diff --git a/Makefile b/Makefile
index bb931d3e61..eebc6e9c96 100644
--- a/Makefile
+++ b/Makefile
@@ -16,6 +16,11 @@ OBJS :=
 
 default: all
 
+ifeq ($(CONF), gmp32)
+FLAGS_COMMON := $(FLAGS_COMMON) -DUSE_NUM_GMP -DUSE_FIELD_10X26
+LIBS := -lgmp
+SECP256K1_FILES := $(SECP256K1_FILES) src/num_gmp.h src/num_gmp.c src/field_10x26.c src/field_10x26.h
+else
 ifeq ($(CONF), openssl)
 FLAGS_COMMON := $(FLAGS_COMMON) -DUSE_NUM_OPENSSL -DUSE_FIELD_INV_BUILTIN
 LIBS := -lcrypto
@@ -39,16 +44,19 @@ SECP256K1_FILES := $(SECP256K1_FILES) src/field_5x52_int128.c
 endif
 endif
 endif
+endif
 
 
 all: src/*.c src/*.asm src/*.h include/*.h
 	+make CONF=openssl all-openssl
 	+make CONF=gmp     all-gmp
+	+make CONF=gmp32   all-gmp32
 	+make CONF=gmpasm  all-gmpasm
 
 clean:
 	+make CONF=openssl clean-openssl
 	+make CONF=gmp     clean-gmp
+	+make CONF=gmp32   clean-gmp32
 	+make CONF=gmpasm  clean-gmpasm
 
 bench-any: bench-$(CONF)
@@ -62,10 +70,10 @@ clean-$(CONF):
 obj/secp256k1-$(CONF).o: $(SECP256K1_FILES) src/secp256k1.c include/secp256k1.h
 	$(CC) $(FLAGS_COMMON) $(FLAGS_PROD) src/secp256k1.c -c -o obj/secp256k1-$(CONF).o
 
-bench-$(CONF): $(OBJS) src/bench.c
+bench-$(CONF): $(OBJS) $(SECP256K1_FILES) src/bench.c
 	$(CC) $(FLAGS_COMMON) $(FLAGS_PROD) src/bench.c $(LIBS) $(OBJS) -o bench-$(CONF)
 
-tests-$(CONF): $(OBJS) src/tests.c
+tests-$(CONF): $(OBJS) $(SECP256K1_FILES) src/tests.c
 	$(CC) $(FLAGS_COMMON) $(FLAGS_TEST) src/tests.c $(LIBS) $(OBJS) -o tests-$(CONF)
 
 libsecp256k1-$(CONF).a: $(OBJS) obj/secp256k1-$(CONF).o
diff --git a/src/field.c b/src/field.c
index d795a9f487..e7c636e05c 100644
--- a/src/field.c
+++ b/src/field.c
@@ -1,5 +1,8 @@
-// just one implementation for now
+#ifdef USE_FIELD_10X26
+#include "field_10x26.c"
+#else
 #include "field_5x52.c"
+#endif
 
 void static secp256k1_fe_get_hex(char *r, int *rlen, const secp256k1_fe_t *a) {
     if (*rlen < 65) {
diff --git a/src/field.h b/src/field.h
index a266b878cb..a2896073c1 100644
--- a/src/field.h
+++ b/src/field.h
@@ -12,8 +12,11 @@
  *    normality.
  */
 
