[crypto/ec] Ladder tweaks

- Convert to affine coords on ladder entry. This lets us use more efficient
  ladder step formulae.

- Convert to affine coords on ladder exit. This prevents the current code
  awkwardness where conversion happens twice during serialization: first to
  fetch the buffer size, then again to fetch the coords.

- Instead of projectively blinding the input point, blind both accumulators
  independently.

Reviewed-by: Nicola Tuveri <nic.tuv@gmail.com>
Reviewed-by: Bernd Edlinger <bernd.edlinger@hotmail.de>
(Merged from https://github.com/openssl/openssl/pull/11435)

2 files changed, 150 insertions, 130 deletions

diff --git a/crypto/ec/ec_mult.c b/crypto/ec/ec_mult.c
index 17aacf877b..2d3fc50acf 100644
--- a/crypto/ec/ec_mult.c
+++ b/crypto/ec/ec_mult.c
@@ -266,17 +266,10 @@ int ec_scalar_mul_ladder(const EC_GROUP *group, EC_POINT *r,
         goto err;
     }
 
-    /*-
-     * Apply coordinate blinding for EC_POINT.
-     *
-     * The underlying EC_METHOD can optionally implement this function:
-     * ec_point_blind_coordinates() returns 0 in case of errors or 1 on
-     * success or if coordinate blinding is not implemented for this
-     * group.
-     */
-    if (!ec_point_blind_coordinates(group, p, ctx)) {
-        ECerr(EC_F_EC_SCALAR_MUL_LADDER, EC_R_POINT_COORDINATES_BLIND_FAILURE);
-        goto err;
+    /* ensure input point is in affine coords for ladder step efficiency */
+    if (!p->Z_is_one && !EC_POINT_make_affine(group, p, ctx)) {
+            ECerr(EC_F_EC_SCALAR_MUL_LADDER, ERR_R_EC_LIB);
+            goto err;
     }
 
     /* Initialize the Montgomery ladder */
diff --git a/crypto/ec/ecp_smpl.c b/crypto/ec/ecp_smpl.c
index 005ab1ec65..9eaa887f21 100644
--- a/crypto/ec/ecp_smpl.c
+++ b/crypto/ec/ecp_smpl.c
@@ -1381,6 +1381,7 @@ int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
  * Computes the multiplicative inverse of a in GF(p), storing the result in r.
  * If a is zero (or equivalent), you'll get a EC_R_CANNOT_INVERT error.
  * Since we don't have a Mont structure here, SCA hardening is with blinding.
+ * NB: "a" must be in _decoded_ form. (i.e. field_decode must precede.)
  */
 int ec_GFp_simple_field_inv(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
                             BN_CTX *ctx)
@@ -1476,77 +1477,96 @@ int ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p,
 }
 
 /*-
- * Set s := p, r := 2p.
+ * Input:
+ * - p: affine coordinates
+ *
+ * Output:
+ * - s := p, r := 2p: blinded projective (homogeneous) coordinates
  *
  * For doubling we use Formula 3 from Izu-Takagi "A fast parallel elliptic curve
- * multiplication resistant against side channel attacks" appendix, as described
- * at
+ * multiplication resistant against side channel attacks" appendix, described at
  * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#doubling-dbl-2002-it-2
+ * simplified for Z1=1.
  *
- * The input point p will be in randomized Jacobian projective coords:
- *      x = X/Z**2, y=Y/Z**3
- *
- * The output points p, s, and r are converted to standard (homogeneous)
- * projective coords:
- *      x = X/Z, y=Y/Z
+ * Blinding uses the equivalence relation (\lambda X, \lambda Y, \lambda Z)
+ * for any non-zero \lambda that holds for projective (homogeneous) coords.
  */
 int ec_GFp_simple_ladder_pre(const EC_GROUP *group,
                              EC_POINT *r, EC_POINT *s,
                              EC_POINT *p, BN_CTX *ctx)
 {
-    BIGNUM *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
+    BIGNUM *t1, *t2, *t3, *t4, *t5 = NULL;
 
