diff --git a/include/secp256k1.h b/include/secp256k1.h
index bcc2be0379..06afd4c65b 100644
--- a/include/secp256k1.h
+++ b/include/secp256k1.h
@@ -328,6 +328,18 @@ SECP256K1_WARN_UNUSED_RESULT int secp256k1_ec_pubkey_tweak_mul(
   const unsigned char *tweak
 ) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2) SECP256K1_ARG_NONNULL(4);
 
+/** Updates the context randomization.
+ *  Returns: 1: randomization successfully updated
+ *           0: error
+ *  In:      ctx:       pointer to a context object (cannot be NULL)
+ *           seed32:    pointer to a 32-byte random seed (NULL resets to initial state)
+ */
+SECP256K1_WARN_UNUSED_RESULT int secp256k1_context_randomize(
+  secp256k1_context_t* ctx,
+  const unsigned char *seed32
+) SECP256K1_ARG_NONNULL(1);
+
+
 # ifdef __cplusplus
 }
 # endif
diff --git a/src/ecmult_gen.h b/src/ecmult_gen.h
index c65e5cd0e0..3745633c47 100644
--- a/src/ecmult_gen.h
+++ b/src/ecmult_gen.h
@@ -24,6 +24,8 @@ typedef struct {
      * the intermediate sums while computing a*G.
      */
     secp256k1_ge_storage_t (*prec)[64][16]; /* prec[j][i] = 16^j * i * G + U_i */
+    secp256k1_scalar_t blind;
+    secp256k1_gej_t initial;
 } secp256k1_ecmult_gen_context_t;
 
 static void secp256k1_ecmult_gen_context_init(secp256k1_ecmult_gen_context_t* ctx);
@@ -36,4 +38,6 @@ static int secp256k1_ecmult_gen_context_is_built(const secp256k1_ecmult_gen_cont
 /** Multiply with the generator: R = a*G */
 static void secp256k1_ecmult_gen(const secp256k1_ecmult_gen_context_t* ctx, secp256k1_gej_t *r, const secp256k1_scalar_t *a);
 
+static void secp256k1_ecmult_gen_blind(secp256k1_ecmult_gen_context_t *ctx, const unsigned char *seed32);
+
 #endif
diff --git a/src/ecmult_gen_impl.h b/src/ecmult_gen_impl.h
index 9c8f7209e7..4697753ac8 100644
--- a/src/ecmult_gen_impl.h
+++ b/src/ecmult_gen_impl.h
@@ -1,5 +1,5 @@
 /**********************************************************************
- * Copyright (c) 2013, 2014 Pieter Wuille                             *
+ * Copyright (c) 2013, 2014, 2015 Pieter Wuille, Gregory Maxwell      *
  * Distributed under the MIT software license, see the accompanying   *
  * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
  **********************************************************************/
@@ -10,6 +10,7 @@
 #include "scalar.h"
 #include "group.h"
 #include "ecmult_gen.h"
+#include "hash_impl.h"
 
 static void secp256k1_ecmult_gen_context_init(secp256k1_ecmult_gen_context_t *ctx) {
     ctx->prec = NULL;
@@ -74,6 +75,7 @@ static void secp256k1_ecmult_gen_context_build(secp256k1_ecmult_gen_context_t *c
             secp256k1_ge_to_storage(&(*ctx->prec)[j][i], &prec[j*16 + i]);
         }
     }
+    secp256k1_ecmult_gen_blind(ctx, NULL);
 }
 
 static int secp256k1_ecmult_gen_context_is_built(const secp256k1_ecmult_gen_context_t* ctx) {
@@ -87,24 +89,31 @@ static void secp256k1_ecmult_gen_context_clone(secp256k1_ecmult_gen_context_t *d
     } else {
         dst->prec = (secp256k1_ge_storage_t (*)[64][16])checked_malloc(sizeof(*dst->prec));
         memcpy(dst->prec, src->prec, sizeof(*dst->prec));
+        dst->initial = src->initial;
+        dst->blind = src->blind;
     }
 }
 
 static void secp256k1_ecmult_gen_context_clear(secp256k1_ecmult_gen_context_t *ctx) {
     free(ctx->prec);
+    secp256k1_scalar_clear(&ctx->blind);
+    secp256k1_gej_clear(&ctx->initial);
     ctx->prec = NULL;
 }
 
