Fix M31::new() modular reduction for values >= 2*P

[Copilot]

The M31::new() function performed single-step reduction (val - P), which only handles values in [0, 2*P). Values >= 2*P were incorrectly reduced:

M31::new(2*P)      // 4294967294 → returned P (2147483647), expected 0
M31::new(u32::MAX) // 4294967295 → returned P+1 (2147483648), expected 1

Changes

• Applied double reduction to handle full u32 range:

  let reduced = if val >= P { val - P } else { val };
  let reduced = if reduced >= P { reduced - P } else { reduced };

• Added boundary tests covering P, 2*P-1, 2*P, and u32::MAX

The fix preserves existing behavior for values < 2*P while correctly reducing the upper boundary cases.

Original prompt

---

This section details on the original issue you should resolve

<issue_title>Bug: M31::new() function performs incorrect modular reduction</issue_title>
<issue_description>We have a function named new in the field, which tries to compute u32 values to a value mod P.

    #[inline]
    pub const fn new(val: u32) -> Self {
        let reduced = if val >= P { val - P } else { val };
        Self(reduced)
    }

Where P = 2^31 -1.

Here's the problem:

When val >= P, the code does val - P, which works only if val < 2*P (i.e., val < 2^32 - 2). For values ≥ 2*P (≥2^32 - 2 ), this will not properly reduce them modulo P:

• Input val = 2*P = 2^32 - 2 → Should output 0, but the code outputs: 2p - p = p (which is P, not 0)

• Input val = u32::MAX = 2^32 - 1 → Should output 1, but the code outputs: 2^32 - 1 - (2^31 - 1) = 2^31 (which is P+1, not 1)

</issue_description>

<agent_instructions>Is the reported issue is valid or invalid in current design ? Can check that specific code also whats the impact and why its currently leads to bug ? create end to end poc to check the bug is valid or not.</agent_instructions>

Comments on the Issue (you are @copilot in this section)

@this-vishalsingh > We have a function named `new` in the [field](https://github.com/ZippelLabs/ZP1/blob/2141f45836728bbc90a9ee9e191e2f11d69f6a66/crates/primitives/src/field.rs#L35), which tries to compute `u32` values to a value mod `P`. > > ``` > #[inline] > pub const fn new(val: u32) -> Self { > let reduced = if val >= P { val - P } else { val }; > Self(reduced) > } > ``` > > Where `P = 2^31 -1`. > > Here's the problem: > > When `val >= P`, the code does `val - P`, which works only if `val < 2*P` (i.e., `val < 2^32 - 2`). For values `≥ 2*P` (`≥2^32 - 2` ), this will not properly reduce them modulo `P`: > > * Input `val = 2*P = 2^32 - 2` → Should output 0, but the code outputs: `2p - p = p` (which is P, not 0) > * Input `val = u32::MAX = 2^32 - 1` → Should output 1, but the code outputs: `2^32 - 1 - (2^31 - 1)` = 2^31 (which is `P+1`, not 1)

@0x6980 Can you create poc that will make easy?</comment_new>

• Fixes Bug: M31::new() function performs incorrect modular reduction #8

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