path: root/include/mcl/fp_tower.hpp

#pragma once
/**
    @file
    @brief finite field extension class
    @author MITSUNARI Shigeo(@herumi)
    @license modified new BSD license
    http://opensource.org/licenses/BSD-3-Clause
*/
#include <mcl/fp.hpp>

namespace mcl {

template<class Fp>
class FpDblT {
    typedef fp::Unit Unit;
    Unit v_[Fp::maxSize * 2];
public:
    static inline size_t getUnitSize() { return Fp::op_.N * 2; }
    void dump() const
    {
        const size_t n = getUnitSize();
        for (size_t i = 0; i < n; i++) {
            printf("%016llx ", (long long)v_[n - 1 - i]);
        }
        printf("\n");
    }
    // QQQ : does not check range of x strictly(use for debug)
    void setMpz(const mpz_class& x)
    {
        if (x < 0) throw cybozu::Exception("FpDblT:_setMpz:negative is not supported") << x;
        const size_t xn = gmp::getUnitSize(x);
        const size_t N2 = getUnitSize();
        if (xn > N2) {
            throw cybozu::Exception("FpDblT:setMpz:too large") << x;
        }
        memcpy(v_, gmp::getUnit(x), xn * sizeof(Unit));
        memset(v_ + xn, 0, (N2 - xn) * sizeof(Unit));
    }
    void getMpz(mpz_class& x) const
    {
        gmp::setArray(x, v_, Fp::op_.N * 2);
    }
    static inline void add(FpDblT& z, const FpDblT& x, const FpDblT& y) { Fp::op_.fpDbl_add(z.v_, x.v_, y.v_); }
    static inline void sub(FpDblT& z, const FpDblT& x, const FpDblT& y) { Fp::op_.fpDbl_sub(z.v_, x.v_, y.v_); }
    static inline void addNC(FpDblT& z, const FpDblT& x, const FpDblT& y) { Fp::op_.fpDbl_addNC(z.v_, x.v_, y.v_); }
    static inline void subNC(FpDblT& z, const FpDblT& x, const FpDblT& y) { Fp::op_.fpDbl_subNC(z.v_, x.v_, y.v_); }
    /*
        mul(z, x, y) = mulPre(xy, x, y) + mod(z, xy)
    */
    static inline void mulPre(FpDblT& xy, const Fp& x, const Fp& y) { Fp::op_.fpDbl_mulPre(xy.v_, x.v_, y.v_); }
    static inline void sqrPre(FpDblT& xx, const Fp& x) { Fp::op_.fpDbl_sqrPre(xx.v_, x.v_); }
    static inline void mod(Fp& z, const FpDblT& xy) { Fp::op_.fpDbl_mod(z.v_, xy.v_); }
};

