */
+
+static int movcheck(mp *r, mp *p, unsigned long B)
+{
+ mpmont mm;
+ mp *r1, *pp = MP_NEW, *t = MP_NEW, *u = MP_NEW, *v = MP_NEW, *tt;
+ struct factor {
+ unsigned long f;
+ unsigned c, e;
+ };
+ DA_DECL(factor_v, struct factor);
+ factor_v fv = DA_INIT;
+ size_t nf;
+ struct factor *ff;
+ primeiter pi;
+ mp *BB;
+ unsigned long d, f;
+ unsigned i, j;
+ int rc = 0;
+
+ /* --- Special case --- *
+ *
+ * If %$r = 2$% then (a) Montgomery reduction won't work, and (b) we have
+ * no security worth checking anyway. Otherwise we're guaranteed that
+ * %$r$% is a prime, so it must be odd.
+ */
+
+ if (MP_EQ(r, MP_TWO))
+ return (0);
+
+ /* --- First factor the %$B%-smooth portion of %$r - 1$% --- *
+ *
+ * We can generate prime numbers up to %$B$% efficiently, so trial division
+ * it is.
+ */
+
+ BB = mp_fromulong(MP_NEW, B);
+ r1 = mp_sub(MP_NEW, r, MP_ONE);
+ primeiter_create(&pi, 0);
+ for (;;) {
+ pp = primeiter_next(&pi, pp);
+ if (MP_CMP(pp, >, BB))
+ break;
+ mp_div(&u, &v, r1, pp);
+ if (!MP_ZEROP(v))
+ continue;
+ i = 0;
+ do {
+ tt = r1; r1 = u; u = tt; i++;
+ mp_div(&u, &v, r1, pp);
+ } while (MP_ZEROP(v));
+ DA_ENSURE(&fv, 1);
+ DA_UNSAFE_EXTEND(&fv, 1);
+ DA_LAST(&fv).f = mp_toulong(pp);
+ DA_LAST(&fv).e = i;
+ DA_LAST(&fv).c = 0;
+ }
+ MP_DROP(BB); MP_DROP(pp); primeiter_destroy(&pi);
+ nf = DA_LEN(&fv); ff = DA(&fv);
+
+ /* --- Now generate divisors of %$r - 1$% less than %$B$% --- *
+ *
+ * For each divisor %$d$%, check whether %$p^d \equiv 1 \pmod{r}$%.
+ */
+
+ mpmont_create(&mm, r);
+ u = mpmont_mul(&mm, u, p, mm.r2);
+ for (;;) {
+
+ /* --- Construct the divisor --- */
+
+ d = 1;
+ for (i = 0; i < nf; i++) {
+ f = ff[i].f; j = ff[i].c; if (!j) continue;
+ for (;;) {
+ if (f >= (B + d - 1)/d) goto toobig;
+ if (j & 1) d *= f;
+ j >>= 1; if (!j) break;
+ f *= f;
+ }
+ }
+ v = mp_fromulong(v, d);
+
+ /* --- Compute %$p^k \bmod r$% and check --- */
+
+ t = mpmont_expr(&mm, t, u, v);
+ if (MP_EQ(t, mm.r)) {
+ rc = -1;
+ break;
+ }
+
+ /* --- Step the divisors along --- */
+
+ toobig:
+ for (i = 0; i < nf; i++) {
+ if (ff[i].c < ff[i].e) {
+ ff[i].c++;
+ goto more;
+ }
+ ff[i].c = 0;
+ }
+ break;
+ more:;
+ }
+
+ /* --- Clear away the debris --- */
+
+ mpmont_destroy(&mm);
+ MP_DROP(t); MP_DROP(u); MP_DROP(v); MP_DROP(r1);
+ DA_DESTROY(&fv);
+ return (rc);
+}
+
/*----- Main code ---------------------------------------------------------*/
/* --- @ec_curveparse@ --- *
* Use: Checks an elliptic curve according to the rules in SEC1.
*/
-static const char *gencheck(const ec_info *ei, grand *gr, mp *q)
+static const char *gencheck(const ec_info *ei, grand *gr, mp *q, mp *ch)
{
ec_curve *c = ei->c;
- field *f = c->f;
- int i, j, n;
+ unsigned long qmbits, rbits, cbits, B;
mp *qq;
mp *nn;
mp *x, *y;
ec p;
int rc;
- /* --- Check %$G \in E$% --- */
+ /* --- Check curve isn't anomalous --- */
+
+ if (MP_EQ(ei->r, q)) return ("curve is anomalous");
+
+ /* --- Check %$G \in E \setminus \{ 0 \}$% --- */
if (EC_ATINF(&ei->g)) return ("generator at infinity");
if (ec_check(c, &ei->g)) return ("generator not on curve");
MP_DROP(qq);
if (!rc) return ("incorrect or ambiguous cofactor");
- /* --- Check %$n G = O$% --- */
+ /* --- Check %$n G = 0$% --- */
EC_CREATE(&p);
ec_mul(c, &p, &ei->g, ei->r);
EC_DESTROY(&p);
if (!rc) return ("incorrect group order");
- /* --- Check %$q^B \not\equiv 1 \pmod{r}$% for %$1 \le B < 20$% --- *
- *
- * Actually, give up if %$q^B \ge 2^{2000}$% because that's probably
- * good enough for jazz.
- */
+ /* --- Check the embedding degree --- */
- x = MP_NEW;
- mp_div(0, &x, q, ei->r);
- n = mp_bits(ei->r) - 1;
- for (i = 0, j = n; i < 20; i++, j += n) {
- if (j >= 2000)
- break;
- if (MP_EQ(x, MP_ONE)) {
- MP_DROP(x);
- return("curve embedding degree too low");
- }
- x = mp_mul(x, x, f->m);
- mp_div(0, &x, x, ei->r);
- }
- MP_DROP(x);
+ rbits = mp_bits(ei->r);
+ cbits = mp_bits(ch);
+ qmbits = keysz_todl(keysz_fromec(rbits * 7/8));
+ B = (qmbits + cbits - 1)/cbits;
+ if (movcheck(ei->r, ch, B))
+ return("curve embedding degree too low");
/* --- Done --- */
/* --- Now do the general checks --- */
- err = gencheck(ei, gr, f->m);
+ err = gencheck(ei, gr, f->m, f->m);
return (err);
}
/* --- Now do the general checks --- */
x = mp_lsl(MP_NEW, MP_ONE, f->nbits);
- err = gencheck(ei, gr, x);
+ err = gencheck(ei, gr, x, MP_TWO);
mp_drop(x);
return (err);
}
projects / u / mdw / catacomb / commitdiff