def rrs_simple(n):
    """
    @type n: integer > 0
    @rtype: list

    @return: a reduced residue system modulo n
    """
    return [x for x in range(n) if gcd(x, n) == 1]

def weak_normalizer(a, d, DE, z=None):
    """
    Weak normalization.

    Given a derivation D on k[t] and f == a/d in k(t), return q in k[t]
    such that f - Dq/q is weakly normalized with respect to t.

    f in k(t) is said to be "weakly normalized" with respect to t if
    residue_p(f) is not a positive integer for any normal irreducible p
    in k[t] such that f is in R_p (Definition 6.1.1).  If f has an
    elementary integral, this is equivalent to no logarithm of
    integral(f) whose argument depends on t has a positive integer
    coefficient, where the arguments of the logarithms not in k(t) are
    in k[t].

    Returns (q, f - Dq/q)
    """
    z = z or Dummy('z')
    dn, ds = splitfactor(d, DE)

    # Compute d1, where dn == d1*d2**2*...*dn**n is a square-free
    # factorization of d.
    g = gcd(dn, dn.diff(DE.t))
    d_sqf_part = dn.quo(g)
    d1 = d_sqf_part.quo(gcd(d_sqf_part, g))

    a1, b = gcdex_diophantine(d.quo(d1).as_poly(DE.t), d1.as_poly(DE.t),
        a.as_poly(DE.t))
    r = (a - Poly(z, DE.t)*derivation(d1, DE)).as_poly(DE.t).resultant(
        d1.as_poly(DE.t))
    r = Poly(r, z)

    if not r.has(z):
        return (Poly(1, DE.t), (a, d))

    N = [i for i in r.real_roots() if i in ZZ and i > 0]

    q = reduce(mul, [gcd(a - Poly(n, DE.t)*derivation(d1, DE), d1) for n in N],
        Poly(1, DE.t))

    dq = derivation(q, DE)
    sn = q*a - d*dq
    sd = q*d
    sn, sd = sn.cancel(sd, include=True)

    return (q, (sn, sd))

    def _deflation(p):
        for y in V:
            if not p.has(y):
                continue

            if _derivation(p) is not S.Zero:
                c, q = p.as_poly(y).primitive()
                return _deflation(c)*gcd(q, q.diff(y)).as_expr()
        else:
            return p

    def deflation(p):
        for y in V:
            if not p.has_any_symbols(y):
                continue

            if derivation(p) is not S.Zero:
                c, q = p.as_poly(y).as_primitive()
                return deflation(c)*gcd(q, q.diff(y))
        else:
            return p

                def extract(f):
                    p = f.args[0]

                    for q in f.args[1:]:
                        p = gcd(p, q, *symbols)

                        if p.is_number:
                            return S.One, f

                    return p, Add(*[quo(g, p, *symbols) for g in f.args])

def rsa(bits):
    p = nextprime(randrange(2**(bits // 2 + 1)))
    q = nextprime(randrange(2**(bits // 2 + 1)))
    n = p * q
    phi_n = (p - 1) * (q - 1)
    while True:
        e = randrange(1, phi_n)
        if gcd(e, phi_n) == 1:
            break
    d = inv_mod(e, phi_n)
    return e, d, n

    def _splitter(p):
        for y in V:
            if not p.has(y):
                continue

            if _derivation(y) is not S.Zero:
                c, q = p.as_poly(y).primitive()

                q = q.as_expr()

                h = gcd(q, _derivation(q), y)
                s = quo(h, gcd(q, q.diff(y), y), y)

                c_split = _splitter(c)

                if s.as_poly(y).degree() == 0:
                    return (c_split[0], q * c_split[1])

                q_split = _splitter(cancel(q / s))

                return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
        else:
            return (S.One, p)

    def splitter(p):
        for y in V:
            if not p.has_any_symbols(y):
                continue

            if derivation(y) is not S.Zero:
                c, q = p.as_poly(y).as_primitive()

                q = q.as_basic()

                h = gcd(q, derivation(q), y)
                s = quo(h, gcd(q, q.diff(y), y), y)

                c_split = splitter(c)

                if s.as_poly(y).degree == 0:
                    return (c_split[0], q * c_split[1])

                q_split = splitter(Poly.cancel((q, s), *V))

                return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
        else:
            return (S.One, p)