/*
    beta = -1
    Fp2 = F[i] / (i^2 + 1)
    x = a + bi
*/
template<class Fp>
class Fp2T : public fp::Operator<Fp2T<Fp> > {
    typedef fp::Unit Unit;
    typedef FpDblT<Fp> FpDbl;
    static Fp xi_a_;
public:
    Fp a, b;
    Fp2T() { }
    Fp2T(int64_t a) : a(a), b(0) { }
    Fp2T(const Fp& a, const Fp& b) : a(a), b(b) { }
    Fp2T(int64_t a, int64_t b) : a(a), b(b) { }
    Fp2T(const std::string& a, const std::string& b, int base = 0) : a(a, base), b(b, base) {}
    Fp* get() { return &a; }
    const Fp* get() const { return &a; }
    void clear()
    {
        a.clear();
        b.clear();
    }
    static inline void add(Fp2T& z, const Fp2T& x, const Fp2T& y) { Fp::op_.fp2_add(z.a.v_, x.a.v_, y.a.v_); }
    static inline void sub(Fp2T& z, const Fp2T& x, const Fp2T& y) { Fp::op_.fp2_sub(z.a.v_, x.a.v_, y.a.v_); }
    static inline void mul(Fp2T& z, const Fp2T& x, const Fp2T& y) { Fp::op_.fp2_mul(z.a.v_, x.a.v_, y.a.v_); }
    static inline void inv(Fp2T& y, const Fp2T& x) { Fp::op_.fp2_inv(y.a.v_, x.a.v_); }
    static inline void neg(Fp2T& y, const Fp2T& x) { Fp::op_.fp2_neg(y.a.v_, x.a.v_); }
    static inline void sqr(Fp2T& y, const Fp2T& x) { Fp::op_.fp2_sqr(y.a.v_, x.a.v_); }
    static inline void mul_xi(Fp2T& y, const Fp2T& x) { Fp::op_.fp2_mul_xi(y.a.v_, x.a.v_); }
    static inline void divBy2(Fp2T& y, const Fp2T& x)
    {
        Fp::divBy2(y.a, x.a);
        Fp::divBy2(y.b, x.b);
    }
    /*
        Fp2T = <a> + ' ' + <b>
    */
    friend inline std::ostream& operator<<(std::ostream& os, const Fp2T& self)
    {
        return os << self.a << ' ' << self.b;
    }
    friend inline std::istream& operator>>(std::istream& is, Fp2T& self)
    {
        return is >> self.a >> self.b;
    }
    std::string getStr(int base = 10, bool withPrefix = false)
    {
        return a.getStr(base, withPrefix) + ' ' + b.getStr(base, withPrefix);
    }
    bool isZero() const { return a.isZero() && b.isZero(); }
    bool isOne() const { return a.isOne() && b.isZero(); }
    bool operator==(const Fp2T& rhs) const { return a == rhs.a && b == rhs.b; }
    bool operator!=(const Fp2T& rhs) const { return !operator==(rhs); }
    void normalize() {} // dummy method
    /*
        this function is for only compressed reprezentation of EC
        isOdd() is not good naming. QQQ
    */
    bool isOdd() const { return a.isOdd(); }
    static inline const Fp& getXi_a() { return xi_a_; }
    static inline void init(uint32_t xi_a)
    {
        assert(Fp::maxSize <= 256);
        xi_a_ = xi_a;
        mcl::fp::Op& op = Fp::op_;
        op.fp2_add = fp2_addW;
        op.fp2_sub = fp2_subW;
        op.fp2_mul = op.isFastMod ? fp2_mulW : fp2_mulUseDblW;
        op.fp2_neg = fp2_negW;
        op.fp2_inv = fp2_invW;
        op.fp2_sqr = fp2_sqrW;
        op.fp2_mul_xi = fp2_mul_xiW;
    }
private:
    /*
        default Fp2T operator
        Fp2T = Fp[i]/(i^2 + 1)
    */
    static inline void fp2_addW(Unit *z, const Unit *x, const Unit *y)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        const Fp *py = reinterpret_cast<const Fp*>(y);
        Fp *pz = reinterpret_cast<Fp*>(z);
        Fp::add(pz[0], px[0], py[0]);
        Fp::add(pz[1], px[1], py[1]);
    }
    static inline void fp2_subW(Unit *z, const Unit *x, const Unit *y)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        const Fp *py = reinterpret_cast<const Fp*>(y);
        Fp *pz = reinterpret_cast<Fp*>(z);
        Fp::sub(pz[0], px[0], py[0]);
        Fp::sub(pz[1], px[1], py[1]);
    }
    static inline void fp2_negW(Unit *y, const Unit *x)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        Fp *py = reinterpret_cast<Fp*>(y);
        Fp::neg(py[0], px[0]);
        Fp::neg(py[1], px[1]);
    }
    /*
        x = a + bi, y = c + di, i^2 = -1
        z = xy = (a + bi)(c + di) = (ac - bd) + (ad + bc)i
        ad+bc = (a + b)(c + d) - ac - bd
        # of mod = 3
    */
    static inline void fp2_mulW(Unit *z, const Unit *x, const Unit *y)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        const Fp *py = reinterpret_cast<const Fp*>(y);
        const Fp& a = px[0];
        const Fp& b = px[1];
        const Fp& c = py[0];
        const Fp& d = py[1];
        Fp *pz = reinterpret_cast<Fp*>(z);
        Fp t1, t2, ac, bd;
        Fp::add(t1, a, b);
        Fp::add(t2, c, d);
        t1 *= t2; // (a + b)(c + d)
        Fp::mul(ac, a, c);
        Fp::mul(bd, b, d);
        Fp::sub(pz[0], ac, bd); // ac - bd
        Fp::sub(pz[1], t1, ac);
        pz[1] -= bd;
    }
    /*
        # of mod = 2
        @note mod of NIST_P192 is fast
    */
    static inline void fp2_mulUseDblW(Unit *z, const Unit *x, const Unit *y)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        const Fp *py = reinterpret_cast<const Fp*>(y);
        const Fp& a = px[0];
        const Fp& b = px[1];
        const Fp& c = py[0];
        const Fp& d = py[1];
        FpDbl d0, d1, d2;
        Fp s, t;
        Fp::addNC(s, a, b);
        Fp::addNC(t, c, d);
        FpDbl::mulPre(d0, s, t); // (a + b)(c + d)
        FpDbl::mulPre(d1, a, c);
        FpDbl::mulPre(d2, b, d);
        FpDbl::subNC(d0, d0, d1); // (a + b)(c + d) - ac
        FpDbl::subNC(d0, d0, d2); // (a + b)(c + d) - ac - bd
        Fp *pz = reinterpret_cast<Fp*>(z);
        FpDbl::mod(pz[1], d0);
        FpDbl::sub(d1, d1, d2); // ac - bd
        FpDbl::mod(pz[0], d1); // set z0
    }
    /*
        x = a + bi, i^2 = -1
        y = x^2 = (a + bi)^2 = (a + b)(a - b) + 2abi
    */
    static inline void fp2_sqrW(Unit *y, const Unit *x)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        Fp *py = reinterpret_cast<Fp*>(y);
        const Fp& a = px[0];
        const Fp& b = px[1];
#if 1 // faster than using FpDbl
        Fp t1, t2, t3;
        Fp::add(t1, b, b); // 2b
        t1 *= a; // 2ab
        Fp::add(t2, a, b); // a + b
        Fp::sub(t3, a, b); // a - b
        Fp::mul(py[0], t2, t3); // (a + b)(a - b)
        py[1] = t1;
#else
        Fp t1, t2;
        FpDbl d1, d2;
        Fp::addNC(t1, b, b); // 2b
        FpDbl::mulPre(d2, t1, a); // 2ab
        Fp::addNC(t1, a, b); // a + b
        Fp::sub(t2, a, b); // a - b
        FpDbl::mulPre(d1, t1, t2); // (a + b)(a - b)
        FpDbl::mod(py[0], d1);
        FpDbl::mod(py[1], d2);
#endif
    }
    /*
        xi = xi_a + i
        x = a + bi
        y = (a + bi)xi = (a + bi)(xi_a + i)
        =(a * x_ic - b) + (a + b xi_a)i
    */
    static inline void fp2_mul_xiW(Unit *y, const Unit *x)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        Fp *py = reinterpret_cast<Fp*>(y);
        const Fp& a = px[0];
        const Fp& b = px[1];
        Fp t;
        Fp::mul(t, a, xi_a_);
        t -= b;
        Fp::mul(py[1], b, xi_a_);
        py[1] += a;
        py[0] = t;
    }
    /*
        x = a + bi
        1 / x = (a - bi) / (a^2 + b^2)
    */
    static inline void fp2_invW(Unit *y, const Unit *x)
    {
        const Fp *px = reinterpret_cast<const Fp*>(x);
        Fp *py = reinterpret_cast<Fp*>(y);
        const Fp& a = px[0];
        const Fp& b = px[1];
        Fp aa, bb;
        Fp::sqr(aa, a);
        Fp::sqr(bb, b);
        aa += bb;
        Fp::inv(aa, aa); // aa = 1 / (a^2 + b^2)
        Fp::mul(py[0], a, aa);
        Fp::mul(py[1], b, aa);
        Fp::neg(py[1], py[1]);
    }
    struct Dbl;
};

