/* -*-c-*-
*
- * $Id: ec-info.c,v 1.3 2004/04/01 21:28:41 mdw Exp $
+ * $Id$
*
* Elliptic curve information management
*
* (c) 2004 Straylight/Edgeware
*/
-/*----- Licensing notice --------------------------------------------------*
+/*----- Licensing notice --------------------------------------------------*
*
* This file is part of Catacomb.
*
* it under the terms of the GNU Library General Public License as
* published by the Free Software Foundation; either version 2 of the
* License, or (at your option) any later version.
- *
+ *
* Catacomb is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Library General Public License for more details.
- *
+ *
* You should have received a copy of the GNU Library General Public
* License along with Catacomb; if not, write to the Free
* Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
* MA 02111-1307, USA.
*/
-/*----- Revision history --------------------------------------------------*
- *
- * $Log: ec-info.c,v $
- * Revision 1.3 2004/04/01 21:28:41 mdw
- * Normal basis support (translates to poly basis internally). Rewrite
- * EC and prime group table generators in awk, so that they can reuse data
- * for repeated constants.
- *
- * Revision 1.2 2004/04/01 12:50:09 mdw
- * Add cyclic group abstraction, with test code. Separate off exponentation
- * functions for better static linking. Fix a buttload of bugs on the way.
- * Generally ensure that negative exponents do inversion correctly. Add
- * table of standard prime-field subgroups. (Binary field subgroups are
- * currently unimplemented but easy to add if anyone ever finds a good one.)
- *
- * Revision 1.1 2004/03/27 17:54:11 mdw
- * Standard curves and curve checking.
- *
- */
-
/*----- 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$% and check --- */
+
+ t = mpmont_expr(&mm, t, u, v);
+ if (MP_EQ(t, mm.r)) {
+ rc = -1;
+ break;
+ }
+
+ /* --- Step the divisors along --- */
+
+ toobig:
+ for (i = 0; i < nf; i++) {
+ if (ff[i].c < ff[i].e) {
+ ff[i].c++;
+ goto more;
+ }
+ ff[i].c = 0;
+ }
+ break;
+ more:;
+ }
+
+ /* --- Clear away the debris --- */
+
+ mpmont_destroy(&mm);
+ MP_DROP(t); MP_DROP(u); MP_DROP(v); MP_DROP(r1);
+ DA_DESTROY(&fv);
+ return (rc);
+}
+
/*----- Main code ---------------------------------------------------------*/
/* --- @ec_curveparse@ --- *
* Use: Parses an elliptic curve description, which has the form
*
* * a field description
- * * an optional `/'
+ * * an optional `;'
* * `prime', `primeproj', `bin', or `binproj'
* * an optional `:'
* * the %$a$% parameter
field *f;
if ((f = field_parse(qd)) == 0) goto fail;
- qd_delim(qd, '/');
+ qd_delim(qd, ';');
switch (qd_enum(qd, "prime,primeproj,bin,binproj")) {
case 0:
if (F_TYPE(f) != FTY_PRIME) {
default:
goto fail;
}
+ if (!c) {
+ qd->e = "bad curve parameters";
+ goto fail;
+ }
if (a) MP_DROP(a);
if (b) MP_DROP(b);
return (c);
return (0);
}
-/* --- @getinfo@ --- *
+/* --- @ec_infofromdata@ --- *
*
* Arguments: @ec_info *ei@ = where to write the information
* @ecdata *ed@ = raw data
* curves.
