Python FiniteField.multiplicative_generator方法代码示例

# 需要导入模块: from sage.rings.finite_rings.finite_field_constructor import FiniteField [as 别名]
# 或者: from sage.rings.finite_rings.finite_field_constructor.FiniteField import multiplicative_generator [as 别名]
def szekeres_difference_set_pair(m, check=True):
    r"""
    Construct Szekeres `(2m+1,m,1)`-cyclic difference family

    Let `4m+3` be a prime power. Theorem 3 in [Sz69]_ contains a construction of a pair
    of *complementary difference sets* `A`, `B` in the subgroup `G` of the quadratic
    residues in `F_{4m+3}^*`. Namely `|A|=|B|=m`, `a\in A` whenever `a-1\in G`, `b\in B`
    whenever `b+1 \in G`. See also Theorem 2.6 in [SWW72]_ (there the formula for `B` is
    correct, as opposed to (4.2) in [Sz69]_, where the sign before `1` is wrong.

    In modern terminology, for `m>1` the sets `A` and `B` form a
    :func:`difference family<sage.combinat.designs.difference_family>` with parameters `(2m+1,m,1)`.
    I.e. each non-identity `g \in G` can be expressed uniquely as `xy^{-1}` for `x,y \in A` or `x,y \in B`.
    Other, specific to this construction, properties of `A` and `B` are: for `a` in `A` one has
    `a^{-1}` not in `A`, whereas for `b` in `B` one has `b^{-1}` in `B`.

    INPUT:

    - ``m`` (integer) -- dimension of the matrix

    - ``check`` (default: ``True``) -- whether to check `A` and `B` for correctness

    EXAMPLES::

        sage: from sage.combinat.matrices.hadamard_matrix import szekeres_difference_set_pair
        sage: G,A,B=szekeres_difference_set_pair(6)
        sage: G,A,B=szekeres_difference_set_pair(7)

    REFERENCE:

    .. [Sz69] \G. Szekeres,
      Tournaments and Hadamard matrices,
      Enseignement Math. (2) 15(1969), 269-278
    """
    from sage.rings.finite_rings.finite_field_constructor import GF
    F = GF(4*m+3)
    t = F.multiplicative_generator()**2
    G = F.cyclotomic_cosets(t, cosets=[F.one()])[0]
    sG = set(G)
    A = filter(lambda a: a-F.one() in sG, G)
    B = filter(lambda b: b+F.one() in sG, G)
    if check:
        from itertools import product, chain
        assert(len(A)==len(B)==m)
        if m>1:
            assert(sG==set([xy[0]/xy[1] for xy in chain(product(A,A), product(B,B))]))
        assert(all(F.one()/b+F.one() in sG for b in B))
        assert(not any(F.one()/a-F.one() in sG for a in A))
    return G,A,B

# 需要导入模块: from sage.rings.finite_rings.finite_field_constructor import FiniteField [as 别名]
# 或者: from sage.rings.finite_rings.finite_field_constructor.FiniteField import multiplicative_generator [as 别名]
    def polynomial(self, n):
        r"""
        Return the pseudo-Conway polynomial of degree `n` in this
        lattice.

        INPUT:

        - ``n`` -- positive integer

        OUTPUT:

        - a pseudo-Conway polynomial of degree `n` for the prime `p`.

        ALGORITHM:

        Uses an algorithm described in [HL99]_, modified to find
        pseudo-Conway polynomials rather than Conway polynomials.  The
        major difference is that we stop as soon as we find a
        primitive polynomial.

        REFERENCE:

        .. [HL99] \L. Heath and N. Loehr (1999).  New algorithms for
           generating Conway polynomials over finite fields.
           Proceedings of the tenth annual ACM-SIAM symposium on
           discrete algorithms, pp. 429-437.

        EXAMPLES::

            sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
            sage: PCL = PseudoConwayLattice(2, use_database=False)
            sage: PCL.polynomial(3)
            x^3 + x + 1
            sage: PCL.polynomial(4)
            x^4 + x^3 + 1
            sage: PCL.polynomial(60)
            x^60 + x^59 + x^58 + x^55 + x^54 + x^53 + x^52 + x^51 + x^48 + x^46 + x^45 + x^42 + x^41 + x^39 + x^38 + x^37 + x^35 + x^32 + x^31 + x^30 + x^28 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^15 + x^14 + x^10 + x^8 + x^7 + x^5 + x^3 + x^2 + x + 1
        """
        if n in self.nodes:
            return self.nodes[n]

        p = self.p
        n = Integer(n)

        if n == 1:
            f = self.ring.gen() - FiniteField(p).multiplicative_generator()
            self.nodes[1] = f
            return f

        # Work in an arbitrary field K of order p**n.
        K = FiniteField(p**n, names='a')

        # TODO: something like the following
        # gcds = [n.gcd(d) for d in self.nodes.keys()]
        # xi = { m: (...) for m in gcds }
        xi = {q: self.polynomial(n//q).any_root(K, -n//q, assume_squarefree=True)
              for q in n.prime_divisors()}

        # The following is needed to ensure that in the concrete instantiation
        # of the "new" extension all previous choices are compatible.
        _frobenius_shift(K, xi)

        # Construct a compatible element having order the lcm of orders
        q, x = xi.popitem()
        v = p**(n//q) - 1
        for q, xitem in six.iteritems(xi):
            w = p**(n//q) - 1
            g, alpha, beta = v.xgcd(w)
            x = x**beta * xitem**alpha
            v = v.lcm(w)

        r = p**n - 1
        # Get the missing part of the order to be primitive
        g = r // v
        # Iterate through g-th roots of x until a primitive one is found
        z = x.nth_root(g)
        root = K.multiplicative_generator()**v
        while z.multiplicative_order() != r:
            z *= root
        # The following should work but tries to create a huge list
        # whose length overflows Python's ints for large parameters
        #Z = x.nth_root(g, all=True)
        #for z in Z:
        #    if z.multiplicative_order() == r:
        #         break
        f = z.minimal_polynomial()
        self.nodes[n] = f
        return f