Replace GFp ladder implementation with ladd-2002-it-4 from EFD

The EFD database does not state that the "ladd-2002-it-3" algorithm
assumes X1 != 0.
Consequently the current implementation, based on it, fails to compute
correctly if the affine x coordinate of the scalar multiplication input
point is 0.

We replace this implementation using the alternative algorithm based on
Eq. (9) and (10) from the same paper, which being derived from the
additive relation of (6) does not incur in this problem, but costs one
extra field multiplication.

The EFD entry for this algorithm is at
https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-ladd-2002-it-4
and the code to implement it was generated with tooling.

Regression tests add one positive test for each named curve that has
such a point. The `SharedSecret` was generated independently from the
OpenSSL codebase with sage.

This bug was originally reported by Dmitry Belyavsky on the
openssl-users maling list:
https://mta.openssl.org/pipermail/openssl-users/2018-August/008540.html

Co-authored-by: Billy Brumley <bbrumley@gmail.com>

Reviewed-by: Andy Polyakov <appro@openssl.org>
Reviewed-by: Tim Hudson <tjh@openssl.org>
(Merged from https://github.com/openssl/openssl/pull/7000)

1 files changed, 33 insertions, 30 deletions

diff --git a/crypto/ec/ecp_smpl.c b/crypto/ec/ecp_smpl.c
index 7ac519ca03..d0c5557ff4 100644
--- a/crypto/ec/ecp_smpl.c
+++ b/crypto/ec/ecp_smpl.c
@@ -1483,10 +1483,10 @@ int ec_GFp_simple_ladder_pre(const EC_GROUP *group,
 }
 
 /*-
- * Differential addition-and-doubling using  Eq. (8) and (10) from Izu-Takagi
+ * 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-3
+ * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-ladd-2002-it-4
  */
 int ec_GFp_simple_ladder_step(const EC_GROUP *group,
                               EC_POINT *r, EC_POINT *s,
@@ -1511,39 +1511,42 @@ int ec_GFp_simple_ladder_step(const EC_GROUP *group,
         || !group->meth->field_mul(group, t2, 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_sub_quick(t4, t0, t4, group->field)
-        || !BN_mod_add_quick(t5, t3, t2, group->field)
-        || !group->meth->field_sqr(group, t4, t4, ctx)
-        || !group->meth->field_mul(group, t5, t1, t5, ctx)
-        || !BN_mod_lshift_quick(t0, group->b, 2, group->field)
-        || !group->meth->field_mul(group, t5, t0, t5, ctx)
-        || !BN_mod_sub_quick(t5, t4, t5, group->field)
+        || !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 */
-        || !group->meth->field_mul(group, s->X, t5, p->Z, ctx)
-        || !BN_mod_sub_quick(t3, t2, t3, group->field)
-        || !group->meth->field_sqr(group, t3, t3, ctx)
+        || !BN_mod_sub_quick(s->X, t0, t3, group->field)
         /* s->Z coord output */
-        || !group->meth->field_mul(group, s->Z, t3, p->X, ctx)
-        || !group->meth->field_sqr(group, t2, r->X, ctx)
-        || !group->meth->field_sqr(group, t4, r->Z, ctx)
-        || !group->meth->field_mul(group, t1, t4, group->a, ctx)
-        || !BN_mod_add_quick(t6, r->X, r->Z, group->field)
+        || !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)
-        || !BN_mod_sub_quick(t6, t6, t2, group->field)
-        || !BN_mod_sub_quick(t6, t6, t4, group->field)
-        || !BN_mod_sub_quick(t7, t2, t1, group->field)
-        || !group->meth->field_sqr(group, t7, t7, ctx)
-        || !group->meth->field_mul(group, t5, t4, t6, ctx)
-        || !group->meth->field_mul(group, t5, t0, t5, 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, t7, t5, group->field)
-        || !BN_mod_add_quick(t2, t2, t1, group->field)
-        || !group->meth->field_sqr(group, t5, t4, ctx)
-        || !group->meth->field_mul(group, t5, t5, t0, ctx)
-        || !group->meth->field_mul(group, t6, t6, t2, ctx)
-        || !BN_mod_lshift1_quick(t6, t6, group->field)
+        || !BN_mod_sub_quick(r->X, t6, t0, 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)
+        || !BN_mod_lshift1_quick(t5, t5, group->field)
         /* r->Z coord output */
-        || !BN_mod_add_quick(r->Z, t5, t6, group->field))
+        || !BN_mod_add_quick(r->Z, t7, t5, group->field))
         goto err;
 
     ret = 1;