*/
-static void getinfo(ec_info *ei, ecdata *ed)
+void ec_infofromdata(ec_info *ei, ecdata *ed)
{
field *f;
abort();
}
+ assert(f); assert(ei->c);
EC_CREATE(&ei->g); ei->g.x = &ed->gx; ei->g.y = &ed->gy; ei->g.z = 0;
ei->r = &ed->r; ei->h = &ed->h;
}
* curve, or it has the form
*
* * elliptic curve description
- * * optional `/'
+ * * optional `;'
* * common point
* * optional `:'
* * group order
for (ee = ectab; ee->name; ee++) {
if (qd_enum(qd, ee->name) >= 0) {
- getinfo(ei, ee->data);
+ ec_infofromdata(ei, ee->data);
goto found;
}
}
+
if ((c = ec_curveparse(qd)) == 0) goto fail;
- qd_delim(qd, '/'); if (!ec_ptparse(qd, &g)) goto fail;
+ qd_delim(qd, ';'); if (!ec_ptparse(qd, &g)) goto fail;
qd_delim(qd, ':'); if ((r = qd_getmp(qd)) == 0) goto fail;
qd_delim(qd, '*'); if ((h = qd_getmp(qd)) == 0) goto fail;
ei->c = c; ei->g = g; ei->r = r; ei->h = h;
* Use: Checks an elliptic curve according to the rules in SEC1.
*/
-static int primeeltp(mp *x, field *f)
+static const char *gencheck(const ec_info *ei, grand *gr, mp *q, mp *ch)
{
- return (!MP_ISNEG(x) && MP_CMP(x, <, f->m));
+ ec_curve *c = ei->c;
+ unsigned long qmbits, rbits, cbits, B;
+ mp *qq;
+ mp *nn;
+ mp *x, *y;
+ ec p;
+ int rc;
+
+ /* --- Check curve isn't anomalous --- */
+
+ if (MP_EQ(ei->r, q)) return ("curve is anomalous");
+
+ /* --- Check %$G \in E \setminus \{ 0 \}$% --- */
+
+ if (EC_ATINF(&ei->g)) return ("generator at infinity");
+ if (ec_check(c, &ei->g)) return ("generator not on curve");
+
+ /* --- Check %$r$% is prime --- */
+
+ if (!pgen_primep(ei->r, gr)) return ("generator order not prime");
+
+ /* --- Check that the cofactor is correct --- *
+ *
+ * Let %$q$% be the size of the field, and let %$n = h r = \#E(\gf{q})$% be
+ * the number of %$\gf{q}$%-rational points on our curve. Hasse's theorem
+ * tells us that
+ *
+ * %$|q + 1 - n| \le 2\sqrt{q}$%
+ *
+ * or, if we square both sides,
+ *
+ * %$(q + 1 - n)^2 \le 4 q$%.
+ *
+ * We'd like the cofactor to be uniquely determined by this equation, which
+ * is possible as long as it's not too big. (If it is, we have to mess
+ * about with Weil pairings, which is no fun.) For this, we need the
+ * following inequalities:
+ *
+ * * %$A = (q + 1 - n)^2 \le 4 q$% (both lower and upper bounds from
+ * Hasse's theorem);
+ *
+ * * %$B = (q + 1 - n - r)^2 > 4 q$% (check %$h - 1$% isn't possible);
+ * and
+ *
+ * * %$C = (q + 1 - n + r)^2 > 4 q$% (check %$h + 1$% isn't possible).
+ */
+
+ rc = 1;
+ qq = mp_add(MP_NEW, q, MP_ONE);
+ nn = mp_mul(MP_NEW, ei->r, ei->h);
+ nn = mp_sub(nn, qq, nn);
+ qq = mp_lsl(qq, q, 2);
+
+ y = mp_sqr(MP_NEW, nn);
+ if (MP_CMP(y, >, qq)) rc = 0;
+
+ x = mp_sub(MP_NEW, nn, ei->r);
+ y = mp_sqr(y, x);
+ if (MP_CMP(y, <=, qq)) rc = 0;
+
+ x = mp_add(x, nn, ei->r);
+ y = mp_sqr(y, x);
+ if (MP_CMP(y, <=, qq)) rc = 0;
+
+ MP_DROP(x);
+ MP_DROP(y);
+ MP_DROP(nn);
+ MP_DROP(qq);
+ if (!rc) return ("incorrect or ambiguous cofactor");
+
+ /* --- Check %$n G = 0$% --- */
+
+ EC_CREATE(&p);
+ ec_mul(c, &p, &ei->g, ei->r);
+ rc = EC_ATINF(&p);
+ EC_DESTROY(&p);
+ if (!rc) return ("incorrect group order");
+
+ /* --- Check the embedding degree --- */
+
+ rbits = mp_bits(ei->r);
+ cbits = mp_bits(ch);
