コード例 #1
0
def new_messege_view(request):
    if request.method == 'POST':
        U = request.user
        server1 = Server
        [K, Lp, LAM1, LAM2, GAMA1,
         GAMA2] = server1.get_security_parameters(server1)
        [N, A, A0, Y, G] = server1.get_public_key(server1)
        Ai = int(U.Ai)
        Ei = int(U.Ei)
        Xi = int(U.Xi)
        M = request.POST['title'] + request.POST['text']

        W = randrange(0, 2**(2 * Lp))

        T1 = (Ai * pow(Y, W, N)) % N
        T2 = pow(G, W, N)
        T3 = pow(T2, Ei, N)

        R1 = getrandbits(GAMA2 + 128 * K)
        R2 = getrandbits(LAM1 + 128 * K)
        R3 = getrandbits(GAMA1 + 2 * Lp + 128 * K + 1)

        INV_A = igcdex(A, N)[0] % N
        INV_Y = igcdex(Y, N)[0] % N
        INV_G = igcdex(G, N)[0] % N

        D1 = (pow(T1, R1, N) * (pow(INV_A, R2, N) * pow(INV_Y, R3, N))) % N
        D2 = (pow(T2, R1, N) * pow(INV_G, R3, N)) % N
        D3 = pow(T2, R1, N)

        hashobj = str(G).encode('utf-8') + str(Y).encode('utf-8') + str(
            A0).encode('utf-8') + str(A).encode('utf-8') + str(T1).encode(
                'utf-8') + str(T2).encode('utf-8') + str(T3).encode(
                    'utf-8') + str(D1).encode('utf-8') + str(D2).encode(
                        'utf-8') + str(D3).encode('utf-8') + M.encode('utf-8')
        C = zlib.crc32(hashobj)

        S1 = R1 - C * (Ei - 2**GAMA1)
        S2 = R2 - C * (Xi - 2**LAM1)
        S3 = R3 - C * Ei * W

        inst = Messeges.objects.create()
        inst.title = request.POST['title']
        inst.text = request.POST['text']
        inst.C = str(C)
        inst.S1 = str(S1)
        inst.S2 = str(S2)
        inst.S3 = str(S3)
        inst.T1 = str(T1)
        inst.T2 = str(T2)
        inst.T3 = str(T3)

        inst.save()
        messages.success(request,
                         ' Messege Signed and Posted successfully !',
                         extra_tags=False)
        return HttpResponseRedirect('/newmessege/')
    else:
        form = NewMessegeform()
        return render(request, 'Groupsign/newmessege.html', {'form': form})
コード例 #2
0
def home(request):
    if request.method == 'POST':
        server1 = Server
        [K, Lp, LAM1, LAM2, GAMA1,
         GAMA2] = server1.get_security_parameters(server1)
        [N, A, A0, Y, G] = server1.get_public_key(server1)
        M_verify = request.POST['title'] + request.POST['text']
        C = int(request.POST['C'])
        S1 = int(request.POST['S1'])
        S2 = int(request.POST['S2'])
        S3 = int(request.POST['S3'])
        T1 = int(request.POST['T1'])
        T2 = int(request.POST['T2'])
        T3 = int(request.POST['T3'])

        T1_INV = igcdex(T1, N)[0] % N
        T2_INV = igcdex(T2, N)[0] % N

        INV_A = igcdex(A, N)[0] % N
        INV_Y = igcdex(Y, N)[0] % N
        INV_G = igcdex(G, N)[0] % N

        Numerator_part = (pow(A0, C, N) * pow(T1, S1, N) *
                          pow(T1_INV, (C * 2**GAMA1), N)) % N
        Denomirator_part = (pow(INV_A, S2, N) * pow(A, C * 2**LAM1, N) *
                            pow(INV_Y, S3, N)) % N
        D1_Cal = (Numerator_part * Denomirator_part) % N

        Numerator_part1 = (pow(T2, S1, N) * pow(T2_INV, C * 2**GAMA1, N)) % N
        Denomirator_part1 = pow(INV_G, S3, N)
        D2_Cal = (Numerator_part1 * Denomirator_part1) % N