-    t1 = r->Z;
-    t2 = r->Y;
+    t1 = s->Z;
+    t2 = r->Z;
     t3 = s->X;
     t4 = r->X;
     t5 = s->Y;
-    t6 = s->Z;
-
-    /* convert p: (X,Y,Z) -> (XZ,Y,Z**3) */
-    if (!group->meth->field_mul(group, p->X, p->X, p->Z, ctx)
-        || !group->meth->field_sqr(group, t1, p->Z, ctx)
-        || !group->meth->field_mul(group, p->Z, p->Z, t1, ctx)
-        /* r := 2p */
-        || !group->meth->field_sqr(group, t2, p->X, ctx)
-        || !group->meth->field_sqr(group, t3, p->Z, ctx)
-        || !group->meth->field_mul(group, t4, t3, group->a, ctx)
-        || !BN_mod_sub_quick(t5, t2, t4, group->field)
-        || !BN_mod_add_quick(t2, t2, t4, group->field)
-        || !group->meth->field_sqr(group, t5, t5, ctx)
-        || !group->meth->field_mul(group, t6, t3, group->b, ctx)
-        || !group->meth->field_mul(group, t1, p->X, p->Z, ctx)
-        || !group->meth->field_mul(group, t4, t1, t6, ctx)
-        || !BN_mod_lshift_quick(t4, t4, 3, group->field)
+
+    if (!p->Z_is_one /* r := 2p */
+        || !group->meth->field_sqr(group, t3, p->X, ctx)
+        || !BN_mod_sub_quick(t4, t3, group->a, group->field)
+        || !group->meth->field_sqr(group, t4, t4, ctx)
+        || !group->meth->field_mul(group, t5, p->X, group->b, ctx)
+        || !BN_mod_lshift_quick(t5, t5, 3, group->field)
         /* r->X coord output */
-        || !BN_mod_sub_quick(r->X, t5, t4, group->field)
-        || !group->meth->field_mul(group, t1, t1, t2, ctx)
-        || !group->meth->field_mul(group, t2, t3, t6, ctx)
-        || !BN_mod_add_quick(t1, t1, t2, group->field)
+        || !BN_mod_sub_quick(r->X, t4, t5, group->field)
+        || !BN_mod_add_quick(t1, t3, group->a, group->field)
+        || !group->meth->field_mul(group, t2, p->X, t1, ctx)
+        || !BN_mod_add_quick(t2, group->b, t2, group->field)
         /* r->Z coord output */
-        || !BN_mod_lshift_quick(r->Z, t1, 2, group->field)
-        || !EC_POINT_copy(s, p))
+        || !BN_mod_lshift_quick(r->Z, t2, 2, group->field))
+        return 0;
+
+    /* make sure lambda (r->Y here for storage) is not zero */
+    do {
+        if (!BN_priv_rand_range_ex(r->Y, group->field, ctx))
+            return 0;
+    } while (BN_is_zero(r->Y));
+
+    /* make sure lambda (s->Z here for storage) is not zero */
+    do {
+        if (!BN_priv_rand_range_ex(s->Z, group->field, ctx))
+            return 0;
+    } while (BN_is_zero(s->Z));
+
+    /* if field_encode defined convert between representations */
+    if (group->meth->field_encode != NULL
+        && (!group->meth->field_encode(group, r->Y, r->Y, ctx)
+            || !group->meth->field_encode(group, s->Z, s->Z, ctx)))
+        return 0;
+
+    /* blind r and s independently */
+    if (!group->meth->field_mul(group, r->Z, r->Z, r->Y, ctx)
+        || !group->meth->field_mul(group, r->X, r->X, r->Y, ctx)
+        || !group->meth->field_mul(group, s->X, p->X, s->Z, ctx)) /* s := p */
         return 0;
 
     r->Z_is_one = 0;
     s->Z_is_one = 0;
-    p->Z_is_one = 0;
 
     return 1;
 }
 
 /*-
- * Differential addition-and-doubling using  Eq. (9) and (10) from Izu-Takagi
+ * Input:
+ * - s, r: projective (homogeneous) coordinates
+ * - p: affine coordinates
+ *
+ * Output:
+ * - s := r + s, r := 2r: projective (homogeneous) coordinates
+ *
+ * Differential addition-and-doubling using Eq. (9) and (10) from Izu-Takagi
  * "A fast parallel elliptic curve multiplication resistant against side channel
  * attacks", as described at
- * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-ladd-2002-it-4
+ * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-mladd-2002-it-4
  */
 int ec_GFp_simple_ladder_step(const EC_GROUP *group,
                               EC_POINT *r, EC_POINT *s,
                               EC_POINT *p, BN_CTX *ctx)
 {
     int ret = 0;
-    BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6, *t7 = NULL;
+    BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
 
