mathtools/internal/lib/lib.go

package lib

import (
	"errors"
	"math/big"
)

func SqrtRepetend(x *big.Int) ([]*big.Int, error) {
	m := big.NewInt(0)
	d := big.NewInt(1)
	a0 := new(big.Int).Sqrt(x)

	s := new(big.Int).Exp(a0, big.NewInt(2), nil)
	if x.Cmp(s) == 0 {
		return nil, errors.New("input is a perfect square")
	}

	repetend := make([]*big.Int, 0)

	a := new(big.Int).Set(a0)
	twoa0 := new(big.Int).Mul(big.NewInt(2), a0)
	for a.Cmp(twoa0) != 0 {
		// m = d * a - m
		tmp := new(big.Int)
		m.Sub(tmp.Mul(d, a), m)

		// d = (x - m^2) // d
		tmp.Exp(m, big.NewInt(2), nil)
		d.Div(tmp.Sub(x, tmp), d)

		// a = (a0 + m) // d
		a.Div(tmp.Add(a0, m), d)

		repetend = append(repetend, new(big.Int).Set(a))
	}

	return repetend, nil
}

func CRTSolution(a1, n1, a2, n2 *big.Int) (*big.Int, *big.Int) {
	// use Bezout's identity to find m1, m2 such that m1*n1 + m2*n2 = 1
	m1 := new(big.Int)
	m2 := new(big.Int)
	tmp := new(big.Int)
	tmp.GCD(m1, m2, n1, n2)

	// x = a1*m2*n2 + a2*m1*n1
	x := new(big.Int).Set(a1)
	x.Mul(x, m2)
	x.Mul(x, n2)

	tmp.Set(a2)
	tmp.Mul(tmp, m1)
	tmp.Mul(tmp, n1)

	x.Add(x, tmp)

	N := new(big.Int).Set(n1)
	N.Mul(N, n2)

	x.Mod(x, N)

	return x, N
}

func ArePairwiseCoprime(moduli []*big.Int) bool {
	z := new(big.Int)
	for i, a := range moduli {
		for _, b := range moduli[i+1:] {
			z.GCD(nil, nil, a, b)
			if z.Cmp(big.NewInt(1)) != 0 {
				return false
			}
		}
	}

	return true
}

func Totient(n *big.Int) *big.Int {
	N := new(big.Int).Set(n)

	phi := new(big.Int).Set(N)

	sqrtn := new(big.Int).Sqrt(N)
	for i := big.NewInt(2); i.Cmp(sqrtn) != 1; i.Add(i, big.NewInt(1)) {
		mod := new(big.Int).Mod(N, i)
		if mod.Cmp(big.NewInt(0)) != 0 {
			continue
		}

		// phi -= phi // i
		tmp := new(big.Int).Div(phi, i)
		phi.Sub(phi, tmp)

		for mod.Cmp(big.NewInt(0)) == 0 {
			N.Div(N, i)
			mod.Mod(N, i)
		}
	}

	if N.Cmp(big.NewInt(1)) == 1 {
		// phi -= phi // N
		tmp := new(big.Int).Div(phi, N)
		phi.Sub(phi, tmp)
	}

	return phi
}

func MultiplicativeOrder(g *big.Int, modulus *big.Int) *big.Int {
	e := new(big.Int).Set(g)
	var k *big.Int
	for k = big.NewInt(1); e.Cmp(big.NewInt(1)) != 0; k.Add(k, big.NewInt(1)) {
		e.Mul(e, g)
		e.Mod(e, modulus)
	}

	return k
}

func PrimitiveRoot(modulus *big.Int) (*big.Int, error) {
	if modulus.Cmp(big.NewInt(1)) == 0 {
		return big.NewInt(0), nil
	}

	phi := Totient(modulus)

	for g := big.NewInt(1); g.Cmp(modulus) == -1; g.Add(g, big.NewInt(1)) {
		gcd := new(big.Int).GCD(nil, nil, g, modulus)
		if gcd.Cmp(big.NewInt(1)) != 0 {
			continue
		}

		order := MultiplicativeOrder(g, modulus)
		if order.Cmp(phi) == 0 {
			return g, nil
		}
	}

	return nil, errors.New("no primitive root")
}

func PrimitiveRootFast(modulus *big.Int, tpf map[string]*big.Int) (*big.Int, error) {
	phi := big.NewInt(1)
	for p, exp := range tpf {
		pow, ok := new(big.Int).SetString(p, 10)
		if !ok {
			return nil, errors.New("invalid factor " + p)
		}

		pow.Exp(pow, exp, nil)
		phi.Mul(phi, pow)
	}

	for g := big.NewInt(1); g.Cmp(modulus) == -1; g.Add(g, big.NewInt(1)) {
		gcd := new(big.Int).GCD(nil, nil, g, modulus)
		if gcd.Cmp(big.NewInt(1)) != 0 {
			continue
		}

		if isPrimitiveRoot(g, modulus, phi, tpf) {
			return g, nil
		}
	}

	return nil, errors.New("no primitive root")
}

func isPrimitiveRoot(g *big.Int, modulus *big.Int, phi *big.Int, tpf map[string]*big.Int) bool {
	for p := range tpf {
		// we already know factors are valid from computing phi
		k, _ := new(big.Int).SetString(p, 10)
		k.Div(phi, k)
		k.Exp(g, k, modulus)
		if k.Cmp(big.NewInt(1)) == 0 {
			return false
		}
	}

	return true
}