-// just one implementation for now
+#ifdef USE_FIELD_10X26
+#include "field_10x26.h"
+#else
 #include "field_5x52.h"
+#endif
 
 typedef struct {
     secp256k1_num_t p;
diff --git a/src/field_10x26.c b/src/field_10x26.c
new file mode 100644
index 0000000000..9f9edb9d69
--- /dev/null
+++ b/src/field_10x26.c
@@ -0,0 +1,475 @@
+#include <stdio.h>
+#include <assert.h>
+#include <string.h>
+#include "num.h"
+#include "field.h"
+
+void static secp256k1_fe_normalize(secp256k1_fe_t *r) {
+//    fog("normalize in: ", r);
+    uint32_t c;
+    c = r->n[0];
+    uint32_t t0 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[1];
+    uint32_t t1 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[2];
+    uint32_t t2 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[3];
+    uint32_t t3 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[4];
+    uint32_t t4 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[5];
+    uint32_t t5 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[6];
+    uint32_t t6 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[7];
+    uint32_t t7 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[8];
+    uint32_t t8 = c & 0x3FFFFFFUL;
+    c = (c >> 26) + r->n[9];
+    uint32_t t9 = c & 0x03FFFFFUL;
+    c >>= 22;
+/*    r->n[0] = t0; r->n[1] = t1; r->n[2] = t2; r->n[3] = t3; r->n[4] = t4;
+    r->n[5] = t5; r->n[6] = t6; r->n[7] = t7; r->n[8] = t8; r->n[9] = t9;
+    fog("         tm1: ", r);
+    fprintf(stderr, "out c= %08lx\n", (unsigned long)c);*/
+
+    // The following code will not modify the t's if c is initially 0.
+    uint32_t d = c * 0x3D1UL + t0;
+    t0 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t1 + c*0x40;
+    t1 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t2;
+    t2 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t3;
+    t3 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t4;
+    t4 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t5;
+    t5 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t6;
+    t6 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t7;
+    t7 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t8;
+    t8 = d & 0x3FFFFFFULL;
+    d = (d >> 26) + t9;
+    t9 = d & 0x03FFFFFULL;
+    assert((d >> 22) == 0);
+/*    r->n[0] = t0; r->n[1] = t1; r->n[2] = t2; r->n[3] = t3; r->n[4] = t4;
+    r->n[5] = t5; r->n[6] = t6; r->n[7] = t7; r->n[8] = t8; r->n[9] = t9;
+    fog("         tm2: ", r); */
+
+    // Subtract p if result >= p
+    uint64_t low = ((uint64_t)t1 << 26) | t0;
+    uint64_t mask = -(int64_t)((t9 < 0x03FFFFFUL) | (t8 < 0x3FFFFFFUL) | (t7 < 0x3FFFFFFUL) | (t6 < 0x3FFFFFFUL) | (t5 < 0x3FFFFFFUL) | (t4 < 0x3FFFFFFUL) | (t3 < 0x3FFFFFFUL) | (t2 < 0x3FFFFFFUL) | (low < 0xFFFFEFFFFFC2FULL));
+    t9 &= mask;
+    t8 &= mask;
+    t7 &= mask;
+    t6 &= mask;
+    t5 &= mask;
+    t4 &= mask;
+    t3 &= mask;
+    t2 &= mask;
+    low -= (~mask & 0xFFFFEFFFFFC2FULL);
+
+    // push internal variables back
+    r->n[0] = low & 0x3FFFFFFUL; r->n[1] = (low >> 26) & 0x3FFFFFFUL; r->n[2] = t2; r->n[3] = t3; r->n[4] = t4;
+    r->n[5] = t5; r->n[6] = t6; r->n[7] = t7; r->n[8] = t8; r->n[9] = t9;
+/*    fog("         out: ", r);*/
+
+#ifdef VERIFY
