fheanor 0.11.2

A library that provides fast implementations of rings commonly used in homomorphic encryption, built on feanor-math.
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use feanor_math::algorithms::zpe_extension::invert_over_local_zn;
use feanor_math::homomorphism::Homomorphism;
use feanor_math::ring::*;
use feanor_math::rings::extension::extension_impl::FreeAlgebraImpl;
use feanor_math::rings::extension::*;
use feanor_math::rings::poly::*;
use feanor_math::rings::zn::ZnReductionMap;

///
/// Interpolation data for a list of moduli `f1, ..., fn` that can be used
/// to derive from remainders `r1, ..., rn` an "interpolation polynomial" `h`
/// such that `h = ri mod fi`.
/// 
/// Clearly this requires that the moduli `fi` are pairwise coprime. Additionally,
/// we currently require that all interpolation unit vectors `ei` (defined
/// by `ei = 1 mod fi`, `ei = 0 mod fj` for `j != i`) exist over the base
/// ring (e.g. they might not be integral, even if the base ring is `Z`).
/// 
pub struct FastPolyInterpolation<P>
    where P: RingStore,
        P::Type: PolyRing,
        <<P::Type as RingExtension>::BaseRing as RingStore>::Type: NiceZn
{
    poly_ring: P,
    input_degree: usize,
    unit_vectors: Vec<Vec<(El<P>, El<P>)>>,
    final_modulus: El<P>,
    n: usize
}

///
/// Computes a polynomial `h` of degree `< deg(fg)` such that `h = 1 mod f` and `h = 0 mod g`.
/// 
fn crt_unit_vectors<P>(poly_ring: P, f: &El<P>, g: &El<P>) -> El<P>
    where P: RingStore,
        P::Type: PolyRing,
        <<P::Type as RingExtension>::BaseRing as RingStore>::Type: NiceZn
{
    assert!(poly_ring.base_ring().is_one(poly_ring.lc(f).unwrap()));
    assert!(poly_ring.base_ring().is_one(poly_ring.lc(g).unwrap()));
    let deg_f = poly_ring.degree(&f).unwrap();

    let mod_f_ring = FreeAlgebraImpl::new(poly_ring.base_ring(), deg_f, (0..deg_f).map(|i| poly_ring.base_ring().negate(poly_ring.base_ring().clone_el(poly_ring.coefficient_at(&f, i)))).collect::<Vec<_>>());
    let g_mod_f = poly_ring.div_rem_monic(poly_ring.clone_el(&g), &f).1;
    let g_mod_f = mod_f_ring.from_canonical_basis((0..deg_f).map(|i| poly_ring.base_ring().clone_el(poly_ring.coefficient_at(&g_mod_f, i))));

    let normalization_factor = invert_over_local_zn(RingRef::new(mod_f_ring.get_ring()), &g_mod_f);
    
    assert!(normalization_factor.is_some(), "crt unit vector modulo {} and {} does not exist", poly_ring.format(f), poly_ring.format(g));
    debug_assert!(mod_f_ring.is_one(&mod_f_ring.mul_ref(normalization_factor.as_ref().unwrap(), &g_mod_f)));
    let g_mod_f_inv = mod_f_ring.poly_repr(&poly_ring, &normalization_factor.unwrap(), poly_ring.base_ring().identity());
    let first_unit_vector = poly_ring.mul_ref_snd(g_mod_f_inv, &g);

    debug_assert!(poly_ring.is_one(&poly_ring.div_rem_monic(poly_ring.clone_el(&first_unit_vector), &f).1));
    debug_assert!(poly_ring.is_zero(&poly_ring.div_rem_monic(poly_ring.clone_el(&first_unit_vector), &g).1));

