diff --git a/src/field_10x26.h b/src/field_10x26.h
index 727c526..5ff03c8 100644
--- a/src/field_10x26.h
+++ b/src/field_10x26.h
@@ -10,7 +10,9 @@
 #include <stdint.h>
 
 typedef struct {
-    /* X = sum(i=0..9, elem[i]*2^26) mod n */
+    /* X = sum(i=0..9, n[i]*2^(i*26)) mod p
+     * where p = 2^256 - 0x1000003D1
+     */
     uint32_t n[10];
 #ifdef VERIFY
     int magnitude;
diff --git a/src/field_10x26_impl.h b/src/field_10x26_impl.h
index 94f8132..4ae4fdc 100644
--- a/src/field_10x26_impl.h
+++ b/src/field_10x26_impl.h
@@ -8,7 +8,6 @@
 #define SECP256K1_FIELD_REPR_IMPL_H
 
 #include "util.h"
-#include "num.h"
 #include "field.h"
 
 #ifdef VERIFY
@@ -486,7 +485,8 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(uint32_t *r, const uint32_t
     VERIFY_BITS(b[9], 26);
 
     /** [... a b c] is a shorthand for ... + a<<52 + b<<26 + c<<0 mod n.
-     *  px is a shorthand for sum(a[i]*b[x-i], i=0..x).
+     *  for 0 <= x <= 9, px is a shorthand for sum(a[i]*b[x-i], i=0..x).
+     *  for 9 <= x <= 18, px is a shorthand for sum(a[i]*b[x-i], i=(x-9)..9)
      *  Note that [x 0 0 0 0 0 0 0 0 0 0] = [x*R1 x*R0].
      */
 
@@ -1069,6 +1069,7 @@ static void secp256k1_fe_mul(secp256k1_fe *r, const secp256k1_fe *a, const secp2
     secp256k1_fe_verify(a);
     secp256k1_fe_verify(b);
     VERIFY_CHECK(r != b);
+    VERIFY_CHECK(a != b);
 #endif
     secp256k1_fe_mul_inner(r->n, a->n, b->n);
 #ifdef VERIFY
diff --git a/src/field_5x52.h b/src/field_5x52.h
index bccd8fe..fc5bfe3 100644
--- a/src/field_5x52.h
+++ b/src/field_5x52.h
@@ -10,7 +10,9 @@
 #include <stdint.h>
 
 typedef struct {
-    /* X = sum(i=0..4, elem[i]*2^52) mod n */
+    /* X = sum(i=0..4, n[i]*2^(i*52)) mod p
+     * where p = 2^256 - 0x1000003D1
+     */
     uint64_t n[5];
 #ifdef VERIFY
     int magnitude;
diff --git a/src/field_5x52_impl.h b/src/field_5x52_impl.h
index 957c61b..f426332 100644
--- a/src/field_5x52_impl.h
+++ b/src/field_5x52_impl.h
@@ -12,7 +12,6 @@
 #endif
 
 #include "util.h"
-#include "num.h"
 #include "field.h"
 
 #if defined(USE_ASM_X86_64)
@@ -422,6 +421,7 @@ static void secp256k1_fe_mul(secp256k1_fe *r, const secp256k1_fe *a, const secp2
     secp256k1_fe_verify(a);
     secp256k1_fe_verify(b);
     VERIFY_CHECK(r != b);
+    VERIFY_CHECK(a != b);
 #endif
     secp256k1_fe_mul_inner(r->n, a->n, b->n);
 #ifdef VERIFY
diff --git a/src/field_5x52_int128_impl.h b/src/field_5x52_int128_impl.h
index 95a0d17..bcbfb92 100644
--- a/src/field_5x52_int128_impl.h
+++ b/src/field_5x52_int128_impl.h
@@ -32,9 +32,11 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(uint64_t *r, const uint64_t
     VERIFY_BITS(b[3], 56);
     VERIFY_BITS(b[4], 52);
     VERIFY_CHECK(r != b);
+    VERIFY_CHECK(a != b);
 
     /*  [... a b c] is a shorthand for ... + a<<104 + b<<52 + c<<0 mod n.
-     *  px is a shorthand for sum(a[i]*b[x-i], i=0..x).
+     *  for 0 <= x <= 4, px is a shorthand for sum(a[i]*b[x-i], i=0..x).
+     *  for 4 <= x <= 8, px is a shorthand for sum(a[i]*b[x-i], i=(x-4)..4)
      *  Note that [x 0 0 0 0 0] = [x*R].
      */
 
diff --git a/src/field_impl.h b/src/field_impl.h
index 2042864..f9f0a1c 100644
--- a/src/field_impl.h
+++ b/src/field_impl.h
@@ -12,6 +12,7 @@
 #endif
 
 #include "util.h"
+#include "num.h"
 
 #if defined(USE_FIELD_10X26)
 #include "field_10x26_impl.h"