🔧 What is Lazy Field Arithmetic?
Lazy field arithmetic delays reductions modulo \(p\) to avoid expensive operations after every add/mul. Instead of immediately reducing results, the library tracks a conservative magnitude, which represents how many times larger than \(p\) a value might be.
📐 What is magnitude?
Magnitude is a conservative estimate of how large a field element is relative to the prime modulus ppp:
If a
FieldElementhasmagnitude = m, it means:
0≤value<m⋅p0 \leq \text{value} < m \cdot p0≤value<m⋅p
This lets the library avoid unnecessary reductions as long as it’s safe.
❓ Why does negate() take a magnitude parameter?
rustCopyEditpub fn negate(&self, magnitude: u32) -> Self {
debug_assert!(self.magnitude <= magnitude);
let new_magnitude = magnitude + 1;
...
}
Answer:
Because negation can increase the magnitude by 1 (e.g., p - a, where a may be close to m·p), the function needs to know a safe upper bound to perform correct lazy negation. The debug_assert ensures the caller isn’t lying about how big the value could be.
common negation trick compute \(-x\) by \(p-x\), or it has magnitude more than 1, then its \(mp-x\)
❓ When should I use .normalize()?
Answer:
Use .normalize() when:
- You are about to invert, serialize, or compare field elements.
- You want to reduce magnitude to 1 (i.e., canonical form).
Avoid overusing it: normalization is expensive and unnecessary if you continue doing lazy operations.
🧠 Practical Guidelines
| Operation | Magnitude Effect |
|---|---|
a + b | max(mag(a), mag(b)) + 1 |
a - b | max(mag(a), mag(b)) + 1 |
a * b | mag(a) * mag(b) |
a * canonical | mag(a) |
a.negate(m) | m + 1 |
.normalize() | resets magnitude to 1 |