+ qmbits = keysz_todl(keysz_fromec(rbits * 7/8));
+ B = (qmbits + cbits - 1)/cbits;
+ if (movcheck(ei->r, ch, B))
+ return("curve embedding degree too low");
+
+ /* --- Done --- */
+
+ return (0);
}
+static int primeeltp(mp *x, field *f)
+ { return (!MP_NEGP(x) && MP_CMP(x, <, f->m)); }
+
static const char *primecheck(const ec_info *ei, grand *gr)
{
ec_curve *c = ei->c;
field *f = c->f;
- int i;
mp *x, *y;
- ec p;
int rc;
+ const char *err;
/* --- Check %$p$% is an odd prime --- */
MP_DROP(y);
if (rc) return ("not an elliptic curve");
- /* --- Check %$G \in E$% --- */
-
- if (EC_ATINF(&ei->g)) return ("generator at infinity");
- if (ec_check(c, &ei->g)) return ("generator not on curve");
-
- /* --- Check %$r$% is prime --- */
-
- if (!pgen_primep(ei->r, gr)) return ("generator order not prime");
-
- /* --- Check %$0 < h \le 4$% --- */
-
- if (MP_CMP(ei->h, <, MP_ONE) || MP_CMP(ei->h, >, MP_FOUR))
- return ("cofactor out of range");
-
- /* --- Check %$h = \lfloor (\sqrt{p} + 1)^2/r \rlfoor$% --- *
- *
- * This seems to work with the approximate-sqrt in the library, but might
- * not be so good in some cases. Throw in some extra significate figures
- * for good measure.
- */
-
- x = mp_lsl(MP_NEW, f->m, 128);
- x = mp_sqrt(x, x);
- y = mp_lsl(MP_NEW, MP_ONE, 64);
- x = mp_add(x, x, y);
- x = mp_sqr(x, x);
- mp_div(&x, 0, x, ei->r);
- x = mp_lsr(x, x, 128);
- rc = MP_EQ(x, ei->h);
- MP_DROP(x);
- MP_DROP(y);
- if (!rc) return ("incorrect cofactor");
-
- /* --- Check %$n G = O$% --- */
-
- EC_CREATE(&p);
- ec_mul(c, &p, &ei->g, ei->r);
- rc = EC_ATINF(&p);
- EC_DESTROY(&p);
- if (!rc) return ("incorrect group order");
-
- /* --- Check that %$p^B \not\equiv 1 \pmod{r}$% for %$1 \le B < 20$% --- *
- *
- * The spec says %$q$%, not %$p$%, but I think that's a misprint.
- */
-
- x = MP_NEW;
- mp_div(0, &x, f->m, ei->r);
- i = 20;
- while (i) {
- if (MP_EQ(x, MP_ONE)) break;
- x = mp_mul(x, x, f->m);
- mp_div(0, &x, x, ei->r);
- i--;
- }
- MP_DROP(x);
- if (i) return ("curve is weak");
-
- /* --- Done --- */
+ /* --- Now do the general checks --- */
- return (0);
+ err = gencheck(ei, gr, f->m, f->m);
+ return (err);
}
static const char *bincheck(const ec_info *ei, grand *gr)
{
ec_curve *c = ei->c;
field *f = c->f;
- int i;
- mp *x, *y;
- ec p;
+ mp *x;
int rc;
+ const char *err;
+
+ /* --- Check that %$m$% is prime --- */
+
+ x = mp_fromuint(MP_NEW, f->nbits);
+ rc = pfilt_smallfactor(x);
+ mp_drop(x);
+ if (rc != PGEN_DONE) return ("degree not prime");
/* --- Check that %$p$% is irreducible --- */
if (F_ZEROP(f, c->b)) return ("b is zero");
- /* --- Check that %$G \in E$% --- */
-
- if (EC_ATINF(&ei->g)) return ("generator at infinity");
- if (ec_check(c, &ei->g)) return ("generator not on curve");
-
- /* --- Check %$r$% is prime --- */
-
- if (!pgen_primep(ei->r, gr)) return ("generator order not prime");
-
- /* --- Check %$0 < h \le 4$% --- */
-
- if (MP_CMP(ei->h, <, MP_ONE) || MP_CMP(ei->h, >, MP_FOUR))
- return ("cofactor out of range");
-
- /* --- Check %$h = \lfloor (\sqrt{2^m} + 1)^2/r \rlfoor$% --- *
- *
- * This seems to work with the approximate-sqrt in the library, but might
- * not be so good in some cases. Throw in some extra significate figures
- * for good measure.