+    r->magnitude = 1;
+    r->normalized = 1;
+#endif
+}
+
+void static inline secp256k1_fe_set_int(secp256k1_fe_t *r, int a) {
+    r->n[0] = a;
+    r->n[1] = r->n[2] = r->n[3] = r->n[4] = r->n[5] = r->n[6] = r->n[7] = r->n[8] = r->n[9] = 0;
+#ifdef VERIFY
+    r->magnitude = 1;
+    r->normalized = 1;
+#endif
+}
+
+// TODO: not constant time!
+int static inline secp256k1_fe_is_zero(const secp256k1_fe_t *a) {
+#ifdef VERIFY
+    assert(a->normalized);
+#endif
+    return (a->n[0] == 0 && a->n[1] == 0 && a->n[2] == 0 && a->n[3] == 0 && a->n[4] == 0 && a->n[5] == 0 && a->n[6] == 0 && a->n[7] == 0 && a->n[8] == 0 && a->n[9] == 0);
+}
+
+int static inline secp256k1_fe_is_odd(const secp256k1_fe_t *a) {
+#ifdef VERIFY
+    assert(a->normalized);
+#endif
+    return a->n[0] & 1;
+}
+
+// TODO: not constant time!
+int static inline secp256k1_fe_equal(const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+#ifdef VERIFY
+    assert(a->normalized);
+    assert(b->normalized);
+#endif
+    return (a->n[0] == b->n[0] && a->n[1] == b->n[1] && a->n[2] == b->n[2] && a->n[3] == b->n[3] && a->n[4] == b->n[4] &&
+            a->n[5] == b->n[5] && a->n[6] == b->n[6] && a->n[7] == b->n[7] && a->n[8] == b->n[8] && a->n[9] == b->n[9]);
+}
+
+void static secp256k1_fe_set_b32(secp256k1_fe_t *r, const unsigned char *a) {
+    r->n[0] = r->n[1] = r->n[2] = r->n[3] = r->n[4] = 0;
+    r->n[5] = r->n[6] = r->n[7] = r->n[8] = r->n[9] = 0;
+    for (int i=0; i<32; i++) {
+        for (int j=0; j<4; j++) {
+            int limb = (8*i+2*j)/26;
+            int shift = (8*i+2*j)%26;
+            r->n[limb] |= (uint32_t)((a[31-i] >> (2*j)) & 0x3) << shift;
+        }
+    }
+#ifdef VERIFY
+    r->magnitude = 1;
+    r->normalized = 1;
+#endif
+}
+
+/** Convert a field element to a 32-byte big endian value. Requires the input to be normalized */
+void static secp256k1_fe_get_b32(unsigned char *r, const secp256k1_fe_t *a) {
+#ifdef VERIFY
+    assert(a->normalized);
+#endif
+    for (int i=0; i<32; i++) {
+        int c = 0;
+        for (int j=0; j<4; j++) {
+            int limb = (8*i+2*j)/26;
+            int shift = (8*i+2*j)%26;
+            c |= ((a->n[limb] >> shift) & 0x3) << (2 * j);
+        }
+        r[31-i] = c;
+    }
+}
+
+void static inline secp256k1_fe_negate(secp256k1_fe_t *r, const secp256k1_fe_t *a, int m) {
+#ifdef VERIFY
+    assert(a->magnitude <= m);
+    r->magnitude = m + 1;
+    r->normalized = 0;
+#endif
+    r->n[0] = 0x3FFFC2FUL * (m + 1) - a->n[0];
+    r->n[1] = 0x3FFFFBFUL * (m + 1) - a->n[1];
+    r->n[2] = 0x3FFFFFFUL * (m + 1) - a->n[2];
+    r->n[3] = 0x3FFFFFFUL * (m + 1) - a->n[3];
+    r->n[4] = 0x3FFFFFFUL * (m + 1) - a->n[4];
+    r->n[5] = 0x3FFFFFFUL * (m + 1) - a->n[5];
+    r->n[6] = 0x3FFFFFFUL * (m + 1) - a->n[6];
+    r->n[7] = 0x3FFFFFFUL * (m + 1) - a->n[7];
+    r->n[8] = 0x3FFFFFFUL * (m + 1) - a->n[8];
+    r->n[9] = 0x03FFFFFUL * (m + 1) - a->n[9];
+}
+
+void static inline secp256k1_fe_mul_int(secp256k1_fe_t *r, int a) {
+#ifdef VERIFY
+    r->magnitude *= a;
+    r->normalized = 0;
+#endif
+    r->n[0] *= a;
+    r->n[1] *= a;
+    r->n[2] *= a;
+    r->n[3] *= a;
+    r->n[4] *= a;
+    r->n[5] *= a;
+    r->n[6] *= a;