        D3_Cal = (pow(T2, S1, N) * pow(T2_INV, C * 2**GAMA1, N) *
                  pow(T3, C, N)) % N

        hashobj2 = str(G).encode('utf-8') + str(Y).encode('utf-8') + str(
            A0).encode('utf-8') + str(A).encode('utf-8') + str(T1).encode(
                'utf-8') + str(T2).encode('utf-8') + str(T3).encode(
                    'utf-8') + str(D1_Cal).encode('utf-8') + str(
                        D2_Cal).encode('utf-8') + str(D3_Cal).encode(
                            'utf-8') + M_verify.encode('utf-8')
        C_DASH = zlib.crc32(hashobj2)
        is_revoked = False
        revoke = Revocation_list.objects.all()
        if revoke is not None:
            for list in revoke:
                if (pow(T2, int(list.Ei), N) == T3):
                    is_revoked = True

        if C == C_DASH and is_revoked is False:
            messages.success(request, ' Messege Is Valid !', extra_tags=False)
            return HttpResponseRedirect('/home/')
        else:
            messages.error(request, ' Messege Is Not Valid !', extra_tags=True)
            return HttpResponseRedirect('/home/')
    else:
        messeges = Messeges.objects.order_by('-time')
        return render(request, 'Groupsign/home.html', {'messeges': messeges})
コード例 #3
0
def _nthroot_mod1(s, q, p, all_roots):
    """
    Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1``

    References
    ==========

    .. [1] A. M. Johnston "A Generalized qth Root Algorithm"

    """
    g = primitive_root(p)
    if not isprime(q):
        r = _nthroot_mod2(s, q, p)
    else:
        f = p - 1
        assert (p - 1) % q == 0
        # determine k
        k = 0
        while f % q == 0:
            k += 1
            f = f // q
        # find z, x, r1
        f1 = igcdex(-f, q)[0] % q
        z = f * f1
        x = (1 + z) // q
        w = pow(g, z, p)
        r1 = pow(s, x, p)
        s1 = pow(s, f, p)
        y = pow(g, f, p)
        h = pow(g, f * q, p)
        # find t discrete log of s1 base h, h**x = s1 mod p
        # used a naive implementation
        # TODO implement using Ref [1]
        pr = 1
        for t in range(p):
            if pr == s1:
                break
            pr = pr * h % p

        g2 = pow(g, z * t, p)
        g3 = igcdex(g2, p)[0]
        r = r1 * g3 % p
        #assert pow(r, q, p) == s
    res = [r]
    h = pow(g, (p - 1) // q, p)
    #assert pow(h, q, p) == 1
    hx = r
    for i in range(q - 1):
        hx = (hx * h) % p
        res.append(hx)
    if all_roots:
        res.sort()
        return res
    return min(res)
コード例 #4
0
def _nthroot_mod1(s, q, p, all_roots):
    """
    Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1``

    References
    ==========

    .. [1] A. M. Johnston "A Generalized qth Root Algorithm"

    """
    g = primitive_root(p)
    if not isprime(q):
        r = _nthroot_mod2(s, q, p)
    else:
        f = p - 1
        assert (p - 1) % q == 0
        # determine k
        k = 0
        while f % q == 0:
            k += 1
            f = f // q
        # find z, x, r1
        f1 = igcdex(-f, q)[0] % q
        z = f*f1
        x = (1 + z) // q
        w = pow(g, z, p)
        r1 = pow(s, x, p)
        s1 = pow(s, f, p)
        y = pow(g, f, p)
        h = pow(g, f*q, p)
        # find t discrete log of s1 base h, h**x = s1 mod p
        # used a naive implementation
        # TODO implement using Ref [1]
        pr = 1
        for t in range(p):
            if pr == s1:
                break
            pr = pr*h % p

        g2 = pow(g, z*t, p)
        g3 = igcdex(g2, p)[0]
        r = r1*g3 % p
        #assert pow(r, q, p) == s
    res = [r]
    h = pow(g, (p - 1) // q, p)
    #assert pow(h, q, p) == 1
    hx = r
    for i in range(q - 1):
        hx = (hx*h) % p
        res.append(hx)
    if all_roots:
        res.sort()
        return res
    return min(res)
コード例 #5
0
ファイル: trigonometric.py プロジェクト: amitjamadagni/sympy
 def migcdex(x):
     # recursive calcuation of gcd and linear combination
     # for a sequence of integers.
     # Given  (x1, x2, x3)
     # Returns (y1, y1, y3, g)
     # such that g is the gcd and x1*y1+x2*y2+x3*y3 - g = 0
     # Note, that this is only one such linear combination.
     if len(x) == 1:
         return (1, x[0])
     if len(x) == 2:
         return igcdex(x[0], x[-1])
     g = migcdex(x[1:])
     u, v, h = igcdex(x[0], g[-1])
     return tuple([u] + [v * i for i in g[0:-1]] + [h])
コード例 #6
0
ファイル: trigonometric.py プロジェクト: bhlegm/sympy
 def migcdex(x):
     # recursive calcuation of gcd and linear combination
     # for a sequence of integers.
     # Given  (x1, x2, x3)
     # Returns (y1, y1, y3, g)
     # such that g is the gcd and x1*y1+x2*y2+x3*y3 - g = 0
     # Note, that this is only one such linear combination.
     if len(x) == 1:
         return (1, x[0])
     if len(x) == 2:
         return igcdex(x[0], x[-1])
     g = migcdex(x[1:])
     u, v, h = igcdex(x[0], g[-1])
     return tuple([u] + [v*i for i in g[0:-1] ] + [h])
コード例 #7
0
ファイル: modular.py プロジェクト: drgnrave/sympy
def crt(m, v, symmetric=False):
    """Chinese Remainder Theorem.

