diff --git a/.travis.yml b/.travis.yml
index 12692deafa..3a85e8cba0 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -19,6 +19,7 @@ env:
     - FIELD=32bit
     - FIELD=32bit     ENDOMORPHISM=yes
     - BIGNUM=none
+    - BIGNUM=none     ENDOMORPHISM=yes
     - BUILD=distcheck
     - EXTRAFLAGS=CFLAGS=-DDETERMINISTIC
 before_script: ./autogen.sh
diff --git a/configure.ac b/configure.ac
index f0d69f93de..6e6fccd7fd 100644
--- a/configure.ac
+++ b/configure.ac
@@ -270,10 +270,7 @@ if test x"$set_field" = x"gmp" || test x"$set_bignum" = x"gmp"; then
 fi
 
 if test x"$use_endomorphism" = x"yes"; then
-  if test x"$set_bignum" = x"none"; then
-    AC_MSG_ERROR([Cannot use endomorphism optimization without a bignum implementation])
-  fi
-  AC_DEFINE(USE_ENDOMORPHISM, 1, [Define this symbol to use endomorphism])
+  AC_DEFINE(USE_ENDOMORPHISM, 1, [Define this symbol to use endomorphism optimization])
 fi
 
 AC_MSG_NOTICE([Using field implementation: $set_field])
diff --git a/src/ecmult_impl.h b/src/ecmult_impl.h
index bbf731c88a..445b81593f 100644
--- a/src/ecmult_impl.h
+++ b/src/ecmult_impl.h
@@ -168,10 +168,10 @@ static void secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const
     secp256k1_scalar_split_lambda_var(&na_1, &na_lam, na);
 
     /* build wnaf representation for na_1 and na_lam. */
-    int wnaf_na_1[129];   int bits_na_1   = secp256k1_ecmult_wnaf(wnaf_na_1,   &na_1,   WINDOW_A);
-    int wnaf_na_lam[129]; int bits_na_lam = secp256k1_ecmult_wnaf(wnaf_na_lam, &na_lam, WINDOW_A);
-    VERIFY_CHECK(bits_na_1 <= 129);
-    VERIFY_CHECK(bits_na_lam <= 129);
+    int wnaf_na_1[130];   int bits_na_1   = secp256k1_ecmult_wnaf(wnaf_na_1,   &na_1,   WINDOW_A);
+    int wnaf_na_lam[130]; int bits_na_lam = secp256k1_ecmult_wnaf(wnaf_na_lam, &na_lam, WINDOW_A);
+    VERIFY_CHECK(bits_na_1 <= 130);
+    VERIFY_CHECK(bits_na_lam <= 130);
     int bits = bits_na_1;
     if (bits_na_lam > bits) bits = bits_na_lam;
 #else
diff --git a/src/scalar_impl.h b/src/scalar_impl.h
index df7a24e1f8..7fc159df77 100644
--- a/src/scalar_impl.h
+++ b/src/scalar_impl.h
@@ -29,7 +29,7 @@ typedef struct {
     secp256k1_num_t order;
 #endif
 #ifdef USE_ENDOMORPHISM
-    secp256k1_num_t a1b2, b1, a2, g1, g2;
+    secp256k1_scalar_t minus_lambda, minus_b1, minus_b2, g1, g2;
 #endif
 } secp256k1_scalar_consts_t;
 