 static void secp256k1_ecmult_gen(const secp256k1_ecmult_gen_context_t *ctx, secp256k1_gej_t *r, const secp256k1_scalar_t *gn) {
     secp256k1_ge_t add;
     secp256k1_ge_storage_t adds;
+    secp256k1_scalar_t gnb;
     int bits;
     int i, j;
     memset(&adds, 0, sizeof(adds));
-    secp256k1_gej_set_infinity(r);
+    *r = ctx->initial;
+    /* Blind scalar/point multiplication by computing (n-b)G + bG instead of nG. */
+    secp256k1_scalar_add(&gnb, gn, &ctx->blind);
     add.infinity = 0;
     for (j = 0; j < 64; j++) {
-        bits = secp256k1_scalar_get_bits(gn, j * 4, 4);
+        bits = secp256k1_scalar_get_bits(&gnb, j * 4, 4);
         for (i = 0; i < 16; i++) {
             /** This uses a conditional move to avoid any secret data in array indexes.
              *   _Any_ use of secret indexes has been demonstrated to result in timing
@@ -123,6 +132,53 @@ static void secp256k1_ecmult_gen(const secp256k1_ecmult_gen_context_t *ctx, secp
     }
     bits = 0;
     secp256k1_ge_clear(&add);
+    secp256k1_scalar_clear(&gnb);
+}
+
+/* Setup blinding values for secp256k1_ecmult_gen. */
+static void secp256k1_ecmult_gen_blind(secp256k1_ecmult_gen_context_t *ctx, const unsigned char *seed32) {
+    secp256k1_scalar_t b;
+    secp256k1_gej_t gb;
+    secp256k1_fe_t s;
+    unsigned char nonce32[32];
+    secp256k1_rfc6979_hmac_sha256_t rng;
+    int retry;
+    if (!seed32) {
+        /* When seed is NULL, reset the initial point and blinding value. */
+        secp256k1_gej_set_ge(&ctx->initial, &secp256k1_ge_const_g);
+        secp256k1_gej_neg(&ctx->initial, &ctx->initial);
+        secp256k1_scalar_set_int(&ctx->blind, 1);
+    }
+    /* The prior blinding value (if not reset) is chained forward by including it in the hash. */
+    secp256k1_scalar_get_b32(nonce32, &ctx->blind);
+    /** Using a CSPRNG allows a failure free interface, avoids needing large amounts of random data,
+     *   and guards against weak or adversarial seeds.  This is a simpler and safer interface than
+     *   asking the caller for blinding values directly and expecting them to retry on failure.
+     */
+    secp256k1_rfc6979_hmac_sha256_initialize(&rng, seed32 ? seed32 : nonce32, 32, nonce32, 32, NULL, 0);
+    /* Retry for out of range results to achieve uniformity. */
+    do {
+        secp256k1_rfc6979_hmac_sha256_generate(&rng, nonce32, 32);
+        retry = !secp256k1_fe_set_b32(&s, nonce32);
+        retry |= secp256k1_fe_is_zero(&s);
+    } while (retry);
+    /* Randomize the projection to defend against multiplier sidechannels. */
+    secp256k1_gej_rescale(&ctx->initial, &s);
+    secp256k1_fe_clear(&s);
+    do {
+        secp256k1_rfc6979_hmac_sha256_generate(&rng, nonce32, 32);
+        secp256k1_scalar_set_b32(&b, nonce32, &retry);
+        /* A blinding value of 0 works, but would undermine the projection hardening. */
+        retry |= secp256k1_scalar_is_zero(&b);
+    } while (retry);
+    secp256k1_rfc6979_hmac_sha256_finalize(&rng);
+    memset(nonce32, 0, 32);
+    secp256k1_ecmult_gen(ctx, &gb, &b);
+    secp256k1_scalar_negate(&b, &b);
+    ctx->blind = b;
+    ctx->initial = gb;
+    secp256k1_scalar_clear(&b);
+    secp256k1_gej_clear(&gb);
 }
 
 #endif
diff --git a/src/group.h b/src/group.h
index d1e5834909..0b08b3b991 100644
--- a/src/group.h
+++ b/src/group.h
@@ -115,4 +115,7 @@ static void secp256k1_ge_from_storage(secp256k1_ge_t *r, const secp256k1_ge_stor
 /** If flag is true, set *r equal to *a; otherwise leave it. Constant-time. */
 static void secp256k1_ge_storage_cmov(secp256k1_ge_storage_t *r, const secp256k1_ge_storage_t *a, int flag);
 
+/** Rescale a jacobian point by b which must be non-zero. Constant-time. */
+static void secp256k1_gej_rescale(secp256k1_gej_t *r, const secp256k1_fe_t *b);
+
 #endif
diff --git a/src/group_impl.h b/src/group_impl.h
index 131393ff7b..0f64576fbb 100644
--- a/src/group_impl.h
+++ b/src/group_impl.h
@@ -396,6 +396,17 @@ static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, c
     r->infinity = infinity;
 }
 