def crack_given_decrypt(n, m):
    n = int(n)
    m = int(m)

    while True:
        if m % 2 != 0:
            break
        divide_out = True
        for _ in range(5):
            a = randrange(1, n)
            if gcd(a, n) == 1:
                if pow(a, m // 2, n) != 1:
                    divide_out = False
                    break
        if divide_out:
            m //= 2
        else:
            break

    while True:
        a = randrange(1, n)
        g = gcd(pow(a, m // 2, n) - 1, n)
        if g != 1 and g != n:
            return g

def _split_gcd(*a):
    """
    split the list of integers ``a`` into a list of integers, ``a1`` having
    ``g = gcd(a1)``, and a list ``a2`` whose elements are not divisible by
    ``g``.  Returns ``g, a1, a2``

    Examples
    ========

    >>> from sympy.simplify.radsimp import _split_gcd
    >>> _split_gcd(55, 35, 22, 14, 77, 10)
    (5, [55, 35, 10], [22, 14, 77])
    """
    g = a[0]
    b1 = [g]
    b2 = []
    for x in a[1:]:
        g1 = gcd(g, x)
        if g1 == 1:
            b2.append(x)
        else:
            g = g1
            b1.append(x)
    return g, b1, b2

def factor_nc(expr):
    """Return the factored form of ``expr`` while handling non-commutative
    expressions.

    **examples**
    >>> from sympy.core.exprtools import factor_nc
    >>> from sympy import Symbol
    >>> from sympy.abc import x
    >>> A = Symbol('A', commutative=False)
    >>> B = Symbol('B', commutative=False)
    >>> factor_nc((x**2 + 2*A*x + A**2).expand())
    (x + A)**2
    >>> factor_nc(((x + A)*(x + B)).expand())
    (x + A)*(x + B)
    """
    from sympy.simplify.simplify import _mexpand
    from sympy.polys import gcd, factor

    expr = sympify(expr)
    if not isinstance(expr, Expr) or not expr.args:
        return expr
    if not expr.is_Add:
        return expr.func(*[factor_nc(a) for a in expr.args])

    expr, rep, nc_symbols = _mask_nc(expr)
    if rep:
        return factor(expr).subs(rep)
    else:
        args = [a.args_cnc() for a in Add.make_args(expr)]
        c = g = l = r = S.One
        hit = False
        # find any commutative gcd term
        for i, a in enumerate(args):
            if i == 0:
                c = Mul._from_args(a[0])
            elif a[0]:
                c = gcd(c, Mul._from_args(a[0]))
            else:
                c = S.One
        if c is not S.One:
            hit = True
            c, g = c.as_coeff_Mul()
            if g is not S.One:
                for i, (cc, _) in enumerate(args):
                    cc = list(Mul.make_args(Mul._from_args(list(cc))/g))
                    args[i][0] = cc
            else:
                for i, (cc, _) in enumerate(args):
                    cc[0] = cc[0]/c
                    args[i][0] = cc
        # find any noncommutative common prefix
        for i, a in enumerate(args):
            if i == 0:
                n = a[1][:]
            else:
                n = common_prefix(n, a[1])
            if not n:
                # is there a power that can be extracted?
                if not args[0][1]:
                    break
                b, e = args[0][1][0].as_base_exp()
                ok = False
                if e.is_Integer:
                    for t in args:
                        if not t[1]:
                            break
                        bt, et = t[1][0].as_base_exp()
                        if et.is_Integer and bt == b:
                            e = min(e, et)
                        else:
                            break
                    else:
                        ok = hit = True
                        l = b**e
                        il = b**-e
                        for i, a in enumerate(args):
                            args[i][1][0] = il*args[i][1][0]
                        break
                if not ok:
                    break
        else:
            hit = True
            lenn = len(n)
            l = Mul(*n)
            for i, a in enumerate(args):
                args[i][1] = args[i][1][lenn:]
        # find any noncommutative common suffix
        for i, a in enumerate(args):
            if i == 0:
                n = a[1][:]
            else:
                n = common_suffix(n, a[1])
            if not n:
                # is there a power that can be extracted?
                if not args[0][1]:
                    break
                b, e = args[0][1][-1].as_base_exp()
                ok = False
                if e.is_Integer:
                    for t in args:
#.........这里部分代码省略.........

def rsolve_ratio(coeffs, f, n, **hints):
    """Given linear recurrence operator L of order 'k' with polynomial
       coefficients and inhomogeneous equation Ly = f, where 'f' is a
       polynomial, we seek for all rational solutions over field K of
       characteristic zero.