+    r->n[7] *= a;
+    r->n[8] *= a;
+    r->n[9] *= a;
+}
+
+void static inline secp256k1_fe_add(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
+#ifdef VERIFY
+    r->magnitude += a->magnitude;
+    r->normalized = 0;
+#endif
+    r->n[0] += a->n[0];
+    r->n[1] += a->n[1];
+    r->n[2] += a->n[2];
+    r->n[3] += a->n[3];
+    r->n[4] += a->n[4];
+    r->n[5] += a->n[5];
+    r->n[6] += a->n[6];
+    r->n[7] += a->n[7];
+    r->n[8] += a->n[8];
+    r->n[9] += a->n[9];
+}
+
+void static inline secp256k1_fe_mul_inner(const uint32_t *a, const uint32_t *b, uint32_t *r) {
+    uint64_t c = (uint64_t)a[0] * b[0];
+    uint32_t t0 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[1] +
+            (uint64_t)a[1] * b[0];
+    uint32_t t1 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[2] +
+            (uint64_t)a[1] * b[1] +
+            (uint64_t)a[2] * b[0];
+    uint32_t t2 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[3] +
+            (uint64_t)a[1] * b[2] +
+            (uint64_t)a[2] * b[1] +
+            (uint64_t)a[3] * b[0];
+    uint32_t t3 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[4] +
+            (uint64_t)a[1] * b[3] +
+            (uint64_t)a[2] * b[2] +
+            (uint64_t)a[3] * b[1] +
+            (uint64_t)a[4] * b[0];
+    uint32_t t4 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[5] +
+            (uint64_t)a[1] * b[4] +
+            (uint64_t)a[2] * b[3] +
+            (uint64_t)a[3] * b[2] +
+            (uint64_t)a[4] * b[1] +
+            (uint64_t)a[5] * b[0];
+    uint32_t t5 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[6] +
+            (uint64_t)a[1] * b[5] +
+            (uint64_t)a[2] * b[4] +
+            (uint64_t)a[3] * b[3] +
+            (uint64_t)a[4] * b[2] +
+            (uint64_t)a[5] * b[1] +
+            (uint64_t)a[6] * b[0];
+    uint32_t t6 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[7] +
+            (uint64_t)a[1] * b[6] +
+            (uint64_t)a[2] * b[5] +
+            (uint64_t)a[3] * b[4] +
+            (uint64_t)a[4] * b[3] +
+            (uint64_t)a[5] * b[2] +
+            (uint64_t)a[6] * b[1] +
+            (uint64_t)a[7] * b[0];
+    uint32_t t7 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[8] +
+            (uint64_t)a[1] * b[7] +
+            (uint64_t)a[2] * b[6] +
+            (uint64_t)a[3] * b[5] +
+            (uint64_t)a[4] * b[4] +
+            (uint64_t)a[5] * b[3] +
+            (uint64_t)a[6] * b[2] +
+            (uint64_t)a[7] * b[1] +
+            (uint64_t)a[8] * b[0];
+    uint32_t t8 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[0] * b[9] +
+            (uint64_t)a[1] * b[8] +
+            (uint64_t)a[2] * b[7] +
+            (uint64_t)a[3] * b[6] +
+            (uint64_t)a[4] * b[5] +
+            (uint64_t)a[5] * b[4] +
+            (uint64_t)a[6] * b[3] +
+            (uint64_t)a[7] * b[2] +
+            (uint64_t)a[8] * b[1] +
+            (uint64_t)a[9] * b[0];
+    uint32_t t9 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[1] * b[9] +
+            (uint64_t)a[2] * b[8] +
+            (uint64_t)a[3] * b[7] +
+            (uint64_t)a[4] * b[6] +
+            (uint64_t)a[5] * b[5] +