+static void secp256k1_gej_rescale(secp256k1_gej_t *r, const secp256k1_fe_t *s) {
+    /* Operations: 4 mul, 1 sqr */
+    secp256k1_fe_t zz;
+    VERIFY_CHECK(!secp256k1_fe_is_zero(s));
+    secp256k1_fe_sqr(&zz, s);
+    secp256k1_fe_mul(&r->x, &r->x, &zz);                /* r->x *= s^2 */
+    secp256k1_fe_mul(&r->y, &r->y, &zz);
+    secp256k1_fe_mul(&r->y, &r->y, s);                  /* r->y *= s^3 */
+    secp256k1_fe_mul(&r->z, &r->z, s);                  /* r->z *= s   */
+}
+
 static void secp256k1_ge_to_storage(secp256k1_ge_storage_t *r, const secp256k1_ge_t *a) {
     secp256k1_fe_t x, y;
     VERIFY_CHECK(!a->infinity);
diff --git a/src/secp256k1.c b/src/secp256k1.c
index 7445ae98e8..d6192dc4ed 100644
--- a/src/secp256k1.c
+++ b/src/secp256k1.c
@@ -410,3 +410,10 @@ int secp256k1_ec_privkey_import(const secp256k1_context_t* ctx, unsigned char *s
     secp256k1_scalar_clear(&key);
     return ret;
 }
+
+int secp256k1_context_randomize(secp256k1_context_t* ctx, const unsigned char *seed32) {
+    DEBUG_CHECK(ctx != NULL);
+    DEBUG_CHECK(secp256k1_ecmult_gen_context_is_built(&ctx->ecmult_gen_ctx));
+    secp256k1_ecmult_gen_blind(&ctx->ecmult_gen_ctx, seed32);
+    return 1;
+}
diff --git a/src/tests.c b/src/tests.c
index d391488b61..31dbc9dd60 100644
--- a/src/tests.c
+++ b/src/tests.c
@@ -907,6 +907,28 @@ void ge_equals_ge(const secp256k1_ge_t *a, const secp256k1_ge_t *b) {
     CHECK(secp256k1_fe_equal_var(&b->y, &b->y));
 }
 
+/* This compares jacobian points including their Z, not just their geometric meaning. */
+int gej_xyz_equals_gej(const secp256k1_gej_t *a, const secp256k1_gej_t *b) {
+    secp256k1_gej_t a2;
+    secp256k1_gej_t b2;
+    int ret = 1;
+    ret &= a->infinity == b->infinity;
+    if (ret && !a->infinity) {
+        a2 = *a;
+        b2 = *b;
+        secp256k1_fe_normalize(&a2.x);
+        secp256k1_fe_normalize(&a2.y);
+        secp256k1_fe_normalize(&a2.z);
+        secp256k1_fe_normalize(&b2.x);
+        secp256k1_fe_normalize(&b2.y);
+        secp256k1_fe_normalize(&b2.z);
+        ret &= secp256k1_fe_cmp_var(&a2.x, &b2.x) == 0;
+        ret &= secp256k1_fe_cmp_var(&a2.y, &b2.y) == 0;
+        ret &= secp256k1_fe_cmp_var(&a2.z, &b2.z) == 0;
+    }
+    return ret;
+}
+
 void ge_equals_gej(const secp256k1_ge_t *a, const secp256k1_gej_t *b) {
     secp256k1_fe_t z2s;
     secp256k1_fe_t u1, u2, s1, s2;
@@ -1033,6 +1055,9 @@ void test_ge(void) {
         secp256k1_ge_t *ge_set_all = (secp256k1_ge_t *)malloc((4 * runs + 1) * sizeof(secp256k1_ge_t));
         secp256k1_ge_set_all_gej_var(4 * runs + 1, ge_set_all, gej);
         for (i = 0; i < 4 * runs + 1; i++) {
+            secp256k1_fe_t s;
+            random_fe_non_zero(&s);
+            secp256k1_gej_rescale(&gej[i], &s);
             ge_equals_gej(&ge_set_all[i], &gej[i]);
         }
         free(ge_set_all);
@@ -1203,6 +1228,87 @@ void run_wnaf(void) {
     }
 }
 
