Exemplo n.º 1
0
def test_schur_partition():
    raises(ValueError, lambda: schur_partition(S.Infinity))
    raises(ValueError, lambda: schur_partition(-1))
    raises(ValueError, lambda: schur_partition(0))
    assert schur_partition(2) == [[1, 2]]

    random_number_generator = _randint(1000)
    for _ in range(5):
        n = random_number_generator(1, 1000)
        result = schur_partition(n)
        t = 0
        numbers = []
        for item in result:
            _sum_free_test(item)
            """
            Checks if the occurance of all numbers  is exactly one
            """
            t += len(item)
            for l in item:
                assert (l in numbers) is False
                numbers.append(l)
        assert n == t

    x = symbols("x")
    raises(ValueError, lambda: schur_partition(x))
Exemplo n.º 2
0
def random_integer_partition(n, seed=None):
    """
    Generates a random integer partition summing to ``n`` as a list
    of reverse-sorted integers.

    Examples
    ========

    >>> from sympy.combinatorics.partitions import random_integer_partition

    For the following, a seed is given so a known value can be shown; in
    practice, the seed would not be given.

    >>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1])
    [85, 12, 2, 1]
    >>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1])
    [5, 3, 1, 1]
    >>> random_integer_partition(1)
    [1]
    """
    from sympy.core.random import _randint

    n = as_int(n)
    if n < 1:
        raise ValueError('n must be a positive integer')

    randint = _randint(seed)

    partition = []
    while (n > 0):
        k = randint(1, n)
        mult = randint(1, n//k)
        partition.append((k, mult))
        n -= k*mult
    partition.sort(reverse=True)
    partition = flatten([[k]*m for k, m in partition])
    return partition
Exemplo n.º 3
0
from sympy.functions.special.hyper import hyper
from sympy.integrals.integrals import Integral
from sympy.series.order import O
from sympy.series.series import series
from sympy.functions.special.bessel import (airyai, airybi,
                                            airyaiprime, airybiprime, marcumq)
from sympy.core.random import (random_complex_number as randcplx,
                                      verify_numerically as tn,
                                      test_derivative_numerically as td,
                                      _randint)
from sympy.simplify import besselsimp
from sympy.testing.pytest import raises

from sympy.abc import z, n, k, x

randint = _randint()


def test_bessel_rand():
    for f in [besselj, bessely, besseli, besselk, hankel1, hankel2]:
        assert td(f(randcplx(), z), z)

    for f in [jn, yn, hn1, hn2]:
        assert td(f(randint(-10, 10), z), z)


def test_bessel_twoinputs():
    for f in [besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn]:
        raises(TypeError, lambda: f(1))
        raises(TypeError, lambda: f(1, 2, 3))
Exemplo n.º 4
0
def _discrete_log_pollard_rho(n, a, b, order=None, retries=10, rseed=None):
    """
    Pollard's Rho algorithm for computing the discrete logarithm of ``a`` to
    the base ``b`` modulo ``n``.

    It is a randomized algorithm with the same expected running time as
    ``_discrete_log_shanks_steps``, but requires a negligible amount of memory.

    Examples
    ========

    >>> from sympy.ntheory.residue_ntheory import _discrete_log_pollard_rho
    >>> _discrete_log_pollard_rho(227, 3**7, 3)
    7

    See Also
    ========

    discrete_log

    References
    ==========

    .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
        Vanstone, S. A. (1997).
    """
    a %= n
    b %= n

    if order is None:
        order = n_order(b, n)
    randint = _randint(rseed)

    for i in range(retries):
        aa = randint(1, order - 1)
        ba = randint(1, order - 1)
        xa = pow(b, aa, n) * pow(a, ba, n) % n

        c = xa % 3
        if c == 0:
            xb = a * xa % n
            ab = aa
            bb = (ba + 1) % order
        elif c == 1:
            xb = xa * xa % n
            ab = (aa + aa) % order
            bb = (ba + ba) % order
        else:
            xb = b * xa % n
            ab = (aa + 1) % order
            bb = ba

        for j in range(order):
            c = xa % 3
            if c == 0:
                xa = a * xa % n
                ba = (ba + 1) % order
            elif c == 1:
                xa = xa * xa % n
                aa = (aa + aa) % order
                ba = (ba + ba) % order
            else:
                xa = b * xa % n
                aa = (aa + 1) % order

            c = xb % 3
            if c == 0:
                xb = a * xb % n
                bb = (bb + 1) % order
            elif c == 1:
                xb = xb * xb % n
                ab = (ab + ab) % order
                bb = (bb + bb) % order
            else:
                xb = b * xb % n
                ab = (ab + 1) % order

