Big Integers

The BigInt module in the standard library exposes some class of integers which do not fit (well) into a Noir native field. It implements modulo arithmetic, modulo a 'big' prime number.

note

The module can currently be considered as Fields with fixed modulo sizes used by a set of elliptic curves, in addition to just the native curve. More work is needed to achieve arbitrarily sized big integers.

Currently 6 classes of integers (i.e 'big' prime numbers) are available in the module, namely:

• BN254 Fq: Bn254Fq
• BN254 Fr: Bn254Fr
• Secp256k1 Fq: Secpk1Fq
• Secp256k1 Fr: Secpk1Fr
• Secp256r1 Fr: Secpr1Fr
• Secp256r1 Fq: Secpr1Fq

Where XXX Fq and XXX Fr denote respectively the order of the base and scalar field of the (usual) elliptic curve XXX. For instance the big integer 'Secpk1Fq' in the standard library refers to integers modulo $2^{256}-2^{32}-977$.

Feel free to explore the source code for the other primes:

big_int_definition

struct BigInt {
    pointer: u32,
    modulus: u32,
}

^{_{Source code: noir_stdlib/src/bigint.nr#L14-L19}}

Example usage

A common use-case is when constructing a big integer from its bytes representation, and performing arithmetic operations on it:

big_int_example

fn big_int_example(x: u8, y: u8) {
    let a = Secpk1Fq::from_le_bytes(&[x, y, 0, 45, 2]);
    let b = Secpk1Fq::from_le_bytes(&[y, x, 9]);
    let c = (a + b) * b / a;
    let d = c.to_le_bytes();
    println(d[0]);
}

^{_{Source code: test_programs/execution_success/bigint/src/main.nr#L70-L78}}

Methods

The available operations for each big integer are:

from_le_bytes

Construct a big integer from its little-endian bytes representation. Example:

 // Construct a big integer from a slice of bytes
 let a = Secpk1Fq::from_le_bytes(&[x, y, 0, 45, 2]);
 // Construct a big integer from an array of 32 bytes
 let a = Secpk1Fq::from_le_bytes_32([1;32]);

Sure, here's the formatted version of the remaining methods:

to_le_bytes

Return the little-endian bytes representation of a big integer. Example:

let bytes = a.to_le_bytes();

add

Add two big integers. Example:

let sum = a + b;

sub

Subtract two big integers. Example:

let difference = a - b;

mul

Multiply two big integers. Example:

let product = a * b;

div

Divide two big integers. Note that division is field division and not euclidean division. Example:

let quotient = a / b;

eq

Compare two big integers. Example:

let are_equal = a == b;