+            (uint64_t)a[6] * b[4] +
+            (uint64_t)a[7] * b[3] +
+            (uint64_t)a[8] * b[2] +
+            (uint64_t)a[9] * b[1];
+    uint32_t t10 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[2] * b[9] +
+            (uint64_t)a[3] * b[8] +
+            (uint64_t)a[4] * b[7] +
+            (uint64_t)a[5] * b[6] +
+            (uint64_t)a[6] * b[5] +
+            (uint64_t)a[7] * b[4] +
+            (uint64_t)a[8] * b[3] +
+            (uint64_t)a[9] * b[2];
+    uint32_t t11 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[3] * b[9] +
+            (uint64_t)a[4] * b[8] +
+            (uint64_t)a[5] * b[7] +
+            (uint64_t)a[6] * b[6] +
+            (uint64_t)a[7] * b[5] +
+            (uint64_t)a[8] * b[4] +
+            (uint64_t)a[9] * b[3];
+    uint32_t t12 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[4] * b[9] +
+            (uint64_t)a[5] * b[8] +
+            (uint64_t)a[6] * b[7] +
+            (uint64_t)a[7] * b[6] +
+            (uint64_t)a[8] * b[5] +
+            (uint64_t)a[9] * b[4];
+    uint32_t t13 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[5] * b[9] +
+            (uint64_t)a[6] * b[8] +
+            (uint64_t)a[7] * b[7] +
+            (uint64_t)a[8] * b[6] +
+            (uint64_t)a[9] * b[5];
+    uint32_t t14 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[6] * b[9] +
+            (uint64_t)a[7] * b[8] +
+            (uint64_t)a[8] * b[7] +
+            (uint64_t)a[9] * b[6];
+    uint32_t t15 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[7] * b[9] +
+            (uint64_t)a[8] * b[8] +
+            (uint64_t)a[9] * b[7];
+    uint32_t t16 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[8] * b[9] +
+            (uint64_t)a[9] * b[8];
+    uint32_t t17 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[9] * b[9];
+    uint32_t t18 = c & 0x3FFFFFFUL; c = c >> 26;
+    uint32_t t19 = c;
+
+    c = t0 + (uint64_t)t10 * 0x3D10UL;
+    t0 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t1 + (uint64_t)t10*0x400UL + (uint64_t)t11 * 0x3D10UL;
+    t1 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t2 + (uint64_t)t11*0x400UL + (uint64_t)t12 * 0x3D10UL;
+    t2 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t3 + (uint64_t)t12*0x400UL + (uint64_t)t13 * 0x3D10UL;
+    r[3] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t4 + (uint64_t)t13*0x400UL + (uint64_t)t14 * 0x3D10UL;
+    r[4] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t5 + (uint64_t)t14*0x400UL + (uint64_t)t15 * 0x3D10UL;
+    r[5] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t6 + (uint64_t)t15*0x400UL + (uint64_t)t16 * 0x3D10UL;
+    r[6] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t7 + (uint64_t)t16*0x400UL + (uint64_t)t17 * 0x3D10UL;
+    r[7] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t8 + (uint64_t)t17*0x400UL + (uint64_t)t18 * 0x3D10UL;
+    r[8] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t9 + (uint64_t)t18*0x400UL + (uint64_t)t19 * 0x1000003D10ULL;
+    r[9] = c & 0x03FFFFFUL; c = c >> 22;
+    uint64_t d = t0 + c * 0x3D1UL;
+    r[0] = d & 0x3FFFFFFUL; d = d >> 26;
+    d = d + t1 + c*0x40;
+    r[1] = d & 0x3FFFFFFUL; d = d >> 26;
+    r[2] = t2 + d;