       The integers in m are assumed to be pairwise coprime.  The output
       is then an integer f, such that f = v_i mod m_i for each pair out
       of v and m.

    """
    mm = 1

    for m_i in m:
        mm *= m_i

    result = 0

    for m_i, v_i in zip(m, v):
        e = mm // m_i
        s, t, g = igcdex(e, m_i)
        c = v_i*s % m_i
        result += c*e

    result %= mm

    if symmetric:
        if result <= mm//2:
            return result
        else:
            return result - mm
    else:
        return result
コード例 #8
0
def div_mod(n, m, p):
    """
    Finds a nonnegative integer x < p such that (m * x) % p == n.
    """
    a, b, c = igcdex(m, p)
    assert c == 1
    return (n * a) % p
コード例 #9
0
def div_mod(n: int, m: int, p: int) -> int:
    """
    Finds a nonnegative integer 0 <= x < p such that (m * x) % p == n
    """
    a, b, c = igcdex(m, p)
    assert c == 1
    return (n * a) % p
コード例 #10
0
def crt(m, v, symmetric=False):
    """Chinese Remainder Theorem.

       The integers in m are assumed to be pairwise coprime.  The output
       is then an integer f, such that f = v_i mod m_i for each pair out
       of v and m.

    """
    mm = 1

    for m_i in m:
        mm *= m_i

    result = 0

    for m_i, v_i in zip(m, v):
        e = mm // m_i
        s, t, g = igcdex(e, m_i)
        c = v_i * s % m_i
        result += c * e

    result %= mm

    if symmetric:
        if result <= mm // 2:
            return result
        else:
            return result - mm
    else:
        return result
コード例 #11
0
def decipher_elgamal(ct, prk):
    r"""
    Decrypt message with private key

    `ct = (c_{1}, c_{2})`

    `prk = (p, r, d)`

    According to extended Eucliden theorem,
    `u c_{1}^{d} + p n = 1`

    `u \equiv 1/{{c_{1}}^d} \pmod p`

    `u c_{2} \equiv \frac{1}{c_{1}^d} c_{2} \equiv \frac{1}{r^{ad}} c_{2} \pmod p`

    `\frac{1}{r^{ad}} m e^a \equiv \frac{1}{r^{ad}} m {r^{d a}} \equiv m \pmod p`

    Examples
    ========

    >>> from sympy.crypto.crypto import decipher_elgamal
    >>> decipher_elgamal((835, 271), (1031, 14, 636))
    100

    """
    u = igcdex(ct[0]**prk[2], prk[0])[0]
    return u * ct[1] % prk[0]
コード例 #12
0
ファイル: crypto.py プロジェクト: Upabjojr/sympy
def decipher_elgamal(ct, prk):
    r"""
    Decrypt message with private key

    `ct = (c_{1}, c_{2})`

    `prk = (p, r, d)`

    According to extended Eucliden theorem,
    `u c_{1}^{d} + p n = 1`

    `u \equiv 1/{{c_{1}}^d} \pmod p`

    `u c_{2} \equiv \frac{1}{c_{1}^d} c_{2} \equiv \frac{1}{r^{ad}} c_{2} \pmod p`

    `\frac{1}{r^{ad}} m e^a \equiv \frac{1}{r^{ad}} m {r^{d a}} \equiv m \pmod p`

    Examples
    ========

    >>> from sympy.crypto.crypto import decipher_elgamal
    >>> decipher_elgamal((835, 271), (1031, 14, 636))
    100

    """
    u = igcdex(ct[0] ** prk[2], prk[0])[0]
    return u * ct[1] % prk[0]
コード例 #13
0
def open_view(request):
    if request.user.is_opening_manager:
        if request.method == 'POST':
            server1 = Server
            [N, A, A0, Y, G] = server1.public_key
            X = Open_Manager.X
            T1 = int(request.POST['T1'])
            T2 = int(request.POST['T2'])
            T2_INV = igcdex(T2, N)[0] % N
            Ai_Check = (T1 * pow(T2_INV, X, N)) % N
            try:
                openuser = User.objects.get(Ai=Ai_Check)
                if openuser is None:
                    messages.error(
                        request,
                        ' No User Is Found ! Please Check Validity Of Message.',
                        extra_tags=True)
                    return HttpResponseRedirect('/open/')
                else:
                    return render(request, 'Groupsign/Ai_show.html', {
                        'Ai_Check': Ai_Check,
                        'openuser': openuser
                    })
            except User.DoesNotExist:
                messages.error(request, ' No User Is Found !', extra_tags=True)
                return HttpResponseRedirect('/open/')
        else:
            form = openform()
            return render(request, 'Groupsign/openform.html', {'form': form})
    else:
        messages.error(request,
                       ' You Do not Have this Privilege !',
                       extra_tags=True)
        return HttpResponseRedirect('/home/')
コード例 #14
0
ファイル: residue_ntheory.py プロジェクト: hridog00/Proyecto
def _nthroot_mod1(s, q, p, all_roots):
    """
    Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1``

