Python MatrixSpace.random_element方法代码示例

# 需要导入模块: from sage.matrix.matrix_space import MatrixSpace [as 别名]
# 或者: from sage.matrix.matrix_space.MatrixSpace import random_element [as 别名]
def RandomLinearCode(n,k,F):
    r"""
    The method used is to first construct a `k \times n`
    matrix using Sage's random_element method for the MatrixSpace
    class. The construction is probabilistic but should only fail
    extremely rarely.

    INPUT: Integers n,k, with `n>k`, and a finite field F

    OUTPUT: Returns a "random" linear code with length n, dimension k
    over field F.

    EXAMPLES::

        sage: C = codes.RandomLinearCode(30,15,GF(2))
        sage: C
        Linear code of length 30, dimension 15 over Finite Field of size 2
        sage: C = codes.RandomLinearCode(10,5,GF(4,'a'))
        sage: C
        Linear code of length 10, dimension 5 over Finite Field in a of size 2^2

    AUTHORS:

    - David Joyner (2007-05)
    """
    MS = MatrixSpace(F,k,n)
    for i in range(50):
        G = MS.random_element()
        if G.rank() == k:
            V = span(G.rows(), F)
            return LinearCodeFromVectorSpace(V)  # may not be in standard form
    MS1 = MatrixSpace(F,k,k)
    MS2 = MatrixSpace(F,k,n-k)
    Ik = MS1.identity_matrix()
    A = MS2.random_element()
    G = Ik.augment(A)
    return LinearCode(G)                          # in standard form

# 需要导入模块: from sage.matrix.matrix_space import MatrixSpace [as 别名]
# 或者: from sage.matrix.matrix_space.MatrixSpace import random_element [as 别名]

#.........这里部分代码省略.........

    .. [A96] Miklos Ajtai.
      Generating hard instances of lattice problems (extended abstract).
      STOC, pp. 99--108, ACM, 1996.

    .. [GM02] Daniel Goldstein and Andrew Mayer.
      On the equidistribution of Hecke points.
      Forum Mathematicum, 15:2, pp. 165--189, De Gruyter, 2003.

    .. [LM06] Vadim Lyubashevsky and Daniele Micciancio.
      Generalized compact knapsacks are collision resistant.
      ICALP, pp. 144--155, Springer, 2006.

    .. [R05] Oded Regev.
      On lattices, learning with errors, random linear codes, and cryptography.
      STOC, pp. 84--93, ACM, 2005.
    """
    from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
    from sage.matrix.constructor import identity_matrix, block_matrix
    from sage.matrix.matrix_space import MatrixSpace
    from sage.rings.integer_ring import IntegerRing
    if seed is not None:
        from sage.misc.randstate import set_random_seed
        set_random_seed(seed)

    if type == 'random':
        if n != 1: raise ValueError('random bases require n = 1')

    ZZ = IntegerRing()
    ZZ_q = IntegerModRing(q)
    A = identity_matrix(ZZ_q, n)

    if type == 'random' or type == 'modular':
        R = MatrixSpace(ZZ_q, m-n, n)
        A = A.stack(R.random_element())

    elif type == 'ideal':
        if quotient is None:
            raise ValueError('ideal bases require a quotient polynomial')
        try:
            quotient = quotient.change_ring(ZZ_q)
        except (AttributeError, TypeError):
            quotient = quotient.polynomial(base_ring=ZZ_q)

        P = quotient.parent()
        # P should be a univariate polynomial ring over ZZ_q
        if not is_PolynomialRing(P):
            raise TypeError("quotient should be a univariate polynomial")
        assert P.base_ring() is ZZ_q

        if quotient.degree() != n:
            raise ValueError('ideal basis requires n = quotient.degree()')
        R = P.quotient(quotient)
        for i in range(m//n):
            A = A.stack(R.random_element().matrix())

    elif type == 'cyclotomic':
        from sage.arith.all import euler_phi
        from sage.misc.functional import cyclotomic_polynomial

        # we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n
        found = False
        for k in range(2*n,n,-1):
            if euler_phi(k) == n:
                found = True
                break
        if not found:
            raise ValueError("cyclotomic bases require that n "
                       "is an image of Euler's totient function")

        R = ZZ_q['x'].quotient(cyclotomic_polynomial(k, 'x'), 'x')
        for i in range(m//n):
            A = A.stack(R.random_element().matrix())

    # switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2
    def minrep(a):
        if abs(a-q) < abs(a): return a-q
        else: return a
    A_prime = A[n:m].lift().apply_map(minrep)

    if not dual:
        B = block_matrix([[ZZ(q), ZZ.zero()], [A_prime, ZZ.one()] ],
                         subdivide=False)
    else:
        B = block_matrix([[ZZ.one(), -A_prime.transpose()],
            [ZZ.zero(), ZZ(q)]], subdivide=False)
        for i in range(m//2):
            B.swap_rows(i,m-i-1)

    if ntl and lattice:
        raise ValueError("Cannot specify ntl=True and lattice=True "
                         "at the same time")

    if ntl:
        return B._ntl_()
    elif lattice:
        from sage.modules.free_module_integer import IntegerLattice
        return IntegerLattice(B)
    else:
        return B

# 需要导入模块: from sage.matrix.matrix_space import MatrixSpace [as 别名]
# 或者: from sage.matrix.matrix_space.MatrixSpace import random_element [as 别名]

#.........这里部分代码省略.........
        [ 0  0  0  0  0 11  0  0  0  0]
        [ 0  0  0  0 11  0  0  0  0  0]
        [ 0  0  0  1 -5 -2 -1  1 -3  5]
        [ 0  0  1  0 -3  4  1  4 -3 -2]
        [ 0  1  0  0 -4  5 -3  3  5  3]
        [ 1  0  0  0 -2 -1  4  2  5  4]

    * Relation of primal and dual bases ::

        sage: B_primal=sage.crypto.gen_lattice(m=10, q=11, seed=42)
        sage: B_dual=sage.crypto.gen_lattice(m=10, q=11, seed=42, dual=True)
        sage: B_dual_alt=transpose(11*B_primal.inverse()).change_ring(ZZ)
        sage: B_dual_alt.hermite_form() == B_dual.hermite_form()
        True

    REFERENCES:

.. [A96] Miklos Ajtai.
   Generating hard instances of lattice problems (extended abstract).
   STOC, pp. 99--108, ACM, 1996.

.. [GM02] Daniel Goldstein and Andrew Mayer.
   On the equidistribution of Hecke points.
   Forum Mathematicum, 15:2, pp. 165--189, De Gruyter, 2003.

.. [LM06] Vadim Lyubashevsky and Daniele Micciancio.
   Generalized compact knapsacks are collision resistant.
   ICALP, pp. 144--155, Springer, 2006.

.. [R05] Oded Regev.
   On lattices, learning with errors, random linear codes, and cryptography.
   STOC, pp. 84--93, ACM, 2005.
    """
    from sage.rings.finite_rings.integer_mod_ring \
        import IntegerModRing
    from sage.matrix.constructor import matrix, \
        identity_matrix, block_matrix
    from sage.matrix.matrix_space import MatrixSpace
    from sage.rings.integer_ring import IntegerRing
    if seed != None:
        from sage.misc.randstate import set_random_seed
        set_random_seed(seed)

    if type == 'random':
        if n != 1: raise ValueError('random bases require n = 1')

    ZZ = IntegerRing()
    ZZ_q = IntegerModRing(q)
    A = identity_matrix(ZZ_q, n)

    if type == 'random' or type == 'modular':
        R = MatrixSpace(ZZ_q, m-n, n)
        A = A.stack(R.random_element())

    elif type == 'ideal':
        if quotient == None: raise \
            ValueError('ideal bases require a quotient polynomial')
        x = quotient.default_variable()
        if n != quotient.degree(x): raise \
            ValueError('ideal bases require n  = quotient.degree()')
        R = ZZ_q[x].quotient(quotient, x)
        for i in range(m//n):
            A = A.stack(R.random_element().matrix())

    elif type == 'cyclotomic':
        from sage.rings.arith import euler_phi
        from sage.misc.functional import cyclotomic_polynomial

        # we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n
        found = False
        for k in range(2*n,n,-1):
            if euler_phi(k) == n:
                found = True
                break
        if not found: raise \
            ValueError('cyclotomic bases require that n is an image of' + \
                       'Euler\'s totient function')

        R = ZZ_q['x'].quotient(cyclotomic_polynomial(k, 'x'), 'x')
        for i in range(m//n):
            A = A.stack(R.random_element().matrix())

    # switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2
    def minrep(a):
        if abs(a-q) < abs(a): return a-q
        else: return a
    A_prime = A[n:m].lift().apply_map(minrep)

    if not dual:
        B = block_matrix([[ZZ(q), ZZ.zero()], [A_prime, ZZ.one()] ], \
                         subdivide=False)
    else:
        B = block_matrix([[ZZ.one(), -A_prime.transpose()], [ZZ.zero(), \
                         ZZ(q)]], subdivide=False)
        for i in range(m//2): B.swap_rows(i,m-i-1)

    if not ntl:
        return B
    else:
        return B._ntl_()