Python galoistools.gf_from_int_poly函数代码示例

def dup_zz_hensel_lift(p, f, f_list, l, K):
    """
    Multifactor Hensel lifting in `Z[x]`.

    Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
    is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
    over `Z[x]` satisfying::

        f = lc(f) f_1 ... f_r (mod p)

    and a positive integer `l`, returns a list of monic polynomials
    `F_1`, `F_2`, ..., `F_r` satisfying::

       f = lc(f) F_1 ... F_r (mod p**l)

       F_i = f_i (mod p), i = 1..r

    References
    ==========

    1. [Gathen99]_

    """
    r = len(f_list)
    lc = dup_LC(f, K)

    if r == 1:
        F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
        return [ dup_trunc(F, p**l, K) ]

    m = p
    k = r // 2
    d = int(_ceil(_log(l, 2)))

    g = gf_from_int_poly([lc], p)

    for f_i in f_list[:k]:
        g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)

    h = gf_from_int_poly(f_list[k], p)

    for f_i in f_list[k + 1:]:
        h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)

    s, t, _ = gf_gcdex(g, h, p, K)

    g = gf_to_int_poly(g, p)
    h = gf_to_int_poly(h, p)
    s = gf_to_int_poly(s, p)
    t = gf_to_int_poly(t, p)

    for _ in range(1, d + 1):
        (g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2

    return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
        + dup_zz_hensel_lift(p, h, f_list[k:], l, K)

def dup_zz_diophantine(F, m, p, K):
    """Wang/EEZ: Solve univariate Diophantine equations. """
    if len(F) == 2:
        a, b = F

        f = gf_from_int_poly(a, p)
        g = gf_from_int_poly(b, p)

        s, t, G = gf_gcdex(g, f, p, K)

        s = gf_lshift(s, m, K)
        t = gf_lshift(t, m, K)

        q, s = gf_div(s, f, p, K)

        t = gf_add_mul(t, q, g, p, K)

        s = gf_to_int_poly(s, p)
        t = gf_to_int_poly(t, p)

        result = [s, t]
    else:
        G = [F[-1]]

        for f in reversed(F[1:-1]):
            G.insert(0, dup_mul(f, G[0], K))

        S, T = [], [[1]]

        for f, g in zip(F, G):
            t, s = dmp_zz_diophantine([g, f], T[-1], [], 0, p, 1, K)
            T.append(t)
            S.append(s)

        result, S = [], S + [T[-1]]

        for s, f in zip(S, F):
            s = gf_from_int_poly(s, p)
            f = gf_from_int_poly(f, p)

            r = gf_rem(gf_lshift(s, m, K), f, p, K)
            s = gf_to_int_poly(r, p)

            result.append(s)

    return result

def test_gf_berlekamp():
    f = gf_from_int_poly([1,-3,1,-3,-1,-3,1], 11)

    Q = [[1, 0, 0, 0, 0, 0],
         [3, 5, 8, 8, 6, 5],
         [3, 6, 6, 1,10, 0],
         [9, 4,10, 3, 7, 9],
         [7, 8,10, 0, 0, 8],
         [8,10, 7, 8,10, 8]]

    V = [[1, 0, 0, 0, 0, 0],
         [0, 1, 1, 1, 1, 0],
         [0, 0, 7, 9, 0, 1]]

    assert gf_Qmatrix(f, 11, ZZ) == Q
    assert gf_Qbasis(Q, 11, ZZ) == V

    assert gf_berlekamp(f, 11, ZZ) == \
        [[1, 1], [1, 5, 3], [1, 2, 3, 4]]

    f = [1,0,1,0,10,10,8,2,8]

    Q = [[1, 0, 0, 0, 0, 0, 0, 0],
         [2, 1, 7,11,10,12, 5,11],
         [3, 6, 4, 3, 0, 4, 7, 2],
         [4, 3, 6, 5, 1, 6, 2, 3],
         [2,11, 8, 8, 3, 1, 3,11],
         [6,11, 8, 6, 2, 7,10, 9],
         [5,11, 7,10, 0,11, 7,12],
         [3, 3,12, 5, 0,11, 9,12]]