    References
    ==========

    .. [1] A. M. Johnston "A Generalized qth Root Algorithm"

    """
    g = primitive_root(p)
    if not isprime(q):
        r = _nthroot_mod2(s, q, p)
    else:
        f = p - 1
        assert (p - 1) % q == 0
        # determine k
        k = 0
        while f % q == 0:
            k += 1
            f = f // q
        # find z, x, r1
        f1 = igcdex(-f, q)[0] % q
        z = f * f1
        x = (1 + z) // q
        w = pow(g, z, p)
        r1 = pow(s, x, p)
        s1 = pow(s, f, p)
        y = pow(g, f, p)
        h = pow(g, f * q, p)
        t = discrete_log(p, s1, h)
        g2 = pow(g, z * t, p)
        g3 = igcdex(g2, p)[0]
        r = r1 * g3 % p
        #assert pow(r, q, p) == s
    res = [r]
    h = pow(g, (p - 1) // q, p)
    #assert pow(h, q, p) == 1
    hx = r
    for i in range(q - 1):
        hx = (hx * h) % p
        res.append(hx)
    if all_roots:
        res.sort()
        return res
    return min(res)
コード例 #15
0
ファイル: galoispolys.py プロジェクト: jcockayne/sympy-rkern
def zp_inv(a, p):
    """Compute multiplicative inverse over GF(p). """
    s, t, g = igcdex(a, p)

    if g == 1:
        return s % p
    else:
        raise ZeroDivisionError("modular division")
コード例 #16
0
ファイル: galoispolys.py プロジェクト: smichr/sympy-live
def zp_inv(a, p):
    """Compute multiplicative inverse over GF(p). """
    s, t, g = igcdex(a, p)

    if g == 1:
        return s % p
    else:
        raise ZeroDivisionError("modular division")
コード例 #17
0
def _nthroot_mod1(s, q, p, all_roots):
    """
    Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1``

    References
    ==========

    .. [1] A. M. Johnston "A Generalized qth Root Algorithm"

    """
    g = primitive_root(p)
    if not isprime(q):
        r = _nthroot_mod2(s, q, p)
    else:
        f = p - 1
        assert (p - 1) % q == 0
        # determine k
        k = 0
        while f % q == 0:
            k += 1
            f = f // q
        # find z, x, r1
        f1 = igcdex(-f, q)[0] % q
        z = f*f1
        x = (1 + z) // q
        w = pow(g, z, p)
        r1 = pow(s, x, p)
        s1 = pow(s, f, p)
        y = pow(g, f, p)
        h = pow(g, f*q, p)
        t = discrete_log(p, s1, h)
        g2 = pow(g, z*t, p)
        g3 = igcdex(g2, p)[0]
        r = r1*g3 % p
        #assert pow(r, q, p) == s
    res = [r]
    h = pow(g, (p - 1) // q, p)
    #assert pow(h, q, p) == 1
    hx = r
    for i in range(q - 1):
        hx = (hx*h) % p
        res.append(hx)
    if all_roots:
        res.sort()
        return res
    return min(res)
コード例 #18
0
ファイル: hnf.py プロジェクト: swist/M4R
def row_one_gcd(A, i, j, l):
    assert i < A.rows
    assert j < l < A.cols
    if A[i, j] != 0 or A[i, l] != 0:
        u, v, d = igcdex(A[i, j], A[i, l])
        reduced = Matrix([[u, -A[i, l] / d], [v, A[i, j] / d]])
        s = A.extract(range(A.rows), [j, l]) * (reduced)
        A[:, j] = s[:, 0]
        A[:, l] = s[:, 1]
    return A
コード例 #19
0
def zzx_hensel_lift(p, f, f_list, l):
    """Multifactor Hensel lifting.

       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

       For more details on the implemented algorithm refer to:

       [1] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
           First Edition, Cambridge University Press, 1999, pp. 424

    """
    r = len(f_list)
    lc = zzx_LC(f)

    if r == 1:
        F = zzx_mul_term(f, igcdex(lc, p**l)[0], 0)
        return [zzx_trunc(F, p**l)]

    m = p
    k = int(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)

    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)

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

    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 = zzx_hensel_step(m, f, g, h, s, t), m**2

    return zzx_hensel_lift(p, g, f_list[:k], l) \
         + zzx_hensel_lift(p, h, f_list[k:], l)
コード例 #20
0
ファイル: integerpolys.py プロジェクト: jcockayne/sympy-rkern
def zzx_hensel_lift(p, f, f_list, l):
    """Multifactor Hensel lifting.