            c = xb % 3
            if c == 0:
                xb = a * xb % n
                bb = (bb + 1) % order
            elif c == 1:
                xb = xb * xb % n
                ab = (ab + ab) % order
                bb = (bb + bb) % order
            else:
                xb = b * xb % n
                ab = (ab + 1) % order

            if xa == xb:
                r = (ba - bb) % order
                try:
                    e = mod_inverse(r, order) * (ab - aa) % order
                    if (pow(b, e, n) - a) % n == 0:
                        return e
                except ValueError:
                    pass
                break
    raise ValueError("Pollard's Rho failed to find logarithm")
Exemplo n.º 5
0
def dmp_zz_wang(f, u, K, mod=None, seed=None):
    r"""
    Factor primitive square-free polynomials in `Z[X]`.

    Given a multivariate polynomial `f` in `Z[x_1,...,x_n]`, which is
    primitive and square-free in `x_1`, computes factorization of `f` into
    irreducibles over integers.

    The procedure is based on Wang's Enhanced Extended Zassenhaus
    algorithm. The algorithm works by viewing `f` as a univariate polynomial
    in `Z[x_2,...,x_n][x_1]`, for which an evaluation mapping is computed::

                      x_2 -> a_2, ..., x_n -> a_n

    where `a_i`, for `i = 2, \dots, n`, are carefully chosen integers.  The
    mapping is used to transform `f` into a univariate polynomial in `Z[x_1]`,
    which can be factored efficiently using Zassenhaus algorithm. The last
    step is to lift univariate factors to obtain true multivariate
    factors. For this purpose a parallel Hensel lifting procedure is used.

    The parameter ``seed`` is passed to _randint and can be used to seed randint
    (when an integer) or (for testing purposes) can be a sequence of numbers.

    References
    ==========

    .. [1] [Wang78]_
    .. [2] [Geddes92]_

    """
    from sympy.core.random import _randint

    randint = _randint(seed)

    ct, T = dmp_zz_factor(dmp_LC(f, K), u - 1, K)

    b = dmp_zz_mignotte_bound(f, u, K)
    p = K(nextprime(b))

    if mod is None:
        if u == 1:
            mod = 2
        else:
            mod = 1

    history, configs, A, r = set(), [], [K.zero] * u, None

    try:
        cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)

        _, H = dup_zz_factor_sqf(s, K)

        r = len(H)

        if r == 1:
            return [f]

        configs = [(s, cs, E, H, A)]
    except EvaluationFailed:
        pass

    eez_num_configs = query('EEZ_NUMBER_OF_CONFIGS')
    eez_num_tries = query('EEZ_NUMBER_OF_TRIES')
    eez_mod_step = query('EEZ_MODULUS_STEP')

    while len(configs) < eez_num_configs:
        for _ in range(eez_num_tries):
            A = [K(randint(-mod, mod)) for _ in range(u)]

            if tuple(A) not in history:
                history.add(tuple(A))
            else:
                continue

            try:
                cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)
            except EvaluationFailed:
                continue

            _, H = dup_zz_factor_sqf(s, K)

            rr = len(H)

            if r is not None:
                if rr != r:  # pragma: no cover
                    if rr < r:
                        configs, r = [], rr
                    else:
                        continue
            else:
                r = rr

            if r == 1:
                return [f]

            configs.append((s, cs, E, H, A))

            if len(configs) == eez_num_configs:
                break
        else:
            mod += eez_mod_step

    s_norm, s_arg, i = None, 0, 0

    for s, _, _, _, _ in configs:
        _s_norm = dup_max_norm(s, K)

        if s_norm is not None:
            if _s_norm < s_norm:
                s_norm = _s_norm
                s_arg = i
        else:
            s_norm = _s_norm

        i += 1

    _, cs, E, H, A = configs[s_arg]
    orig_f = f

    try:
        f, H, LC = dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K)
        factors = dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K)
    except ExtraneousFactors:  # pragma: no cover
        if query('EEZ_RESTART_IF_NEEDED'):
            return dmp_zz_wang(orig_f, u, K, mod + 1)
        else:
            raise ExtraneousFactors(
                "we need to restart algorithm with better parameters")

    result = []

    for f in factors:
        _, f = dmp_ground_primitive(f, u, K)

        if K.is_negative(dmp_ground_LC(f, u, K)):
            f = dmp_neg(f, u, K)

        result.append(f)

    return result