+void test_ecmult_constants(void) {
+    /* Test ecmult_gen() for [0..36) and [order-36..0). */
+    secp256k1_scalar_t x;
+    secp256k1_gej_t r;
+    secp256k1_ge_t ng;
+    int i;
+    int j;
+    secp256k1_ge_neg(&ng, &secp256k1_ge_const_g);
+    for (i = 0; i < 36; i++ ) {
+        secp256k1_scalar_set_int(&x, i);
+        secp256k1_ecmult_gen(&ctx->ecmult_gen_ctx, &r, &x);
+        for (j = 0; j < i; j++) {
+            if (j == i - 1) {
+                ge_equals_gej(&secp256k1_ge_const_g, &r);
+            }
+            secp256k1_gej_add_ge(&r, &r, &ng);
+        }
+        CHECK(secp256k1_gej_is_infinity(&r));
+    }
+    for (i = 1; i <= 36; i++ ) {
+        secp256k1_scalar_set_int(&x, i);
+        secp256k1_scalar_negate(&x, &x);
+        secp256k1_ecmult_gen(&ctx->ecmult_gen_ctx, &r, &x);
+        for (j = 0; j < i; j++) {
+            if (j == i - 1) {
+                ge_equals_gej(&ng, &r);
+            }
+            secp256k1_gej_add_ge(&r, &r, &secp256k1_ge_const_g);
+        }
+        CHECK(secp256k1_gej_is_infinity(&r));
+    }
+}
+
+void run_ecmult_constants(void) {
+    test_ecmult_constants();
+}
+
+void test_ecmult_gen_blind(void) {
+    /* Test ecmult_gen() blinding and confirm that the blinding changes, the affline points match, and the z's don't match. */
+    secp256k1_scalar_t key;
+    secp256k1_scalar_t b;
+    unsigned char seed32[32];
+    secp256k1_gej_t pgej;
+    secp256k1_gej_t pgej2;
+    secp256k1_gej_t i;
+    secp256k1_ge_t pge;
+    random_scalar_order_test(&key);
+    secp256k1_ecmult_gen(&ctx->ecmult_gen_ctx, &pgej, &key);
+    secp256k1_rand256(seed32);
+    b = ctx->ecmult_gen_ctx.blind;
+    i = ctx->ecmult_gen_ctx.initial;
+    secp256k1_ecmult_gen_blind(&ctx->ecmult_gen_ctx, seed32);
+    CHECK(!secp256k1_scalar_eq(&b, &ctx->ecmult_gen_ctx.blind));
+    secp256k1_ecmult_gen(&ctx->ecmult_gen_ctx, &pgej2, &key);
+    CHECK(!gej_xyz_equals_gej(&pgej, &pgej2));
+    CHECK(!gej_xyz_equals_gej(&i, &ctx->ecmult_gen_ctx.initial));
+    secp256k1_ge_set_gej(&pge, &pgej);
+    ge_equals_gej(&pge, &pgej2);
+}
+
+void test_ecmult_gen_blind_reset(void) {
+    /* Test ecmult_gen() blinding reset and confirm that the blinding is consistent. */
+    secp256k1_scalar_t b;
+    secp256k1_gej_t initial;
+    secp256k1_ecmult_gen_blind(&ctx->ecmult_gen_ctx, 0);
+    b = ctx->ecmult_gen_ctx.blind;
+    initial = ctx->ecmult_gen_ctx.initial;
+    secp256k1_ecmult_gen_blind(&ctx->ecmult_gen_ctx, 0);
+    CHECK(secp256k1_scalar_eq(&b, &ctx->ecmult_gen_ctx.blind));
+    CHECK(gej_xyz_equals_gej(&initial, &ctx->ecmult_gen_ctx.initial));
+}
+
+void run_ecmult_gen_blind(void) {
+    int i;
+    test_ecmult_gen_blind_reset();
+    for (i = 0; i < 10; i++) {
+        test_ecmult_gen_blind();
+    }
+}
+
+
 void random_sign(secp256k1_ecdsa_sig_t *sig, const secp256k1_scalar_t *key, const secp256k1_scalar_t *msg, int *recid) {
     secp256k1_scalar_t nonce;
     do {
@@ -1913,6 +2019,11 @@ int main(int argc, char **argv) {
     run_context_tests();
     ctx = secp256k1_context_create(SECP256K1_CONTEXT_SIGN | SECP256K1_CONTEXT_VERIFY);
 
+    if (secp256k1_rand32() & 1) {
+        secp256k1_rand256(run32);
+        CHECK(secp256k1_context_randomize(ctx, secp256k1_rand32() & 1 ? run32 : NULL));
+    }
+
     run_sha256_tests();
     run_hmac_sha256_tests();
     run_rfc6979_hmac_sha256_tests();
@@ -1941,6 +2052,8 @@ int main(int argc, char **argv) {
     run_wnaf();
     run_point_times_order();
     run_ecmult_chain();
+    run_ecmult_constants();
+    run_ecmult_gen_blind();
 
     /* ecdsa tests */
     run_random_pubkeys();