p521: Improve Debug for unsaturated math (#949)

When using p521, the limbs can be populated differently which when
deriving [`Debug`] shows up as different values for equal FieldElements,
this can lead to confusion.

See:
-
#946 (comment),
equal field elements were considered as unequal.
-
#946 (comment),
assert_eq! wasn't meaningful.

Signed-off-by: Arvind Mukund <[email protected]>

p521/Cargo.toml

@@ -22,6 +22,7 @@ sha2 = { version = "0.10", optional = true, default-features = false }
 # optional dependencies
 hex-literal = { version = "0.4", optional = true }
 primeorder = { version = "0.13.1", optional = true, path = "../primeorder" }
+base16ct = "0.2.0"
 
 [dev-dependencies]
 hex-literal = "0.4"

p521/src/arithmetic/field.rs

@@ -51,9 +51,43 @@ const MODULUS_HEX: &str = "00000000000001fffffffffffffffffffffffffffffffffffffff
 const MODULUS: U576 = U576::from_be_hex(MODULUS_HEX);
 
 /// Element of the secp521r1 base field used for curve coordinates.
-#[derive(Clone, Copy, Debug)]
+#[derive(Clone, Copy)]
 pub struct FieldElement(fiat_p521_tight_field_element);
 
+impl core::fmt::Debug for FieldElement {
+    /// Formatting machinery for [`FieldElement`]
+    ///
+    /// # Why
+    /// ```ignore
+    /// let fe1 = FieldElement([9, 0, 0, 0, 0, 0, 0, 0, 0]);
+    /// let fe2 = FieldElement([
+    ///     8,
+    ///     0,
+    ///     288230376151711744,
+    ///     288230376151711743,
+    ///     288230376151711743,
+    ///     288230376151711743,
+    ///     288230376151711743,
+    ///     288230376151711743,
+    ///     144115188075855871,
+    /// ]);
+    /// ```
+    ///
+    /// For the above example, deriving [`core::fmt::Debug`] will result in returning 2 different strings,
+    /// which are in reality the same due to p521's unsaturated math, instead print the output as a hex string in
+    /// big-endian
+    ///
+    /// This makes debugging easier.
+    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
+        let mut bytes = fiat_p521_to_bytes(&self.0);
+        bytes.reverse();
+        let formatter = base16ct::HexDisplay(&bytes);
+        f.debug_tuple("FieldElement")
+            .field(&format_args!("0x{formatter:X}"))
+            .finish()
+    }
+}
+
 impl FieldElement {
     /// Zero element.
     pub const ZERO: Self = Self::from_u64(0);