       This procedure accepts only polynomials, however if you are
       interested in solving recurrence with rational coefficients
       then use rsolve() which will pre-process the given equation
       and run this procedure with polynomial arguments.

       The algorithm performs two basic steps:

           (1) Compute polynomial v(n) which can be used as universal
               denominator of any rational solution of equation Ly = f.

           (2) Construct new linear difference equation by substitution
               y(n) = u(n)/v(n) and solve it for u(n) finding all its
               polynomial solutions. Return None if none were found.

       Algorithm implemented here is a revised version of the original
       Abramov's algorithm, developed in 1989. The new approach is much
       simpler to implement and has better overall efficiency. This
       method can be easily adapted to q-difference equations case.

       Besides finding rational solutions alone, this functions is
       an important part of Hyper algorithm were it is used to find
       particular solution of inhomogeneous part of a recurrence.

       For more information on the implemented algorithm refer to:

       [1] S. A. Abramov, Rational solutions of linear difference
           and q-difference equations with polynomial coefficients,
           in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York,
           1995, 285-289

    """
    f = sympify(f)

    if not f.is_polynomial(n):
        return None

    coeffs = map(sympify, coeffs)

    r = len(coeffs)-1

    A, B = coeffs[r], coeffs[0]
    A = A.subs(n, n-r).expand()

    h = Symbol('h', dummy=True)

    res = resultant(A, B.subs(n, n+h), n)

    if not res.is_polynomial(h):
        p, q = res.as_numer_denom()
        res = exquo(p, q, h)

    nni_roots = roots(res, h, filter='Z',
        predicate=lambda r: r >= 0).keys()

    if not nni_roots:
        return rsolve_poly(coeffs, f, n, **hints)
    else:
        C, numers = S.One, [S.Zero]*(r+1)

        for i in xrange(int(max(nni_roots)), -1, -1):
            d = gcd(A, B.subs(n, n+i), n)

            A = exquo(A, d, n)
            B = exquo(B, d.subs(n, n-i), n)

            C *= Mul(*[ d.subs(n, n-j) for j in xrange(0, i+1) ])

        denoms = [ C.subs(n, n+i) for i in range(0, r+1) ]

        for i in range(0, r+1):
            g = gcd(coeffs[i], denoms[i], n)

            numers[i] = exquo(coeffs[i], g, n)
            denoms[i] = exquo(denoms[i], g, n)

        for i in xrange(0, r+1):
            numers[i] *= Mul(*(denoms[:i] + denoms[i+1:]))

        result = rsolve_poly(numers, f * Mul(*denoms), n, **hints)

        if result is not None:
            if hints.get('symbols', False):
                return (simplify(result[0] / C), result[1])
            else:
                return simplify(result / C)
        else:
            return None

def rsolve_ratio(coeffs, f, n, **hints):
    """
    Given linear recurrence operator `\operatorname{L}` of order `k`
    with polynomial coefficients and inhomogeneous equation
    `\operatorname{L} y = f`, where `f` is a polynomial, we seek
    for all rational solutions over field `K` of characteristic zero.

    This procedure accepts only polynomials, however if you are
    interested in solving recurrence with rational coefficients
    then use ``rsolve`` which will pre-process the given equation
    and run this procedure with polynomial arguments.

    The algorithm performs two basic steps:

        (1) Compute polynomial `v(n)` which can be used as universal
            denominator of any rational solution of equation
            `\operatorname{L} y = f`.

        (2) Construct new linear difference equation by substitution
            `y(n) = u(n)/v(n)` and solve it for `u(n)` finding all its
            polynomial solutions. Return ``None`` if none were found.

    Algorithm implemented here is a revised version of the original
    Abramov's algorithm, developed in 1989. The new approach is much
    simpler to implement and has better overall efficiency. This
    method can be easily adapted to q-difference equations case.

    Besides finding rational solutions alone, this functions is
    an important part of Hyper algorithm were it is used to find
    particular solution of inhomogeneous part of a recurrence.