    return first_unit_vector;
}

impl<P> FastPolyInterpolation<P>
    where P: RingStore,
        P::Type: PolyRing,
        <<P::Type as RingExtension>::BaseRing as RingStore>::Type: NiceZn
{
    #[instrument(skip_all)]
    pub fn new(poly_ring: P, moduli: Vec<El<P>>) -> Self {
        let n = moduli.len();
        let input_degree = moduli.iter().map(|f| poly_ring.degree(f).unwrap_or(0)).max().unwrap();
        let mut current = moduli.iter().map(|f| poly_ring.clone_el(f)).collect::<Vec<_>>();
        let mut result = Vec::new();
        while current.len() > 1 {
            let mut result_part = Vec::new();
            let mut new = Vec::new();
            let (chunks, remainder) = current.as_chunks::<2>();
            for [f, g] in chunks {
                let new_modulus = poly_ring.mul_ref(f, g);
                let unit_vectors = (crt_unit_vectors(&poly_ring, f, g), crt_unit_vectors(&poly_ring, g, f));
                result_part.push(unit_vectors);
                new.push(new_modulus);
            }
            if let Some(last) = remainder.get(0) {
                new.push(poly_ring.clone_el(last));
            }
            current = new;
            result.push(result_part);
        }
        return Self {
            final_modulus: current.pop().unwrap(),
            n: n,
            input_degree: input_degree,
            poly_ring: poly_ring,
            unit_vectors: result
        };
    }

    pub fn change_modulus<PNew>(&self, new_poly_ring: PNew) -> FastPolyInterpolation<PNew>
        where PNew: RingStore,
            PNew::Type: PolyRing,
            <<PNew::Type as RingExtension>::BaseRing as RingStore>::Type: NiceZn
    {
        let red_map = ZnReductionMap::new(self.poly_ring().base_ring(), new_poly_ring.base_ring()).unwrap();
        let lifted_red_map = new_poly_ring.lifted_hom(self.poly_ring(), &red_map);
        let unit_vectors = self.unit_vectors.iter().map(|list| list.iter().map(|(e0, e1)| (lifted_red_map.map_ref(e0), lifted_red_map.map_ref(e1))).collect()).collect();
        let final_modulus = lifted_red_map.map_ref(&self.final_modulus);
        return FastPolyInterpolation {
            final_modulus: final_modulus,
            input_degree: self.input_degree,
            poly_ring: new_poly_ring,
            n: self.n,
            unit_vectors: unit_vectors
        };
    }

    pub fn allocation_size(&self) -> usize {
        self.unit_vectors.iter().map(|vec| vec.iter().map(|(l, r)|
            (self.poly_ring().degree(l).unwrap_or(0) + self.poly_ring().degree(r).unwrap_or(0) + 2) * size_of::<El<<P::Type as RingExtension>::BaseRing>>()
        ).sum::<usize>()).sum()
    }

    pub fn poly_ring(&self) -> &P {
        &self.poly_ring
    }

    pub fn final_modulus(&self) -> &El<P> {
        &self.final_modulus
    }

    ///
    /// Computes a polynomial of degree `< 2 * deg(prod(moduli))` that is congruent
    /// to `remainders[i]` modulo `moduli[i]`.
    /// 
    /// It is unreduced, since we can reduce its degree to `< deg(prod(moduli))` by
    /// taking the remainder modulo `prod(moduli)`.
    /// 
    /// However, this can be computed really fast, in time `n log(n)^2` if FFT-based
    /// polynomial multiplication is used by the underlying polynomial ring. It is also
    /// very fast in practice, since we don't perform any polynomial division.
    /// 
    #[instrument(skip_all)]
    pub fn interpolate_unreduced(&self, remainders: Vec<El<P>>) -> El<P> {
        assert_eq!(self.n, remainders.len());
        for i in 0..self.n {
            assert!(self.poly_ring.degree(&remainders[i]).unwrap_or(0) < self.input_degree);
        }
        let mut current = remainders;
        for i in 0..self.unit_vectors.len() {
            let mut new = Vec::new();
            let (chunks, remainer) = current.as_chunks::<2>();
            for ([f0, f1], (e0, e1)) in chunks.iter().zip(self.unit_vectors[i].iter()) {
                new.push(self.poly_ring.add(
                    self.poly_ring.mul_ref(f0, e0),
                    self.poly_ring.mul_ref(f1, e1)
                ));
            }
            if let Some(last) = remainer.get(0) {
                new.push(self.poly_ring.clone_el(last));
            }
            current = new;
        }
        debug_assert_eq!(1, current.len());
        let result = current.pop().unwrap();
        debug_assert!(self.poly_ring.degree(&result).unwrap_or(0) < (self.input_degree << (self.unit_vectors.len() +  1)));
        return result;
    }
}