     BN_CTX_start(ctx);
     t0 = BN_CTX_get(ctx);
@@ -1556,50 +1576,47 @@ int ec_GFp_simple_ladder_step(const EC_GROUP *group,
     t4 = BN_CTX_get(ctx);
     t5 = BN_CTX_get(ctx);
     t6 = BN_CTX_get(ctx);
-    t7 = BN_CTX_get(ctx);
 
-    if (t7 == NULL
-        || !group->meth->field_mul(group, t0, r->X, s->X, ctx)
-        || !group->meth->field_mul(group, t1, r->Z, s->Z, ctx)
-        || !group->meth->field_mul(group, t2, r->X, s->Z, ctx)
+    if (t6 == NULL
+        || !group->meth->field_mul(group, t6, r->X, s->X, ctx)
+        || !group->meth->field_mul(group, t0, r->Z, s->Z, ctx)
+        || !group->meth->field_mul(group, t4, r->X, s->Z, ctx)
         || !group->meth->field_mul(group, t3, r->Z, s->X, ctx)
-        || !group->meth->field_mul(group, t4, group->a, t1, ctx)
-        || !BN_mod_add_quick(t0, t0, t4, group->field)
-        || !BN_mod_add_quick(t4, t3, t2, group->field)
-        || !group->meth->field_mul(group, t0, t4, t0, ctx)
-        || !group->meth->field_sqr(group, t1, t1, ctx)
-        || !BN_mod_lshift_quick(t7, group->b, 2, group->field)
-        || !group->meth->field_mul(group, t1, t7, t1, ctx)
-        || !BN_mod_lshift1_quick(t0, t0, group->field)
-        || !BN_mod_add_quick(t0, t1, t0, group->field)
-        || !BN_mod_sub_quick(t1, t2, t3, group->field)
-        || !group->meth->field_sqr(group, t1, t1, ctx)
-        || !group->meth->field_mul(group, t3, t1, p->X, ctx)
-        || !group->meth->field_mul(group, t0, p->Z, t0, ctx)
-        /* s->X coord output */
-        || !BN_mod_sub_quick(s->X, t0, t3, group->field)
-        /* s->Z coord output */
-        || !group->meth->field_mul(group, s->Z, p->Z, t1, ctx)
-        || !group->meth->field_sqr(group, t3, r->X, ctx)
-        || !group->meth->field_sqr(group, t2, r->Z, ctx)
-        || !group->meth->field_mul(group, t4, t2, group->a, ctx)
-        || !BN_mod_add_quick(t5, r->X, r->Z, group->field)
-        || !group->meth->field_sqr(group, t5, t5, ctx)
-        || !BN_mod_sub_quick(t5, t5, t3, group->field)
-        || !BN_mod_sub_quick(t5, t5, t2, group->field)
-        || !BN_mod_sub_quick(t6, t3, t4, group->field)
-        || !group->meth->field_sqr(group, t6, t6, ctx)
-        || !group->meth->field_mul(group, t0, t2, t5, ctx)
-        || !group->meth->field_mul(group, t0, t7, t0, ctx)
-        /* r->X coord output */
-        || !BN_mod_sub_quick(r->X, t6, t0, group->field)
+        || !group->meth->field_mul(group, t5, group->a, t0, ctx)
+        || !BN_mod_add_quick(t5, t6, t5, group->field)
         || !BN_mod_add_quick(t6, t3, t4, group->field)
-        || !group->meth->field_sqr(group, t3, t2, ctx)
-        || !group->meth->field_mul(group, t7, t3, t7, ctx)
-        || !group->meth->field_mul(group, t5, t5, t6, ctx)
+        || !group->meth->field_mul(group, t5, t6, t5, ctx)
+        || !group->meth->field_sqr(group, t0, t0, ctx)
+        || !BN_mod_lshift_quick(t2, group->b, 2, group->field)
+        || !group->meth->field_mul(group, t0, t2, t0, ctx)
         || !BN_mod_lshift1_quick(t5, t5, group->field)
+        || !BN_mod_sub_quick(t3, t4, t3, group->field)
+        /* s->Z coord output */
+        || !group->meth->field_sqr(group, s->Z, t3, ctx)
+        || !group->meth->field_mul(group, t4, s->Z, p->X, ctx)
+        || !BN_mod_add_quick(t0, t0, t5, group->field)
+        /* s->X coord output */
+        || !BN_mod_sub_quick(s->X, t0, t4, group->field)
+        || !group->meth->field_sqr(group, t4, r->X, ctx)
+        || !group->meth->field_sqr(group, t5, r->Z, ctx)
+        || !group->meth->field_mul(group, t6, t5, group->a, ctx)
+        || !BN_mod_add_quick(t1, r->X, r->Z, group->field)
+        || !group->meth->field_sqr(group, t1, t1, ctx)
+        || !BN_mod_sub_quick(t1, t1, t4, group->field)
+        || !BN_mod_sub_quick(t1, t1, t5, group->field)
+        || !BN_mod_sub_quick(t3, t4, t6, group->field)
+        || !group->meth->field_sqr(group, t3, t3, ctx)
+        || !group->meth->field_mul(group, t0, t5, t1, ctx)
+        || !group->meth->field_mul(group, t0, t2, t0, ctx)
+        /* r->X coord output */
+        || !BN_mod_sub_quick(r->X, t3, t0, group->field)
+        || !BN_mod_add_quick(t3, t4, t6, group->field)
+        || !group->meth->field_sqr(group, t4, t5, ctx)
+        || !group->meth->field_mul(group, t4, t4, t2, ctx)
+        || !group->meth->field_mul(group, t1, t1, t3, ctx)
+        || !BN_mod_lshift1_quick(t1, t1, group->field)
         /* r->Z coord output */
-        || !BN_mod_add_quick(r->Z, t7, t5, group->field))
+        || !BN_mod_add_quick(r->Z, t4, t1, group->field))
         goto err;
 
