Function utils_fq12_square

bitvm/src/chunk/taps_mul.rs:289–346

Compute square of an Fq12 elements in normalized form (1 + c J) <- (1 + a J) x (1 + a J). Input a, and output c. In this form (1 + c J) <- 1 + {(2a)/(1 + a^2 J^2)} => c . (1 + a^2 J^2) =?= 2a a c are passed as input to the script and the above equation is validated to show that c is the correct output Assumes input a is valid i.e (1 + a^2 J ^2) != 0

(a: ark_bn254::Fq6)

// a c are passed as input to the script and the above equation is validated to show that c is the correct output
// Assumes input a is valid i.e (1 + a^2 J ^2) != 0
pub(crate) fn utils_fq12_square(a: ark_bn254::Fq6) -> (ark_bn254::Fq6, bool, Script, Vec<Hint>) {
    let mut hints = vec![];

    // compute ab in Script
    let (asq_scr, ab_hints) = Fq6::hinted_square(a);
    hints.extend_from_slice(&ab_hints);

    // calculate denom
    let beta_sq = ark_bn254::Fq12Config::NONRESIDUE;
    let denom = ark_bn254::Fq6::ONE + a * a * beta_sq;

    // is input valid ? output: mock_output
    let (denom_mul_c_scr, c) = {
        let c = (a + a) / denom;
        let res = Fq6::hinted_mul(6, denom, 0, c);
        hints.extend_from_slice(&res.1);
        (res.0, c)
    };

    let mul_by_beta_sq_scr = script! {
        {Fq6::mul_fq2_by_nonresidue()}
        {Fq2::roll(4)} {Fq2::roll(4)}
    };

    let scr = script! {
        // [hints a, c] []
        {Fq6::toaltstack()}
        // [hints, a], [c]
        {Fq6::copy(0)}
        // [hints a, a] [c]
        {asq_scr}
        // [hints, a, asq]
        {mul_by_beta_sq_scr}
        // [hints, a, asq*beta_sq]
        {Fq6::push(ark_bn254::Fq6::ONE)}
        {Fq6::add(6, 0)}
        // [hints, a, denom] [c]
        {Fq6::fromaltstack()}
        // [hints, a, denom, c]
        {Fq6::copy(0)}
        // [hints, a, denom, c, c]
        {Fq6::roll(12)} {Fq6::roll(12)}
        // [hints, a, c, denom, c]
        {denom_mul_c_scr}
        // [a, c, denom_c]
        {Fq6::copy(12)}
        // [a, c, denom_c, a]
        {Fq6::double(0)}
        // [a, c, denom_c, 2a]
        {Fq6::equalverify()}
        // [a,c] []
        {1}
        // [a, c, 0/1]
    };

    let input_is_valid = denom != ark_bn254::Fq6::ZERO;
    (c, input_is_valid, scr, hints)
}