#[cfg(test)]
use feanor_math::assert_el_eq;
#[cfg(test)]
use feanor_math::rings::local::AsLocalPIR;
#[cfg(test)]
use feanor_math::rings::poly::dense_poly::DensePolyRing;
#[cfg(test)]
use feanor_math::rings::zn::zn_64::Zn;
use tracing::instrument;

use crate::NiceZn;

#[test]
fn test_interpolate() {
    feanor_tracing::DelayedLogger::init_test();
    let base_ring = AsLocalPIR::from_zn(Zn::new(257)).unwrap();
    let poly_ring = DensePolyRing::new(base_ring, "X");

    let moduli = poly_ring.with_wrapped_indeterminate(|X| [
        X.pow_ref(2) - 1,
        X.pow_ref(3) + X - 1,
        X.pow_ref(2) - X + 2,
        X.pow_ref(2) - 2 * X
    ]);
    let remainders = poly_ring.with_wrapped_indeterminate(|X| [
        -1 * X + 3,
        -5 * X.pow_ref(2) + 21 * X - 12,
        -728 * X + 16,
        720896 * X
    ]);
    let interpolation = FastPolyInterpolation::new(&poly_ring, moduli.iter().map(|f| poly_ring.clone_el(f)).collect());
    let result = interpolation.interpolate_unreduced(remainders.iter().map(|f| poly_ring.clone_el(f)).collect());
    for i in 0..4 {
        assert_el_eq!(&poly_ring, &remainders[i], poly_ring.div_rem_monic(poly_ring.clone_el(&result), &moduli[i]).1);
    }

    let moduli = poly_ring.with_wrapped_indeterminate(|X| [
        X.pow_ref(2) - 1,
        X.pow_ref(3) + X - 1,
        X.pow_ref(2) - X + 2
    ]);
    let remainders = poly_ring.with_wrapped_indeterminate(|X| [
        -1 * X + 3,
        -5 * X.pow_ref(2) + 21 * X - 12,
        -728 * X + 16
    ]);
    let interpolation = FastPolyInterpolation::new(&poly_ring, moduli.iter().map(|f| poly_ring.clone_el(f)).collect());
    let result = interpolation.interpolate_unreduced(remainders.iter().map(|f| poly_ring.clone_el(f)).collect());
    for i in 0..3 {
        assert_el_eq!(&poly_ring, &remainders[i], poly_ring.div_rem_monic(poly_ring.clone_el(&result), &moduli[i]).1);
    }
    
    let moduli = poly_ring.with_wrapped_indeterminate(|X| [
        X.pow_ref(2) - 1,
        X.pow_ref(2) - 2,
        X.pow_ref(2) - 3,
        X.pow_ref(2) - 4,
        X.pow_ref(2) - 5,
        X.pow_ref(2) - 6,
        X.pow_ref(2) - 7,
        X.pow_ref(2) - 8,
        X.pow_ref(2) - 9,
        X.pow_ref(2) - 10,
    ]);
    let remainders = poly_ring.with_wrapped_indeterminate(|X| [
        5 * X + 10,
        5 * X + 9,
        5 * X + 8,
        5 * X + 7,
        5 * X + 6,
        5 * X + 5,
        5 * X + 4,
        5 * X + 3,
        5 * X + 2,
        5 * X + 1,
    ]);
    let interpolation = FastPolyInterpolation::new(&poly_ring, moduli.iter().map(|f| poly_ring.clone_el(f)).collect());
    let result = interpolation.interpolate_unreduced(remainders.iter().map(|f| poly_ring.clone_el(f)).collect());
    for i in 0..10 {
        assert_el_eq!(&poly_ring, &remainders[i], poly_ring.div_rem_monic(poly_ring.clone_el(&result), &moduli[i]).1);
    }
}