@@ -52,20 +52,30 @@ static void secp256k1_scalar_start(void) {
     secp256k1_num_set_bin(&ret->order, secp256k1_scalar_consts_order, sizeof(secp256k1_scalar_consts_order));
 #endif
 #ifdef USE_ENDOMORPHISM
-    static const unsigned char secp256k1_scalar_consts_a1b2[] = {
-        0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,
-        0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15
-    };
-    static const unsigned char secp256k1_scalar_consts_b1[] = {
-        0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,
-        0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3
-    };
-    static const unsigned char secp256k1_scalar_consts_a2[] = {
-        0x01,
-        0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,
-        0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8
+    /**
+     * Lambda is a scalar which has the property for secp256k1 that point multiplication by
+     * it is efficiently computable (see secp256k1_gej_mul_lambda). */
+    static const unsigned char secp256k1_scalar_consts_lambda[32] = {
+         0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,
+         0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
+         0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,
+         0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72
     };
     /**
+     * "Guide to Elliptic Curve Cryptography" (Hankerson, Menezes, Vanstone) gives an algorithm
+     * (algorithm 3.74) to find k1 and k2 given k, such that k1 + k2 * lambda == k mod n, and k1
+     * and k2 have a small size.
+     * It relies on constants a1, b1, a2, b2. These constants for the value of lambda above are:
+     *
+     * - a1 =      {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
+     * - b1 =     -{0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3}
+     * - a2 = {0x01,0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8}
+     * - b2 =      {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
+     *
+     * The algorithm then computes c1 = round(b1 * k / n) and c2 = round(b2 * k / n), and gives
+     * k1 = k - (c1*a1 + c2*a2) and k2 = -(c1*b1 + c2*b2). Instead, we use modular arithmetic, and
+     * compute k1 as k - k2 * lambda, avoiding the need for constants a1 and a2.
+     *
      * g1, g2 are precomputed constants used to replace division with a rounded multiplication
      * when decomposing the scalar for an endomorphism-based point multiplication.
      *
@@ -82,21 +92,38 @@ static void secp256k1_scalar_start(void) {
      * (Note that 'd' is also equal to the curve order here because [a1,b1] and [a2,b2] are found
      * as outputs of the Extended Euclidean Algorithm on inputs 'order' and 'lambda').
      */
-    static const unsigned char secp256k1_scalar_consts_g1[] = {
-        0x30,0x86,
+    static const unsigned char secp256k1_scalar_consts_minus_b1[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,
+        0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3
+    };
+    static const unsigned char secp256k1_scalar_consts_b2[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,
+        0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15
+    };
+    static const unsigned char secp256k1_scalar_consts_g1[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0x30,0x86,
         0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,
         0x90,0xe4,0x92,0x84,0xeb,0x15,0x3d,0xab
     };
-    static const unsigned char secp256k1_scalar_consts_g2[] = {
-        0xe4,0x43,
+    static const unsigned char secp256k1_scalar_consts_g2[32] = {
+        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
+        0x00,0x00,0x00,0x00,0x00,0x00,0xe4,0x43,
         0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,
         0x7f,0xa9,0x0a,0xbf,0xe4,0xc4,0x22,0x12
     };
-    secp256k1_num_set_bin(&ret->a1b2, secp256k1_scalar_consts_a1b2, sizeof(secp256k1_scalar_consts_a1b2));
-    secp256k1_num_set_bin(&ret->a2, secp256k1_scalar_consts_a2, sizeof(secp256k1_scalar_consts_a2));
-    secp256k1_num_set_bin(&ret->b1, secp256k1_scalar_consts_b1, sizeof(secp256k1_scalar_consts_b1));
-    secp256k1_num_set_bin(&ret->g1, secp256k1_scalar_consts_g1, sizeof(secp256k1_scalar_consts_g1));
-    secp256k1_num_set_bin(&ret->g2, secp256k1_scalar_consts_g2, sizeof(secp256k1_scalar_consts_g2));
+
+    secp256k1_scalar_set_b32(&ret->minus_lambda, secp256k1_scalar_consts_lambda, NULL);
+    secp256k1_scalar_negate(&ret->minus_lambda, &ret->minus_lambda);
+    secp256k1_scalar_set_b32(&ret->minus_b1, secp256k1_scalar_consts_minus_b1, NULL);
+    secp256k1_scalar_set_b32(&ret->minus_b2, secp256k1_scalar_consts_b2, NULL);
+    secp256k1_scalar_negate(&ret->minus_b2, &ret->minus_b2);
+    secp256k1_scalar_set_b32(&ret->g1, secp256k1_scalar_consts_g1, NULL);
+    secp256k1_scalar_set_b32(&ret->g2, secp256k1_scalar_consts_g2, NULL);
 #endif
 
     /* Set the global pointer. */
@@ -293,47 +320,16 @@ static void secp256k1_scalar_inverse_var(secp256k1_scalar_t *r, const secp256k1_
 
 #ifdef USE_ENDOMORPHISM
 static void secp256k1_scalar_split_lambda_var(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) {
-    unsigned char b[32];
-    secp256k1_scalar_get_b32(b, a);
-    secp256k1_num_t na;
-    secp256k1_num_set_bin(&na, b, 32);
-
-    secp256k1_num_t rn1, rn2;
-
-    const secp256k1_scalar_consts_t *c = secp256k1_scalar_consts;
-    secp256k1_num_t d1, d2, t, one;
-    unsigned char cone[1] = {0x01};
-    secp256k1_num_set_bin(&one, cone, 1);
-
-    secp256k1_num_mul(&d1, &na, &c->g1);
-    secp256k1_num_shift(&d1, 271);
-    secp256k1_num_add(&d1, &d1, &one);
-    secp256k1_num_shift(&d1, 1);
-
-    secp256k1_num_mul(&d2, &na, &c->g2);
-    secp256k1_num_shift(&d2, 271);
-    secp256k1_num_add(&d2, &d2, &one);
-    secp256k1_num_shift(&d2, 1);
-
-    secp256k1_num_mul(&t, &d1, &c->a1b2);
-    secp256k1_num_sub(&rn1, &na, &t);
-    secp256k1_num_mul(&t, &d2, &c->a2);
-    secp256k1_num_sub(&rn1, &rn1, &t);
-
-    secp256k1_num_mul(&rn2, &d1, &c->b1);
-    secp256k1_num_mul(&t, &d2, &c->a1b2);
-    secp256k1_num_sub(&rn2, &rn2, &t);
-
-    secp256k1_num_get_bin(b, 32, &rn1);
-    secp256k1_scalar_set_b32(r1, b, NULL);
-    if (secp256k1_num_is_neg(&rn1)) {
-        secp256k1_scalar_negate(r1, r1);
-    }
-    secp256k1_num_get_bin(b, 32, &rn2);
-    secp256k1_scalar_set_b32(r2, b, NULL);
-    if (secp256k1_num_is_neg(&rn2)) {
-        secp256k1_scalar_negate(r2, r2);
-    }
+    VERIFY_CHECK(r1 != a);
+    VERIFY_CHECK(r2 != a);
+    secp256k1_scalar_t c1, c2;
+    secp256k1_scalar_mul_shift_var(&c1, a, &secp256k1_scalar_consts->g1, 272);
+    secp256k1_scalar_mul_shift_var(&c2, a, &secp256k1_scalar_consts->g2, 272);
+    secp256k1_scalar_mul(&c1, &c1, &secp256k1_scalar_consts->minus_b1);
+    secp256k1_scalar_mul(&c2, &c2, &secp256k1_scalar_consts->minus_b2);
+    secp256k1_scalar_add(r2, &c1, &c2);
+    secp256k1_scalar_mul(r1, r2, &secp256k1_scalar_consts->minus_lambda);
+    secp256k1_scalar_add(r1, r1, a);
 }
 #endif