template<class Fp>
struct Fp2T<Fp>::Dbl {
    typedef fp::Unit Unit;
    typedef typename Fp::Dbl FpDbl;
    FpDbl a, b;
    static inline void add(Dbl& z, const Dbl& x, const Dbl& y)
    {
        FpDbl::add(z.a, x.a, y.a);
        FpDbl::add(z.b, x.b, y.b);
    }
    static inline void addNC(Dbl& z, const Dbl& x, const Dbl& y)
    {
        FpDbl::addNC(z.a, x.a, y.a);
        FpDbl::addNC(z.b, x.b, y.b);
    }
    static inline void sub(Dbl& z, const Dbl& x, const Dbl& y)
    {
        FpDbl::sub(z.a, x.a, y.a);
        FpDbl::sub(z.b, x.b, y.b);
    }
    static inline void subNC(Dbl& z, const Dbl& x, const Dbl& y)
    {
        FpDbl::subNC(z.a, x.a, y.a);
        FpDbl::subNC(z.b, x.b, y.b);
    }
    static inline void neg(Dbl& y, const Dbl& x)
    {
        FpDbl::neg(y.a, x.a);
        FpDbl::neg(y.b, x.b);
    }
    static inline void sqr(Dbl& y, const Fp2T& x)
    {
        Fp t1, t2;
        Fp::addNC(t1, x.b, x.b); // 2b
        FpDbl::mulPre(y.b, t1, x.a); // 2ab
        Fp::addNC(t1, x.a, x.b); // a + b
        Fp::sub(t2, x.a, x.b); // a - b
        FpDbl::mulPre(y.a, t1, t2); // (a + b)(a - b)
    }
    static inline void mod(Fp2T& y, const Dbl& x)
    {
        FpDbl::mod(y.a, x.a);
        FpDbl::mod(y.b, x.b);
    }
};