    V = [[1, 0, 0, 0, 0, 0, 0, 0],
         [0, 5, 5, 0, 9, 5, 1, 0],
         [0, 9,11, 9,10,12, 0, 1]]

    assert gf_Qmatrix(f, 13, ZZ) == Q
    assert gf_Qbasis(Q, 13, ZZ) == V

    assert gf_berlekamp(f, 13, ZZ) == \
        [[1, 3], [1, 8, 4, 12], [1, 2, 3, 4, 6]]

def dup_zz_zassenhaus(f, K):
    """Factor primitive square-free polynomials in `Z[x]`. """
    n = dup_degree(f)

    if n == 1:
        return [f]

    A = dup_max_norm(f, K)
    b = dup_LC(f, K)
    B = int(abs(K.sqrt(K(n+1))*2**n*A*b))
    C = int((n+1)**(2*n)*A**(2*n-1))
    gamma = int(ceil(2*log(C, 2)))
    bound = int(2*gamma*log(gamma))

    for p in xrange(3, bound+1):
        if not isprime(p) or b % p == 0:
            continue

        p = K.convert(p)

        F = gf_from_int_poly(f, p)

        if gf_sqf_p(F, p, K):
            break

    l = int(ceil(log(2*B + 1, p)))

    modular = []

    for ff in gf_factor_sqf(F, p, K)[1]:
        modular.append(gf_to_int_poly(ff, p))

    g = dup_zz_hensel_lift(p, f, modular, l, K)

    T = set(range(len(g)))
    factors, s = [], 1

    while 2*s <= len(T):
        for S in subsets(T, s):
            G, H = [b], [b]

            S = set(S)

            for i in S:
                G = dup_mul(G, g[i], K)
            for i in T-S:
                H = dup_mul(H, g[i], K)

            G = dup_trunc(G, p**l, K)
            H = dup_trunc(H, p**l, K)

            G_norm = dup_l1_norm(G, K)
            H_norm = dup_l1_norm(H, K)

            if G_norm*H_norm <= B:
                T = T - S

                G = dup_primitive(G, K)[1]
                f = dup_primitive(H, K)[1]

                factors.append(G)
                b = dup_LC(f, K)

                break
        else:
            s += 1

    return factors + [f]

def test_gf_factor():
    assert gf_factor([], 11, ZZ) == (0, [])
    assert gf_factor([1], 11, ZZ) == (1, [])
    assert gf_factor([1,1], 11, ZZ) == (1, [([1, 1], 1)])

    assert gf_factor_sqf([], 11, ZZ) == (0, [])
    assert gf_factor_sqf([1], 11, ZZ) == (1, [])
    assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])

    config.setup('GF_FACTOR_METHOD', 'berlekamp')

    assert gf_factor_sqf([], 11, ZZ) == (0, [])
    assert gf_factor_sqf([1], 11, ZZ) == (1, [])
    assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])

    config.setup('GF_FACTOR_METHOD', 'zassenhaus')

    assert gf_factor_sqf([], 11, ZZ) == (0, [])
    assert gf_factor_sqf([1], 11, ZZ) == (1, [])
    assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])

    config.setup('GF_FACTOR_METHOD', 'shoup')

    assert gf_factor_sqf([], 11, ZZ) == (0, [])
    assert gf_factor_sqf([1], 11, ZZ) == (1, [])
    assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])

    f, p = [1,0,0,1,0], 2

    g = (1, [([1, 0], 1),
             ([1, 1], 1),
             ([1, 1, 1], 1)])

    config.setup('GF_FACTOR_METHOD', 'berlekamp')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'shoup')
    assert gf_factor(f, p, ZZ) == g

    g = (1, [[1, 0],
             [1, 1],
             [1, 1, 1]])

    config.setup('GF_FACTOR_METHOD', 'berlekamp')
    assert gf_factor_sqf(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
    assert gf_factor_sqf(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'shoup')
    assert gf_factor_sqf(f, p, ZZ) == g

    f, p = gf_from_int_poly([1,-3,1,-3,-1,-3,1], 11), 11

    g = (1, [([1, 1], 1),
             ([1, 5, 3], 1),
             ([1, 2, 3, 4], 1)])