       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

       For more details on the implemented algorithm refer to:

       [1] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
           First Edition, Cambridge University Press, 1999, pp. 424

    """
    r = len(f_list)
    lc = zzx_LC(f)

    if r == 1:
        F = zzx_mul_term(f, igcdex(lc, p**l)[0], 0)
        return [ zzx_trunc(F, p**l) ]

    m = p
    k = int(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)

    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)

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

    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 = zzx_hensel_step(m, f, g, h, s, t), m**2

    return zzx_hensel_lift(p, g, f_list[:k], l) \
         + zzx_hensel_lift(p, h, f_list[k:], l)
コード例 #21
0
 def get_member_cert(self, Num):
     if legendre_symbol(Num, self.P) and legendre_symbol(Num, self.Q) == 1:
         Ei = randprime((2**self.GAMA1 - 2**self.GAMA2),
                        (2**self.GAMA1 + 2**self.GAMA2))
         INVE_Ei = igcdex(Ei,
                          ((self.P - 1) *
                           (self.Q - 1)))[0] % ((self.P - 1) * (self.Q - 1))
         Ai = pow((Num * self.A0), INVE_Ei, self.N)
         return [Ai, Ei]
     else:
         return False, False
コード例 #22
0
def crt1(m):
    """First part of chines remainder theorem, for multiple application. """
    mm, e, s = 1, [], []

    for m_i in m:
        mm *= m_i

    for m_i in m:
        e.append(mm // m_i)
        s.append(igcdex(e[-1], m_i)[0])

    return mm, e, s
コード例 #23
0
ファイル: modular.py プロジェクト: drgnrave/sympy
def crt1(m):
    """First part of Chinese Remainder Theorem, for multiple application. """
    mm, e, s = 1, [], []

    for m_i in m:
        mm *= m_i

    for m_i in m:
        e.append(mm // m_i)
        s.append(igcdex(e[-1], m_i)[0])

    return mm, e, s
 def A(s, a_val, b_val, a, b, d, x, y):
     # In this method you should generate an answer based on the particular
     # parameters and variable names generated for this QA instance.
     # It is ideal to rely on sympy and other libraries to do the solving
     # for you.
     seqg, choiceg = s.seqg, s.choiceg
     x_val, y_val, d_val = igcdex(a_val, b_val)
     a1 = seqg(sym.Eq(d, d_val), sym.Eq(x, x_val), sym.Eq(y, y_val))
     a2 = seqg("The greatest common divisor of", a, "and", b, "is",
               sym.Eq(d, d_val), "with coefficients", sym.Eq(x, x_val),
               sym.Eq(y, y_val))
     # try to generate at least 2 or 3 different wordings of the answer.
     return choiceg(a1, a2)
コード例 #25
0
ファイル: modular.py プロジェクト: mokarted/sympy
def mod_inverse(a, m):
    """
    Return the number c such that, ( a * c ) % m == 1.
    This means that if a is multiplied by c
    then the remainder obtained on division with m is 1.
    The value of m need not be a prime.
    Return the value of c if some c exists.
    Otherwise, raise an exception stating that the
    modular inverse does not exist.

    Examples
    ========

    >>> from sympy.ntheory.modular import mod_inverse
    >>> mod_inverse(3,11)
    4

    Suppose we wish to find multiplicative inverse x of
    3 modulo 11. This is the same as finding x such
    that 3 * x = 1 (mod 11). One value of x that satisfies
    this congruence is 4. Because 3 * 4 = 12 and 12 = 1 mod(11).

    Similarly

    >>> mod_inverse(5,11)
    9

    Exception Cases

    >>> mod_inverse(2,4)
    Traceback (most recent call last):
    ...
    ValueError: modular inverse does not exist

    References
    ==========
    - https://en.wikipedia.org/wiki/Modular_multiplicative_inverse
    - https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
    """
    a, m = as_int(a), as_int(m)
    x, y, g = igcdex(a, m)
    if g != 1:
        raise ValueError('modular inverse does not exist')
    else:
        return x % m
コード例 #26
0
ファイル: modular.py プロジェクト: Sam-Tygier/sympy
def mod_inverse(a, m):
    """
    Return the number c such that, ( a * c ) % m == 1.
    This means that if a is multiplied by c
    then the remainder obtained on division with m is 1.
    The value of m need not be a prime.
    Return the value of c if some c exists.
    Otherwise, raise an exception stating that the
    modular inverse does not exist.

    Examples
    ========

    >>> from sympy.ntheory.modular import mod_inverse
    >>> mod_inverse(3,11)
    4

    Suppose we wish to find multiplicative inverse x of
    3 modulo 11. This is the same as finding x such
    that 3 * x = 1 (mod 11). One value of x that satisfies
    this congruence is 4. Because 3 * 4 = 12 and 12 = 1 mod(11).