template<class Fp> Fp Fp2T<Fp>::xi_a_;

/*
    Fp6T = Fp2[v] / (v^3 - xi)
    x = a + b v + c v^2
*/
template<class Fp>
struct Fp6T : public fp::Operator<Fp6T<Fp> > {
    typedef Fp2T<Fp> Fp2;
    Fp2 a, b, c;
    Fp6T() { }
    Fp6T(int64_t a) : a(a) , b(0) , c(0) { }
    Fp6T(const Fp2& a, const Fp2& b, const Fp2& c) : a(a) , b(b) , c(c) { }
    void clear()
    {
        a.clear();
        b.clear();
        c.clear();
    }
    Fp* get() { return a.get(); }
    const Fp* get() const { return a.get(); }
    Fp2* getFp2() { return &a; }
    const Fp2* getFp2() const { return &a; }
    bool isZero() const
    {
        return a.isZero() && b.isZero() && c.isZero();
    }
    bool operator==(const Fp6T& rhs) const
    {
        return a == rhs.a && b == rhs.b && c == rhs.c;
    }
    bool operator!=(const Fp6T& rhs) const { return !operator==(rhs); }
    friend std::ostream& operator<<(std::ostream& os, const Fp6T& x)
    {
        return os << x.a << ' ' << x.b << ' ' << x.c;
    }
    friend std::istream& operator>>(std::istream& is, Fp6T& x)
    {
        return is >> x.a >> x.b >> x.c;
    }
    static inline void add(Fp6T& z, const Fp6T& x, const Fp6T& y)
    {
        Fp2::add(z.a, x.a, y.a);
        Fp2::add(z.b, x.b, y.b);
        Fp2::add(z.c, x.c, y.c);
    }
    static inline void sub(Fp6T& z, const Fp6T& x, const Fp6T& y)
    {
        Fp2::sub(z.a, x.a, y.a);
        Fp2::sub(z.b, x.b, y.b);
        Fp2::sub(z.c, x.c, y.c);
    }
    static inline void neg(Fp6T& y, const Fp6T& x)
    {
        Fp2::neg(y.a, x.a);
        Fp2::neg(y.b, x.b);
        Fp2::neg(y.c, x.c);
    }
    /*
        x = a + bv + cv^2, v^3 = xi
        x^2 = (a^2 + 2bc xi) + (c^2 xi + 2ab)v + (b^2 + 2ac)v^2

