X-Git-Url: https://git.distorted.org.uk/u/mdw/catacomb/blobdiff_plain/45c0fd363937c6e9b05da04a9167e9912c05ca0c..30ac115b90b0ed66eed17b722a76d3e7e6e4531c:/ec-info.c

diff --git a/ec-info.c b/ec-info.c
index 474691b..0e225a7 100644
--- a/ec-info.c
+++ b/ec-info.c
@@ -328,46 +328,17 @@ void ec_freeinfo(ec_info *ei)
  * Use:		Checks an elliptic curve according to the rules in SEC1.
  */
 
-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)
+static const char *gencheck(const ec_info *ei, grand *gr, mp *q)
 {
   ec_curve *c = ei->c;
   field *f = c->f;
-  int i;
+  int i, j, n;
+  mp *qq;
+  mp *nn;
   mp *x, *y;
   ec p;
   int rc;
 
-  /* --- Check %$p$% is an odd prime --- */
-
-  if (!pgen_primep(f->m, gr)) return ("p not prime");
-
-  /* --- Check %$a$%, %$b$%, %$G_x$% and %$G_y$% are in %$[0, p)$% --- */
-
-  if (!primeeltp(c->a, f)) return ("a out of range");
-  if (!primeeltp(c->b, f)) return ("b out of range");
-  if (!primeeltp(ei->g.x, f)) return ("G_x out of range");
-  if (!primeeltp(ei->g.x, f)) return ("G_y out of range");
-
-  /* --- Check %$4 a^3 + 27 b^2 \not\equiv 0 \pmod{p}$% --- */
-
-  x = F_SQR(f, MP_NEW, c->a);
-  x = F_MUL(f, x, x, c->a);
-  x = F_QDL(f, x, x);
-  y = F_SQR(f, MP_NEW, c->b);
-  y = F_TPL(f, y, y);
-  y = F_TPL(f, y, y);
-  y = F_TPL(f, y, y);
-  x = F_ADD(f, x, x, y);
-  rc = F_ZEROP(f, x);
-  MP_DROP(x);
-  MP_DROP(y);
-  if (rc) return ("not an elliptic curve");
-
   /* --- Check %$G \in E$% --- */
 
   if (EC_ATINF(&ei->g)) return ("generator at infinity");
@@ -377,24 +348,54 @@ static const char *primecheck(const ec_info *ei, grand *gr)
 
   if (!pgen_primep(ei->r, gr)) return ("generator order not prime");
 
-  /* --- Check %$h = \lfloor (\sqrt{p} + 1)^2/r \rlfoor$% --- *
+  /* --- 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);
    *
-   * 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.
+   *   * %$B = (q + 1 - n - r)^2 > 4 q$% (check %$h - 1$% isn't possible);
+   *	 and
+   *
+   *   * %$C = (q + 1 - n + r)^2 > 4 q$% (check %$h + 1$% isn't possible).
    */
 
-  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);
+  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);
-  if (!rc) return ("incorrect cofactor");
+  MP_DROP(nn);
+  MP_DROP(qq);
+  if (!rc) return ("incorrect or ambiguous cofactor");
 
   /* --- Check %$n G = O$% --- */
 
@@ -404,41 +405,82 @@ static const char *primecheck(const ec_info *ei, grand *gr)
   EC_DESTROY(&p);
   if (!rc) return ("incorrect group order");
 
-  /* --- Check that %$p^B \not\equiv 1 \pmod{r}$% for %$1 \le B < 20$% --- *
+  /* --- Check %$q^B \not\equiv 1 \pmod{r}$% for %$1 \le B < 20$% --- *
    *
-   * The spec says %$q$%, not %$p$%, but I think that's a misprint.
+   * Actually, give up if %$q^B \ge 2^{2000}$% because that's probably
+   * good enough for jazz.
    */
 
   x = MP_NEW;
-  mp_div(0, &x, f->m, ei->r);
-  i = 20;
-  while (i) {
-    if (MP_EQ(x, MP_ONE)) break;
+  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);
-    i--;
   }
   MP_DROP(x);
-  if (i) return ("curve is weak");
-
-  /* --- Check %$0 < h \le 4$% --- */
-
-  if (MP_CMP(ei->h, <, MP_ONE) || MP_CMP(ei->h, >, MP_FOUR))
-    return ("cofactor out of range");
 
   /* --- Done --- */
 
   return (0);
 }
 
-static const char *bincheck(const ec_info *ei, grand *gr)
+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 --- */
+
+  if (!pgen_primep(f->m, gr)) return ("p not prime");
+
+  /* --- Check %$a$%, %$b$%, %$G_x$% and %$G_y$% are in %$[0, p)$% --- */
+
+  if (!primeeltp(c->a, f)) return ("a out of range");
+  if (!primeeltp(c->b, f)) return ("b out of range");
+  if (!primeeltp(ei->g.x, f)) return ("G_x out of range");
+  if (!primeeltp(ei->g.x, f)) return ("G_y out of range");
+
+  /* --- Check %$4 a^3 + 27 b^2 \not\equiv 0 \pmod{p}$% --- */
+
+  x = F_SQR(f, MP_NEW, c->a);
+  x = F_MUL(f, x, x, c->a);
+  x = F_QDL(f, x, x);
+  y = F_SQR(f, MP_NEW, c->b);
+  y = F_TPL(f, y, y);
+  y = F_TPL(f, y, y);
+  y = F_TPL(f, y, y);
+  x = F_ADD(f, x, x, y);
+  rc = F_ZEROP(f, x);
+  MP_DROP(x);
+  MP_DROP(y);
+  if (rc) return ("not an elliptic curve");
+
+  /* --- Now do the general checks --- */
+
+  err = gencheck(ei, gr, f->m);
+  return (err);
+}
+
+static const char *bincheck(const ec_info *ei, grand *gr)
+{
+  ec_curve *c = ei->c;
+  field *f = c->f;
+  mp *x;
+  int rc;
+  const char *err;
 
   /* --- Check that %$m$% is prime --- */
 
@@ -462,64 +504,12 @@ static const char *bincheck(const ec_info *ei, grand *gr)
 
   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 %$h = \lfloor (\sqrt{2^m} + 1)^2/r \rlfoor$% --- *
-   *
-   * 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");
-
-  /* --- Check %$0 < h \le 4$% --- */
-
-  if (MP_CMP(ei->h, <, MP_ONE) || MP_CMP(ei->h, >, MP_FOUR))
-    return ("cofactor out of range");
-
-  /* --- Done --- */
-
-  return (0);
+  err = gencheck(ei, gr, x);
+  mp_drop(x);
+  return (err);
 }
 
 const char *ec_checkinfo(const ec_info *ei, grand *gr)
@@ -561,6 +551,7 @@ int main(int argc, char *argv[])
 	  ok = 0;
 	}
       }
+      assert(mparena_count(MPARENA_GLOBAL) == 0);
     }
   } else {
     fputs("checking standard curves:", stdout);
@@ -576,6 +567,7 @@ int main(int argc, char *argv[])
       } else
 	printf(" %s", ee->name);
       fflush(stdout);
+      assert(mparena_count(MPARENA_GLOBAL) == 0);
     }
     fputs(ok ? " ok\n" : " failed\n", stdout);
   }