    Similarly

    >>> mod_inverse(5,11)
    9

    Exception Cases

    >>> mod_inverse(2,4)
    Traceback (most recent call last):
    ...
    ValueError: modular inverse does not exist

    References
    ==========
    - https://en.wikipedia.org/wiki/Modular_multiplicative_inverse
    - https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
    """
    a, m = as_int(a), as_int(m)
    x, y, g = igcdex(a, m)
    if g != 1:
        raise ValueError('modular inverse does not exist')
    else:
        return x % m
コード例 #27
0
ファイル: modular.py プロジェクト: 101man/sympy
    def combine(c1, c2):
        """Return the tuple (a, m) which satisfies the requirement
        that n = a + i*m satisfy n = a1 + j*m1 and n = a2 = k*m2.

        Reference:
           http://en.wikipedia.org/wiki/Method_of_successive_substitution
        """
        from sympy.core.numbers import igcdex
        a1, m1 = c1
        a2, m2 = c2
        a, b, c = m1, a2 - a1, m2
        g = reduce(igcd, [a, b, c])
        a, b, c = [i//g for i in [a, b, c]]
        if a != 1:
            inv_a, _, g = igcdex(a, c)
            if g != 1:
                return None
            b *= inv_a
        a, m = a1 + m1*b, m1*c
        return a, m
コード例 #28
0
ファイル: modular.py プロジェクト: fhelmli/homeNOWG2
    def combine(c1, c2):
        """Return the tuple (a, m) which satisfies the requirement
        that n = a + i*m satisfy n = a1 + j*m1 and n = a2 = k*m2.

        References
        ==========

        - http://en.wikipedia.org/wiki/Method_of_successive_substitution
        """
        a1, m1 = c1
        a2, m2 = c2
        a, b, c = m1, a2 - a1, m2
        g = reduce(igcd, [a, b, c])
        a, b, c = [i // g for i in [a, b, c]]
        if a != 1:
            inv_a, _, g = igcdex(a, c)
            if g != 1:
                return None
            b *= inv_a
        a, m = a1 + m1 * b, m1 * c
        return a, m
コード例 #29
0
ファイル: test_numbers.py プロジェクト: 101man/sympy
def test_igcdex():
    assert igcdex(2, 3) == (-1, 1, 1)
    assert igcdex(10, 12) == (-1, 1, 2)
    assert igcdex(100, 2004) == (-20, 1, 4)
コード例 #30
0
ファイル: residue_ntheory.py プロジェクト: vprusso/sympy
def nthroot_mod(a, n, p, all_roots=False):
    """
    find the solutions to ``x**n = a mod p``

    Parameters
    ==========

    a : integer
    n : positive integer
    p : positive integer
    all_roots : if False returns the smallest root, else the list of roots

    Examples
    ========

    >>> from sympy.ntheory.residue_ntheory import nthroot_mod
    >>> nthroot_mod(11, 4, 19)
    8
    >>> nthroot_mod(11, 4, 19, True)
    [8, 11]
    >>> nthroot_mod(68, 3, 109)
    23
    """
    from sympy.core.numbers import igcdex
    if n == 2:
        return sqrt_mod(a, p , all_roots)
    f = totient(p)
    # see Hackman "Elementary Number Theory" (2009), page 76
    if pow(a, f // igcd(f, n), p) != 1:
        return None
    if not isprime(p):
        raise NotImplementedError

    if (p - 1) % n == 0:
        return _nthroot_mod1(a, n, p, all_roots)
    # The roots of ``x**n - a = 0 (mod p)`` are roots of
    # ``gcd(x**n - a, x**(p - 1) - 1) = 0 (mod p)``
    pa = n
    pb = p - 1
    b = 1
    if pa < pb:
        a, pa, b, pb = b, pb, a, pa
    while pb:
        # x**pa - a = 0; x**pb - b = 0
        # x**pa - a = x**(q*pb + r) - a = (x**pb)**q * x**r - a =
        #             b**q * x**r - a; x**r - c = 0; c = b**-q * a mod p
        q, r = divmod(pa, pb)
        c = pow(b, q, p)
        c = igcdex(c, p)[0]
        c = (c * a) % p
        pa, pb = pb, r
        a, b = b, c
    if pa == 1:
        if all_roots:
            res = [a]
        else:
            res = a
    elif pa == 2:
        return sqrt_mod(a, p , all_roots)
    else:
        res = _nthroot_mod1(a, pa, p, all_roots)
    return res
コード例 #31
0
ファイル: residue_ntheory.py プロジェクト: vprusso/sympy
def _sqrt_mod_prime_power(a, p, k):
    """
    find the solutions to ``x**2 = a mod p**k`` when ``a % p != 0``