        b^2 + 2ac = (a + b + c)^2 - a^2 - 2bc - c^2 - 2ab
    */
    static inline void sqr(Fp6T& y, const Fp6T& x)
    {
        Fp2 t1, t2, t3;
        Fp2::mul(t1, x.a, x.b);
        t1 += t1; // 2ab
        Fp2::mul(t2, x.b, x.c);
        t2 += t2; // 2bc
        Fp2::sqr(t3, x.c); // c^2
        Fp2::add(y.c, x.a, x.c); // a + c, destroy y.c
        y.c += x.b; // a + b + c
        Fp2::sqr(y.b, y.c); // (a + b + c)^2, destroy y.b
        y.b -= t2; // (a + b + c)^2 - 2bc
        Fp2::mul_xi(t2, t2); // 2bc xi
        Fp2::sqr(y.a, x.a); // a^2, destroy y.a
        y.b -= y.a; // (a + b + c)^2 - 2bc - a^2
        y.a += t2; // a^2 + 2bc xi
        Fp2::sub(y.c, y.b, t3); // (a + b + c)^2 - 2bc - a^2 - c^2
        Fp2::mul_xi(y.b, t3); // c^2 xi
        y.b += t1; // c^2 xi + 2ab
        y.c -= t1; // b^2 + 2ac
    }
    /*
        x = a + bv + cv^2, y = d + ev + fv^2, v^3 = xi
        xy = (ad + (bf + ce)xi) + ((ae + bd) + cf xi)v + ((af + cd) + be)v^2
        bf + ce = (b + c)(e + f) - be - cf
        ae + bd = (a + b)(e + d) - ad - be
        af + cd = (a + c)(d + f) - ad - cf
    */
    static inline void mul(Fp6T& z, const Fp6T& x, const Fp6T& y)
    {
        const Fp2& a = x.a;
        const Fp2& b = x.b;
        const Fp2& c = x.c;
        const Fp2& d = y.a;
        const Fp2& e = y.b;
        const Fp2& f = y.c;
        Fp2 ad, be, cf;
        Fp2::mul(ad, a, d);
        Fp2::mul(be, b, e);
        Fp2::mul(cf, c, f);

        Fp2 t1, t2, t3, t4;
        Fp2::add(t1, b, c);
        Fp2::add(t2, e, f);
        t1 *= t2;
        t1 -= be;
        t1 -= cf;
        Fp2::mul_xi(t1, t1);

        Fp2::add(t2, a, b);
        Fp2::add(t3, e, d);
        t2 *= t3;
        t2 -= ad;
        t2 -= be;

        Fp2::add(t3, a, c);
        Fp2::add(t4, d, f);
        t3 *= t4;
        t3 -= ad;
        t3 -= cf;

        Fp2::add(z.a, ad, t1);
        Fp2::mul_xi(z.b, cf);
        z.b += t2;
        Fp2::add(z.c, t3, be);
    }
    /*
        x = a + bv + cv^2, v^3 = xi
        y = 1/x = p/q where
        p = (a^2 - bc xi) + (c^2 xi - ab)v + (b^2 - ac)v^2
        q = c^3 xi^2 + b(b^2 - 3ac)xi + a^3
          = (a^2 - bc xi)a + ((c^2 xi - ab)c + (b^2 - ac)b) xi
    */
    static inline void inv(Fp6T& y, const Fp6T& x)
    {
        const Fp2& a = x.a;
        const Fp2& b = x.b;
        const Fp2& c = x.c;
        Fp2 aa, bb, cc, ab, bc, ac;
        Fp2::sqr(aa, a);
        Fp2::sqr(bb, b);
        Fp2::sqr(cc, c);
        Fp2::mul(ab, a, b);
        Fp2::mul(bc, b, c);
        Fp2::mul(ac, c, a);

        Fp6T p;
        Fp2::mul_xi(p.a, bc);
        Fp2::sub(p.a, aa, p.a); // a^2 - bc xi
        Fp2::mul_xi(p.b, cc);
        p.b -= ab; // c^2 xi - ab
        Fp2::sub(p.c, bb, ac); // b^2 - ac
        Fp2 q, t;
        Fp2::mul(q, p.b, c);
        Fp2::mul(t, p.c, b);
        q += t;
        Fp2::mul_xi(q, q);
        Fp2::mul(t, p.a, a);
        q += t;
        Fp2::inv(q, q);

        Fp2::mul(y.a, p.a, q);
        Fp2::mul(y.b, p.b, q);
        Fp2::mul(y.c, p.c, q);
    }
    void normalize() {} // dummy
};

/*
    Fp12T = Fp6[w] / (w^2 - v)
    x = a + b w
*/
template<class Fp>
struct Fp12T : public fp::Operator<Fp12T<Fp> > {
    typedef Fp2T<Fp> Fp2;
    typedef Fp6T<Fp> Fp6;
    Fp6 a, b;
    Fp12T() {}
    Fp12T(int64_t a) : a(a), b(0) {}
    Fp12T(const Fp6& a, const Fp6& b) : a(a), b(b) {}
    void clear()
    {
        a.clear();
        b.clear();
    }