    Examples
    ========

    >>> from sympy.abc import x
    >>> from sympy.solvers.recurr import rsolve_ratio
    >>> rsolve_ratio([-2*x**3 + x**2 + 2*x - 1, 2*x**3 + x**2 - 6*x,
    ... - 2*x**3 - 11*x**2 - 18*x - 9, 2*x**3 + 13*x**2 + 22*x + 8], 0, x)
    C2*(2*x - 3)/(2*(x**2 - 1))

    References
    ==========

    .. [1] S. A. Abramov, Rational solutions of linear difference
           and q-difference equations with polynomial coefficients,
           in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York,
           1995, 285-289

    See Also
    ========

    rsolve_hyper
    """
    f = sympify(f)

    if not f.is_polynomial(n):
        return None

    coeffs = map(sympify, coeffs)

    r = len(coeffs) - 1

    A, B = coeffs[r], coeffs[0]
    A = A.subs(n, n - r).expand()

    h = Dummy('h')

    res = resultant(A, B.subs(n, n + h), n)

    if not res.is_polynomial(h):
        p, q = res.as_numer_denom()
        res = quo(p, q, h)

    nni_roots = roots(res, h, filter='Z',
        predicate=lambda r: r >= 0).keys()

    if not nni_roots:
        return rsolve_poly(coeffs, f, n, **hints)
    else:
        C, numers = S.One, [S.Zero]*(r + 1)

        for i in xrange(int(max(nni_roots)), -1, -1):
            d = gcd(A, B.subs(n, n + i), n)

            A = quo(A, d, n)
            B = quo(B, d.subs(n, n - i), n)

            C *= Mul(*[ d.subs(n, n - j) for j in xrange(0, i + 1) ])

        denoms = [ C.subs(n, n + i) for i in range(0, r + 1) ]

        for i in range(0, r + 1):
            g = gcd(coeffs[i], denoms[i], n)

            numers[i] = quo(coeffs[i], g, n)
            denoms[i] = quo(denoms[i], g, n)

        for i in xrange(0, r + 1):
            numers[i] *= Mul(*(denoms[:i] + denoms[i + 1:]))

#.........这里部分代码省略.........

 def nc_gcd(aa, bb):
     a, b = [i.as_coeff_Mul() for i in [aa, bb]]
     c = gcd(a[0], b[0]).as_numer_denom()[0]
     g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0]))
     return _keep_coeff(c, g)

        def gcd(f, g):
            from sympy.polys import gcd

            return f.__class__(gcd(f.ex, f.__class__(g).ex))

def normal(f, g, n=None):
    """Given relatively prime univariate polynomials 'f' and 'g',
       rewrite their quotient to a normal form defined as follows:

                       f(n)       A(n) C(n+1)
                       ----  =  Z -----------
                       g(n)       B(n)  C(n)

       where Z is arbitrary constant and A, B, C are monic
       polynomials in 'n' with following properties:

           (1) gcd(A(n), B(n+h)) = 1 for all 'h' in N
           (2) gcd(B(n), C(n+1)) = 1
           (3) gcd(A(n), C(n)) = 1

       This normal form, or rational factorization in other words,
       is crucial step in Gosper's algorithm and in difference
       equations solving. It can be also used to decide if two
       hypergeometric are similar or not.

       This procedure will return a tuple containing elements
       of this factorization in the form (Z*A, B, C). For example:

       >>> from sympy import Symbol, normal
       >>> n = Symbol('n', integer=True)

       >>> normal(4*n+5, 2*(4*n+1)*(2*n+3), n)
       (1/4, 3/2 + n, 1/4 + n)

    """
    f, g = map(sympify, (f, g))

    p = f.as_poly(n, field=True)
    q = g.as_poly(n, field=True)

    a, p = p.LC(), p.monic()
    b, q = q.LC(), q.monic()

    A = p.as_basic()
    B = q.as_basic()

    C, Z = S.One, a / b

    h = Dummy('h')

    res = resultant(A, B.subs(n, n+h), n)

    nni_roots = roots(res, h, filter='Z',
        predicate=lambda r: r >= 0).keys()

    if not nni_roots:
        return (f, g, S.One)
    else:
        for i in sorted(nni_roots):
            d = gcd(A, B.subs(n, n+i), n)

            A = quo(A, d, n)
            B = quo(B, d.subs(n, n-i), n)

            C *= Mul(*[ d.subs(n, n-j) for j in xrange(1, i+1) ])

        return (Z*A, B, C)