    config.setup('GF_FACTOR_METHOD', 'berlekamp')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'shoup')
    assert gf_factor(f, p, ZZ) == g

    f, p = [1, 5, 8, 4], 11

    g = (1, [([1, 1], 1), ([1, 2], 2)])

    config.setup('GF_FACTOR_METHOD', 'berlekamp')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'shoup')
    assert gf_factor(f, p, ZZ) == g

    f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11

    g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])

    config.setup('GF_FACTOR_METHOD', 'berlekamp')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
    assert gf_factor(f, p, ZZ) == g

    config.setup('GF_FACTOR_METHOD', 'shoup')
    assert gf_factor(f, p, ZZ) == g

    f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11

    g = (1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
             ([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
#.........这里部分代码省略.........

def test_gf_from_to_int_poly():
    assert gf_from_int_poly([1,0,7,2,20], 5) == [1,0,2,2,0]
    assert gf_to_int_poly([1,0,4,2,3], 5) == [1,0,-1,2,-2]

    assert gf_to_int_poly([10], 11, symmetric=True) == [-1]
    assert gf_to_int_poly([10], 11, symmetric=False) == [10]

def dup_zz_zassenhaus(f, K):
    """Factor primitive square-free polynomials in `Z[x]`. """
    n = dup_degree(f)

    if n == 1:
        return [f]

    fc = f[-1]
    A = dup_max_norm(f, K)
    b = dup_LC(f, K)
    B = int(abs(K.sqrt(K(n + 1))*2**n*A*b))
    C = int((n + 1)**(2*n)*A**(2*n - 1))
    gamma = int(_ceil(2*_log(C, 2)))
    bound = int(2*gamma*_log(gamma))
    a = []
    # choose a prime number `p` such that `f` be square free in Z_p
    # if there are many factors in Z_p, choose among a few different `p`
    # the one with fewer factors
    for px in range(3, bound + 1):
        if not isprime(px) or b % px == 0:
            continue

        px = K.convert(px)

        F = gf_from_int_poly(f, px)

        if not gf_sqf_p(F, px, K):
            continue
        fsqfx = gf_factor_sqf(F, px, K)[1]
        a.append((px, fsqfx))
        if len(fsqfx) < 15 or len(a) > 4:
            break
    p, fsqf = min(a, key=lambda x: len(x[1]))

    l = int(_ceil(_log(2*B + 1, p)))

    modular = [gf_to_int_poly(ff, p) for ff in fsqf]

    g = dup_zz_hensel_lift(p, f, modular, l, K)

    sorted_T = range(len(g))
    T = set(sorted_T)
    factors, s = [], 1
    pl = p**l

    while 2*s <= len(T):
        for S in subsets(sorted_T, s):
            # lift the constant coefficient of the product `G` of the factors
            # in the subset `S`; if it is does not divide `fc`, `G` does
            # not divide the input polynomial

            if b == 1:
                q = 1
                for i in S:
                    q = q*g[i][-1]
                q = q % pl
                if not _test_pl(fc, q, pl):
                    continue
            else:
                G = [b]
                for i in S:
                    G = dup_mul(G, g[i], K)
                G = dup_trunc(G, pl, K)
                G = dup_primitive(G, K)[1]
                q = G[-1]
                if q and fc % q != 0:
                    continue

            H = [b]
            S = set(S)
            T_S = T - S

            if b == 1:
                G = [b]
                for i in S:
                    G = dup_mul(G, g[i], K)
                G = dup_trunc(G, pl, K)

            for i in T_S:
                H = dup_mul(H, g[i], K)

            H = dup_trunc(H, pl, K)

            G_norm = dup_l1_norm(G, K)
            H_norm = dup_l1_norm(H, K)

            if G_norm*H_norm <= B:
                T = T_S
                sorted_T = [i for i in sorted_T if i not in S]

                G = dup_primitive(G, K)[1]
                f = dup_primitive(H, K)[1]

                factors.append(G)
                b = dup_LC(f, K)

                break
        else:
            s += 1

#.........这里部分代码省略.........