    Fp* get() { return a.get(); }
    const Fp* get() const { return a.get(); }
    Fp2* getFp2() { return a.getFp2(); }
    const Fp2* getFp2() const { return a.getFp2(); }
    void set(const Fp2& v0, const Fp2& v1, const Fp2& v2, const Fp2& v3, const Fp2& v4, const Fp2& v5)
    {
        a.set(v0, v1, v2);
        b.set(v3, v4, v5);
    }

    bool isZero() const
    {
        return a.isZero() && b.isZero();
    }
    bool operator==(const Fp12T& rhs) const
    {
        return a == rhs.a && b == rhs.b;
    }
    bool operator!=(const Fp12T& rhs) const { return !operator==(rhs); }
    static inline void add(Fp12T& z, const Fp12T& x, const Fp12T& y)
    {
        Fp6::add(z.a, x.a, y.a);
        Fp6::add(z.b, x.b, y.b);
    }
    static inline void sub(Fp12T& z, const Fp12T& x, const Fp12T& y)
    {
        Fp6::sub(z.a, x.a, y.a);
        Fp6::sub(z.b, x.b, y.b);
    }
    static inline void neg(Fp12T& z, const Fp12T& x)
    {
        Fp6::neg(z.a, x.a);
        Fp6::neg(z.b, x.b);
    }
    /*
        z = x v + y
        in Fp6 : (a + bv + cv^2)v = cv^3 + av + bv^2 = cxi + av + bv^2
    */
    static inline void mulVadd(Fp6& z, const Fp6& x, const Fp6& y)
    {
        Fp2 t;
        Fp2::mul_xi(t, x.c);
        Fp2::add(z.c, x.b, y.c);
        Fp2::add(z.b, x.a, y.b);
        Fp2::add(z.a, t, y.a);
    }
    /*
        x = a + bw, y = c + dw, w^2 = v
        z = xy = (a + bw)(c + dw) = (ac + bdv) + (ad + bc)w
        ad+bc = (a + b)(c + d) - ac - bd

        in Fp6 : (a + bv + cv^2)v = cv^3 + av + bv^2 = cxi + av + bv^2
    */
    static inline void mul(Fp12T& z, const Fp12T& x, const Fp12T& y)
    {
        const Fp6& a = x.a;
        const Fp6& b = x.b;
        const Fp6& c = y.a;
        const Fp6& d = y.b;
        Fp6 t1, t2, ac, bd;
        Fp6::add(t1, a, b);
        Fp6::add(t2, c, d);
        t1 *= t2; // (a + b)(c + d)
        Fp6::mul(ac, a, c);
        Fp6::mul(bd, b, d);
        mulVadd(z.a, bd, ac);
        t1 -= ac;
        Fp6::sub(z.b, t1, bd);
    }
    /*
        x = a + bw, w^2 = v
        y = x^2 = (a + bw)^2 = (a^2 + b^2v) + 2abw
        a^2 + b^2v = (a + b)(bv + a) - (abv + ab)
    */
    static inline void sqr(Fp12T& y, const Fp12T& x)
    {
        const Fp6& a = x.a;
        const Fp6& b = x.b;
        Fp6 t0, t1;
        Fp6::add(t0, a, b); // a + b
        mulVadd(t1, b, a); // bv + a
        t0 *= t1; // (a + b)(bv + a)
        Fp6::mul(t1, a, b); // ab
        Fp6::add(y.b, t1, t1); // 2ab
        mulVadd(y.a, t1, t1); // abv + ab
        Fp6::sub(y.a, t0, y.a);
    }
    /*
        x = a + bw, w^2 = v
        y = 1/x = (a - bw) / (a^2 - b^2v)
    */
    static inline void inv(Fp12T& y, const Fp12T& x)
    {
        const Fp6& a = x.a;
        const Fp6& b = x.b;
        Fp6 t0, t1;
        Fp6::sqr(t0, a);
        Fp6::sqr(t1, b);
        Fp2::mul_xi(t1.c, t1.c);
        t0.a -= t1.c;
        t0.b -= t1.a;
        t0.c -= t1.b; // t0 = a^2 - b^2v
        Fp6::inv(t0, t0);
        Fp6::mul(y.a, x.a, t0);
        Fp6::mul(y.b, x.b, t0);
        Fp6::neg(y.b, y.b);
    }
    friend inline std::ostream& operator<<(std::ostream& os, const Fp12T& self)
    {
        return os << self.a << ' ' << self.b;
    }
    friend inline std::istream& operator>>(std::istream& is, Fp12T& self)
    {
        return is >> self.a >> self.b;
    }
    void normalize() {} // dummy
};

} // mcl