     ret = 1;
@@ -1610,17 +1627,23 @@ int ec_GFp_simple_ladder_step(const EC_GROUP *group,
 }
 
 /*-
+ * Input:
+ * - s, r: projective (homogeneous) coordinates
+ * - p: affine coordinates
+ *
+ * Output:
+ * - r := (x,y): affine coordinates
+ *
  * Recovers the y-coordinate of r using Eq. (8) from Brier-Joye, "Weierstrass
- * Elliptic Curves and Side-Channel Attacks", modified to work in projective
- * coordinates and return r in Jacobian projective coordinates.
+ * Elliptic Curves and Side-Channel Attacks", modified to work in mixed
+ * projective coords, i.e. p is affine and (r,s) in projective (homogeneous)
+ * coords, and return r in affine coordinates.
  *
- * X4 = two*Y1*X2*Z3*Z2*Z1;
- * Y4 = two*b*Z3*SQR(Z2*Z1) + Z3*(a*Z2*Z1+X1*X2)*(X1*Z2+X2*Z1) - X3*SQR(X1*Z2-X2*Z1);
- * Z4 = two*Y1*Z3*SQR(Z2)*Z1;
+ * X4 = two*Y1*X2*Z3*Z2;
+ * Y4 = two*b*Z3*SQR(Z2) + Z3*(a*Z2+X1*X2)*(X1*Z2+X2) - X3*SQR(X1*Z2-X2);
+ * Z4 = two*Y1*Z3*SQR(Z2);
  *
  * Z4 != 0 because:
- *  - Z1==0 implies p is at infinity, which would have caused an early exit in
- *    the caller;
  *  - Z2==0 implies r is at infinity (handled by the BN_is_zero(r->Z) branch);
  *  - Z3==0 implies s is at infinity (handled by the BN_is_zero(s->Z) branch);
  *  - Y1==0 implies p has order 2, so either r or s are infinity and handled by
@@ -1637,11 +1660,7 @@ int ec_GFp_simple_ladder_post(const EC_GROUP *group,
         return EC_POINT_set_to_infinity(group, r);
 
     if (BN_is_zero(s->Z)) {
-        /* (X,Y,Z) -> (XZ,YZ**2,Z) */
-        if (!group->meth->field_mul(group, r->X, p->X, p->Z, ctx)
-            || !group->meth->field_sqr(group, r->Z, p->Z, ctx)
-            || !group->meth->field_mul(group, r->Y, p->Y, r->Z, ctx)
-            || !BN_copy(r->Z, p->Z)
+        if (!EC_POINT_copy(r, p)
             || !EC_POINT_invert(group, r, ctx))
             return 0;
         return 1;
@@ -1657,38 +1676,46 @@ int ec_GFp_simple_ladder_post(const EC_GROUP *group,
     t6 = BN_CTX_get(ctx);
 