    Parameters
    ==========

    a : integer
    p : prime number
    k : positive integer

    References
    ==========

    [1] P. Hackman "Elementary Number Theory" (2009),  page 160
    [2] http://www.numbertheory.org/php/squareroot.html
    [3] [Gathen99]_

    Examples
    ========

    >>> from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
    >>> _sqrt_mod_prime_power(11, 43, 1)
    [21, 22]
    """
    from sympy.core.numbers import igcdex
    from sympy.polys.domains import ZZ

    pk = p**k
    a = a % pk

    if k == 1:
        if p == 2:
            return [ZZ(a)]
        if not is_quad_residue(a, p):
            return None

        if p % 4 == 3:
            res = pow(a, (p + 1) // 4, p)
        elif p % 8 == 5:
            sign = pow(a, (p - 1) // 4, p)
            if sign == 1:
                res = pow(a, (p + 3) // 8, p)
            else:
                b = pow(4*a, (p - 5) // 8, p)
                x =  (2*a*b) % p
                if pow(x, 2, p) == a:
                    res = x
        else:
            res = _sqrt_mod_tonelli_shanks(a, p)

        # ``_sqrt_mod_tonelli_shanks(a, p)`` is not deterministic;
        # sort to get always the same result
        return sorted([ZZ(res), ZZ(p - res)])

    if k > 1:
        f = factorint(a)
        # see Ref.[2]
        if p == 2:
            if a % 8 != 1:
                return None
            if k <= 3:
               s = set()
               for i in xrange(0, pk, 4):
                    s.add(1 + i)
                    s.add(-1 + i)
               return list(s)
            # according to Ref.[2] for k > 2 there are two solutions
            # (mod 2**k-1), that is four solutions (mod 2**k), which can be
            # obtained from the roots of x**2 = 0 (mod 8)
            rv = [ZZ(1), ZZ(3), ZZ(5), ZZ(7)]
            # hensel lift them to solutions of x**2 = 0 (mod 2**k)
            # if r**2 - a = 0 mod 2**nx but not mod 2**(nx+1)
            # then r + 2**(nx - 1) is a root mod 2**(nx+1)
            n = 3
            res = []
            for r in rv:
                nx = n
                while nx < k:
                    r1 = (r**2 - a) >> nx
                    if r1 % 2:
                        r = r + (1 << (nx - 1))
                    #assert (r**2 - a)% (1 << (nx + 1)) == 0
                    nx += 1
                if r not in res:
                    res.append(r)
                x = r + (1 << (k - 1))
                #assert (x**2 - a) % pk == 0
                if x < (1 << nx) and x not in res:
                    if (x**2 - a) % pk == 0:
                        res.append(x)
            return res
        rv = _sqrt_mod_prime_power(a, p, 1)
        if not rv:
            return None
        r = rv[0]
        fr = r**2 - a
        # hensel lifting with Newton iteration, see Ref.[3] chapter 9
        # with f(x) = x**2 - a; one has f'(a) != 0 (mod p) for p != 2
        n = 1
        px = p
        while 1:
            n1 = n
            n1 *= 2
            if n1 > k:
                break
            n = n1
            px = px**2
            frinv = igcdex(2*r, px)[0]
            r = (r - fr*frinv) % px
            fr = r**2 - a
        if n < k:
            px = p**k
            frinv = igcdex(2*r, px)[0]
            r = (r - fr*frinv) % px
        return [r, px - r]
コード例 #32
0
def test_igcdex():
    assert igcdex(2, 3) == (-1, 1, 1)
    assert igcdex(10, 12) == (-1, 1, 2)
    assert igcdex(100, 2004) == (-20, 1, 4)
コード例 #33
0
ファイル: residue_ntheory.py プロジェクト: xiangyazi24/sympy
def nthroot_mod(a, n, p, all_roots=False):
    """
    Find the solutions to ``x**n = a mod p``

    Parameters
    ==========

    a : integer
    n : positive integer
    p : positive integer
    all_roots : if False returns the smallest root, else the list of roots

    Examples
    ========

    >>> from sympy.ntheory.residue_ntheory import nthroot_mod
    >>> nthroot_mod(11, 4, 19)
    8
    >>> nthroot_mod(11, 4, 19, True)
    [8, 11]
    >>> nthroot_mod(68, 3, 109)
    23
    """
    from sympy.core.numbers import igcdex
    a, n, p = as_int(a), as_int(n), as_int(p)
    if n == 2:
        return sqrt_mod(a, p, all_roots)
    # see Hackman "Elementary Number Theory" (2009), page 76
    if not is_nthpow_residue(a, n, p):
        return None
    if primitive_root(p) is None:
        raise NotImplementedError(
            "Not Implemented for m without primitive root")