- */
-
- x = mp_lsl(MP_NEW, MP_ONE, f->nbits + 128);
- x = mp_sqrt(x, x);
- y = mp_lsl(MP_NEW, MP_ONE, 64);
- x = mp_add(x, x, y);
- x = mp_sqr(x, x);
- mp_div(&x, 0, x, ei->r);
- x = mp_lsr(x, x, 128);
- rc = MP_EQ(x, ei->h);
- MP_DROP(x);
- MP_DROP(y);
- if (!rc) return ("incorrect cofactor");
-
- /* --- Check %$n G = O$% --- */
-
- EC_CREATE(&p);
- ec_mul(c, &p, &ei->g, ei->r);
- rc = EC_ATINF(&p);
- EC_DESTROY(&p);
- if (!rc) return ("incorrect group order");
-
- /* --- Check %$2^{m B} \not\equiv 1 \pmod{r}$% for %$1 \le B < 20$% --- */
+ /* --- Now do the general checks --- */
x = mp_lsl(MP_NEW, MP_ONE, f->nbits);
- mp_div(0, &x, x, ei->r);
- i = 20;
- while (i) {
- if (MP_EQ(x, MP_ONE)) break;
- x = mp_mul(x, x, f->m);
- mp_div(0, &x, x, ei->r);
- i--;
- }
- MP_DROP(x);
- if (i) return ("curve is weak");
-
- /* --- Done --- */
-
- return (0);
+ err = gencheck(ei, gr, x, MP_TWO);
+ mp_drop(x);
+ return (err);
}
const char *ec_checkinfo(const ec_info *ei, grand *gr)
#include "fibrand.h"
-int main(void)
+int main(int argc, char *argv[])
{
const ecentry *ee;
const char *e;
int ok = 1;
+ int i;
grand *gr;
gr = fibrand_create(0);
- fputs("checking standard curves: ", stdout);
- for (ee = ectab; ee->name; ee++) {
- ec_info ei;
- getinfo(&ei, ee->data);
- e = ec_checkinfo(&ei, gr);
- ec_freeinfo(&ei);
- if (e) {
- fprintf(stderr, "\n*** curve %s fails: %s\n", ee->name, e);
- ok = 0;
+ if (argc > 1) {
+ for (i = 1; i < argc; i++) {
+ ec_info ei;
+ if ((e = ec_getinfo(&ei, argv[i])) != 0)
+ fprintf(stderr, "bad curve spec `%s': %s\n", argv[i], e);
+ else {
+ e = ec_checkinfo(&ei, gr);
+ ec_freeinfo(&ei);
+ if (!e)
+ printf("OK %s\n", argv[i]);
+ else {
+ printf("BAD %s: %s\n", argv[i], e);
+ ok = 0;
+ }
+ }
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
}
- putchar('.');
+ } else {
+ fputs("checking standard curves:", stdout);
fflush(stdout);
+ for (ee = ectab; ee->name; ee++) {
+ ec_info ei;
+ ec_infofromdata(&ei, ee->data);
+ e = ec_checkinfo(&ei, gr);
+ ec_freeinfo(&ei);
+ if (e) {
+ printf(" [%s fails: %s]", ee->name, e);
+ ok = 0;
+ } else
+ printf(" %s", ee->name);
+ fflush(stdout);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ }
+ fputs(ok ? " ok\n" : " failed\n", stdout);
}
gr->ops->destroy(gr);
- fputs(ok ? " ok\n" : " failed\n", stdout);
return (!ok);
}
projects / u / mdw / catacomb / blobdiff