     if (t6 == NULL
-        || !BN_mod_lshift1_quick(t0, p->Y, group->field)
-        || !group->meth->field_mul(group, t1, r->X, p->Z, ctx)
-        || !group->meth->field_mul(group, t2, r->Z, s->Z, ctx)
-        || !group->meth->field_mul(group, t2, t1, t2, ctx)
-        || !group->meth->field_mul(group, t3, t2, t0, ctx)
-        || !group->meth->field_mul(group, t2, r->Z, p->Z, ctx)
-        || !group->meth->field_sqr(group, t4, t2, ctx)
-        || !BN_mod_lshift1_quick(t5, group->b, group->field)
-        || !group->meth->field_mul(group, t4, t4, t5, ctx)
-        || !group->meth->field_mul(group, t6, t2, group->a, ctx)
-        || !group->meth->field_mul(group, t5, r->X, p->X, ctx)
-        || !BN_mod_add_quick(t5, t6, t5, group->field)
-        || !group->meth->field_mul(group, t6, r->Z, p->X, ctx)
-        || !BN_mod_add_quick(t2, t6, t1, group->field)
-        || !group->meth->field_mul(group, t5, t5, t2, ctx)
-        || !BN_mod_sub_quick(t6, t6, t1, group->field)
-        || !group->meth->field_sqr(group, t6, t6, ctx)
-        || !group->meth->field_mul(group, t6, t6, s->X, ctx)
-        || !BN_mod_add_quick(t4, t5, t4, group->field)
-        || !group->meth->field_mul(group, t4, t4, s->Z, ctx)
-        || !BN_mod_sub_quick(t4, t4, t6, group->field)
-        || !group->meth->field_sqr(group, t5, r->Z, ctx)
-        || !group->meth->field_mul(group, r->Z, p->Z, s->Z, ctx)
-        || !group->meth->field_mul(group, r->Z, t5, r->Z, ctx)
-        || !group->meth->field_mul(group, r->Z, r->Z, t0, ctx)
-        /* t3 := X, t4 := Y */
-        /* (X,Y,Z) -> (XZ,YZ**2,Z) */
-        || !group->meth->field_mul(group, r->X, t3, r->Z, ctx)
+        || !BN_mod_lshift1_quick(t4, p->Y, group->field)
+        || !group->meth->field_mul(group, t6, r->X, t4, ctx)
+        || !group->meth->field_mul(group, t6, s->Z, t6, ctx)
+        || !group->meth->field_mul(group, t5, r->Z, t6, ctx)
+        || !BN_mod_lshift1_quick(t1, group->b, group->field)
+        || !group->meth->field_mul(group, t1, s->Z, t1, ctx)
         || !group->meth->field_sqr(group, t3, r->Z, ctx)
-        || !group->meth->field_mul(group, r->Y, t4, t3, ctx))
+        || !group->meth->field_mul(group, t2, t3, t1, ctx)
+        || !group->meth->field_mul(group, t6, r->Z, group->a, ctx)
+        || !group->meth->field_mul(group, t1, p->X, r->X, ctx)
+        || !BN_mod_add_quick(t1, t1, t6, group->field)
+        || !group->meth->field_mul(group, t1, s->Z, t1, ctx)
+        || !group->meth->field_mul(group, t0, p->X, r->Z, ctx)
+        || !BN_mod_add_quick(t6, r->X, t0, group->field)
+        || !group->meth->field_mul(group, t6, t6, t1, ctx)
+        || !BN_mod_add_quick(t6, t6, t2, group->field)
+        || !BN_mod_sub_quick(t0, t0, r->X, group->field)
+        || !group->meth->field_sqr(group, t0, t0, ctx)
+        || !group->meth->field_mul(group, t0, t0, s->X, ctx)
+        || !BN_mod_sub_quick(t0, t6, t0, group->field)
+        || !group->meth->field_mul(group, t1, s->Z, t4, ctx)
+        || !group->meth->field_mul(group, t1, t3, t1, ctx)
+        || (group->meth->field_decode != NULL
+            && !group->meth->field_decode(group, t1, t1, ctx))
+        || !group->meth->field_inv(group, t1, t1, ctx)
+        || (group->meth->field_encode != NULL
+            && !group->meth->field_encode(group, t1, t1, ctx))
+        || !group->meth->field_mul(group, r->X, t5, t1, ctx)
+        || !group->meth->field_mul(group, r->Y, t0, t1, ctx))
         goto err;
 
+    if (group->meth->field_set_to_one != NULL) {
+        if (!group->meth->field_set_to_one(group, r->Z, ctx))
+            goto err;
+    } else {
+        if (!BN_one(r->Z))
+            goto err;
+    }
+
+    r->Z_is_one = 1;
     ret = 1;
 
  err: