Merge magnitude/normalized fields, move/improve comments

Also split secp256k1_fe_verify into a generic and an implementation
specific part.

src/field_10x26.h

@@ -9,15 +9,28 @@
 
 #include <stdint.h>
 
+/** This field implementation represents the value as 10 uint32_t limbs in base
+ *  2^26. */
 typedef struct {
-    /* X = sum(i=0..9, n[i]*2^(i*26)) mod p
-     * where p = 2^256 - 0x1000003D1
-     */
+   /* A field element f represents the sum(i=0..9, f.n[i] << (i*26)) mod p,
+    * where p is the field modulus, 2^256 - 2^32 - 977.
+    *
+    * The individual limbs f.n[i] can exceed 2^26; the field's magnitude roughly
+    * corresponds to how much excess is allowed. The value
+    * sum(i=0..9, f.n[i] << (i*26)) may exceed p, unless the field element is
+    * normalized. */
     uint32_t n[10];
-#ifdef VERIFY
-    int magnitude;
-    int normalized;
-#endif
+    /*
+     * Magnitude m requires:
+     *     n[i] <= 2 * m * (2^26 - 1) for i=0..8
+     *     n[9] <= 2 * m * (2^22 - 1)
+     *
+     * Normalized requires:
+     *     n[i] <= (2^26 - 1) for i=0..8
+     *     sum(i=0..9, n[i] << (i*26)) < p
+     *     (together these imply n[9] <= 2^22 - 1)
+     */
+    SECP256K1_FE_VERIFY_FIELDS
 } secp256k1_fe;
 
 /* Unpacks a constant into a overlapping multi-limbed FE element. */