+}
+
+void static inline secp256k1_fe_sqr_inner(const uint32_t *a, uint32_t *r) {
+    uint64_t c = (uint64_t)a[0] * a[0];
+    uint32_t t0 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[1];
+    uint32_t t1 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[2] +
+            (uint64_t)a[1] * a[1];
+    uint32_t t2 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[3] +
+            (uint64_t)(a[1]*2) * a[2];
+    uint32_t t3 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[4] +
+            (uint64_t)(a[1]*2) * a[3] +
+            (uint64_t)a[2] * a[2];
+    uint32_t t4 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[5] +
+            (uint64_t)(a[1]*2) * a[4] +
+            (uint64_t)(a[2]*2) * a[3];
+    uint32_t t5 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[6] +
+            (uint64_t)(a[1]*2) * a[5] +
+            (uint64_t)(a[2]*2) * a[4] +
+            (uint64_t)a[3] * a[3];
+    uint32_t t6 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[7] +
+            (uint64_t)(a[1]*2) * a[6] +
+            (uint64_t)(a[2]*2) * a[5] +
+            (uint64_t)(a[3]*2) * a[4];
+    uint32_t t7 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[8] +
+            (uint64_t)(a[1]*2) * a[7] +
+            (uint64_t)(a[2]*2) * a[6] +
+            (uint64_t)(a[3]*2) * a[5] +
+            (uint64_t)a[4] * a[4];
+    uint32_t t8 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[0]*2) * a[9] +
+            (uint64_t)(a[1]*2) * a[8] +
+            (uint64_t)(a[2]*2) * a[7] +
+            (uint64_t)(a[3]*2) * a[6] +
+            (uint64_t)(a[4]*2) * a[5];
+    uint32_t t9 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[1]*2) * a[9] +
+            (uint64_t)(a[2]*2) * a[8] +
+            (uint64_t)(a[3]*2) * a[7] +
+            (uint64_t)(a[4]*2) * a[6] +
+            (uint64_t)a[5] * a[5];
+    uint32_t t10 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[2]*2) * a[9] +
+            (uint64_t)(a[3]*2) * a[8] +
+            (uint64_t)(a[4]*2) * a[7] +
+            (uint64_t)(a[5]*2) * a[6];
+    uint32_t t11 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[3]*2) * a[9] +
+            (uint64_t)(a[4]*2) * a[8] +
+            (uint64_t)(a[5]*2) * a[7] +
+            (uint64_t)a[6] * a[6];
+    uint32_t t12 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[4]*2) * a[9] +
+            (uint64_t)(a[5]*2) * a[8] +
+            (uint64_t)(a[6]*2) * a[7];
+    uint32_t t13 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[5]*2) * a[9] +
+            (uint64_t)(a[6]*2) * a[8] +
+            (uint64_t)a[7] * a[7];
+    uint32_t t14 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[6]*2) * a[9] +
+            (uint64_t)(a[7]*2) * a[8];
+    uint32_t t15 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[7]*2) * a[9] +
+            (uint64_t)a[8] * a[8];
+    uint32_t t16 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)(a[8]*2) * a[9];
+    uint32_t t17 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + (uint64_t)a[9] * a[9];
+    uint32_t t18 = c & 0x3FFFFFFUL; c = c >> 26;
+    uint32_t t19 = c;
+
+    c = t0 + (uint64_t)t10 * 0x3D10UL;