    if (p - 1) % n == 0:
        return _nthroot_mod1(a, n, p, all_roots)
    # The roots of ``x**n - a = 0 (mod p)`` are roots of
    # ``gcd(x**n - a, x**(p - 1) - 1) = 0 (mod p)``
    pa = n
    pb = p - 1
    b = 1
    if pa < pb:
        a, pa, b, pb = b, pb, a, pa
    while pb:
        # x**pa - a = 0; x**pb - b = 0
        # x**pa - a = x**(q*pb + r) - a = (x**pb)**q * x**r - a =
        #             b**q * x**r - a; x**r - c = 0; c = b**-q * a mod p
        q, r = divmod(pa, pb)
        c = pow(b, q, p)
        c = igcdex(c, p)[0]
        c = (c * a) % p
        pa, pb = pb, r
        a, b = b, c
    if pa == 1:
        if all_roots:
            res = [a]
        else:
            res = a
    elif pa == 2:
        return sqrt_mod(a, p, all_roots)
    else:
        res = _nthroot_mod1(a, pa, p, all_roots)
    return res
コード例 #34
0
ファイル: residue_ntheory.py プロジェクト: xiangyazi24/sympy
def _sqrt_mod_prime_power(a, p, k):
    """
    Find the solutions to ``x**2 = a mod p**k`` when ``a % p != 0``

    Parameters
    ==========

    a : integer
    p : prime number
    k : positive integer

    Examples
    ========

    >>> from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
    >>> _sqrt_mod_prime_power(11, 43, 1)
    [21, 22]

    References
    ==========

    .. [1] P. Hackman "Elementary Number Theory" (2009), page 160
    .. [2] http://www.numbertheory.org/php/squareroot.html
    .. [3] [Gathen99]_
    """
    from sympy.core.numbers import igcdex
    from sympy.polys.domains import ZZ

    pk = p**k
    a = a % pk

    if k == 1:
        if p == 2:
            return [ZZ(a)]
        if not is_quad_residue(a, p):
            return None

        if p % 4 == 3:
            res = pow(a, (p + 1) // 4, p)
        elif p % 8 == 5:
            sign = pow(a, (p - 1) // 4, p)
            if sign == 1:
                res = pow(a, (p + 3) // 8, p)
            else:
                b = pow(4 * a, (p - 5) // 8, p)
                x = (2 * a * b) % p
                if pow(x, 2, p) == a:
                    res = x
        else:
            res = _sqrt_mod_tonelli_shanks(a, p)

        # ``_sqrt_mod_tonelli_shanks(a, p)`` is not deterministic;
        # sort to get always the same result
        return sorted([ZZ(res), ZZ(p - res)])

    if k > 1:
        # see Ref.[2]
        if p == 2:
            if a % 8 != 1:
                return None
            if k <= 3:
                s = set()
                for i in range(0, pk, 4):
                    s.add(1 + i)
                    s.add(-1 + i)
                return list(s)
            # according to Ref.[2] for k > 2 there are two solutions
            # (mod 2**k-1), that is four solutions (mod 2**k), which can be
            # obtained from the roots of x**2 = 0 (mod 8)
            rv = [ZZ(1), ZZ(3), ZZ(5), ZZ(7)]
            # hensel lift them to solutions of x**2 = 0 (mod 2**k)
            # if r**2 - a = 0 mod 2**nx but not mod 2**(nx+1)
            # then r + 2**(nx - 1) is a root mod 2**(nx+1)
            n = 3
            res = []
            for r in rv:
                nx = n
                while nx < k:
                    r1 = (r**2 - a) >> nx
                    if r1 % 2:
                        r = r + (1 << (nx - 1))
                    #assert (r**2 - a)% (1 << (nx + 1)) == 0
                    nx += 1
                if r not in res:
                    res.append(r)
                x = r + (1 << (k - 1))
                #assert (x**2 - a) % pk == 0
                if x < (1 << nx) and x not in res:
                    if (x**2 - a) % pk == 0:
                        res.append(x)
            return res
        rv = _sqrt_mod_prime_power(a, p, 1)
        if not rv:
            return None
        r = rv[0]
        fr = r**2 - a
        # hensel lifting with Newton iteration, see Ref.[3] chapter 9
        # with f(x) = x**2 - a; one has f'(a) != 0 (mod p) for p != 2
        n = 1
        px = p
        while 1:
            n1 = n
            n1 *= 2
            if n1 > k:
                break
            n = n1
            px = px**2
            frinv = igcdex(2 * r, px)[0]
            r = (r - fr * frinv) % px
            fr = r**2 - a
        if n < k:
            px = p**k
            frinv = igcdex(2 * r, px)[0]
            r = (r - fr * frinv) % px
        return [r, px - r]
コード例 #35
0
def ejercicio_7_a(a):
    return igcdex(a, 101)[0] % 101