+    t0 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t1 + (uint64_t)t10*0x400UL + (uint64_t)t11 * 0x3D10UL;
+    t1 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t2 + (uint64_t)t11*0x400UL + (uint64_t)t12 * 0x3D10UL;
+    t2 = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t3 + (uint64_t)t12*0x400UL + (uint64_t)t13 * 0x3D10UL;
+    r[3] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t4 + (uint64_t)t13*0x400UL + (uint64_t)t14 * 0x3D10UL;
+    r[4] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t5 + (uint64_t)t14*0x400UL + (uint64_t)t15 * 0x3D10UL;
+    r[5] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t6 + (uint64_t)t15*0x400UL + (uint64_t)t16 * 0x3D10UL;
+    r[6] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t7 + (uint64_t)t16*0x400UL + (uint64_t)t17 * 0x3D10UL;
+    r[7] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t8 + (uint64_t)t17*0x400UL + (uint64_t)t18 * 0x3D10UL;
+    r[8] = c & 0x3FFFFFFUL; c = c >> 26;
+    c = c + t9 + (uint64_t)t18*0x400UL + (uint64_t)t19 * 0x1000003D10ULL;
+    r[9] = c & 0x03FFFFFUL; c = c >> 22;
+    uint64_t d = t0 + c * 0x3D1UL;
+    r[0] = d & 0x3FFFFFFUL; d = d >> 26;
+    d = d + t1 + c*0x40;
+    r[1] = d & 0x3FFFFFFUL; d = d >> 26;
+    r[2] = t2 + d;
+}
+
+
+void static secp256k1_fe_mul(secp256k1_fe_t *r, const secp256k1_fe_t *a, const secp256k1_fe_t *b) {
+#ifdef VERIFY
+    assert(a->magnitude <= 8);
+    assert(b->magnitude <= 8);
+    r->magnitude = 1;
+    r->normalized = 0;
+#endif
+    secp256k1_fe_mul_inner(a->n, b->n, r->n);
+}
+
+void static secp256k1_fe_sqr(secp256k1_fe_t *r, const secp256k1_fe_t *a) {
+#ifdef VERIFY
+    assert(a->magnitude <= 8);
+    r->magnitude = 1;
+    r->normalized = 0;
+#endif
+    secp256k1_fe_sqr_inner(a->n, r->n);
+}
diff --git a/src/field_10x26.h b/src/field_10x26.h
new file mode 100644
index 0000000000..11f6cf9999
--- /dev/null
+++ b/src/field_10x26.h
@@ -0,0 +1,15 @@
+#ifndef _SECP256K1_FIELD_10x26_
+#define _SECP256K1_FIELD_10x26_
+
+#include <stdint.h>
+
+typedef struct {
+    // X = sum(i=0..9, elem[i]*2^26) mod n
+    uint32_t n[10];
+#ifdef VERIFY
+    int magnitude;
+    int normalized;
+#endif
+} secp256k1_fe_t;
+
+#endif
diff --git a/src/field_5x52.c b/src/field_5x52.c
index 8b99501406..0d53352fb5 100644
--- a/src/field_5x52.c
+++ b/src/field_5x52.c
@@ -42,6 +42,7 @@ void static secp256k1_fe_normalize(secp256k1_fe_t *r) {
     t3 = c & 0xFFFFFFFFFFFFFULL;
     c = (c >> 52) + t4;
     t4 = c & 0x0FFFFFFFFFFFFULL;
+    assert((c >> 48) == 0);
 
     // Subtract p if result >= p
     uint64_t mask = -(int64_t)((t4 < 0xFFFFFFFFFFFFULL) | (t3 < 0xFFFFFFFFFFFFFULL) | (t2 < 0xFFFFFFFFFFFFFULL) | (t1 < 0xFFFFFFFFFFFFFULL) | (t0 < 0xFFFFEFFFFFC2FULL));
diff --git a/src/field_5x52_int128.c b/src/field_5x52_int128.c
index 740e6f1d91..6ad3fa127c 100644
--- a/src/field_5x52_int128.c
+++ b/src/field_5x52_int128.c
@@ -1,7 +1,7 @@
 #include "field.h"
 
 void static inline secp256k1_fe_mul_inner(const uint64_t *a, const uint64_t *b, uint64_t *r) {
-    unsigned __int128 c = (__int128)a[0] * b[0];
+    __int128 c = (__int128)a[0] * b[0];
     uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0FFFFFFFFFFFFFE0
     c = c + (__int128)a[0] * b[1] +
             (__int128)a[1] * b[0];