lwe.py

# -*- coding: utf-8 -*-
"""
(Ring-)LWE oracle generators

The Learning with Errors problem (LWE) is solving linear systems of equations
where the right hand side has been disturbed 'slightly' where 'slightly' is made
precise by a noise distribution - typically a discrete Gaussian
distribution. See [Reg09]_ for details.

The Ring Learning with Errors problem (LWE) is solving a set of univariate
polynomial equations - typically in a cyclotomic field - where the right hand
side was disturbed 'slightly'. See [LPR10]_ for details.

This module implements generators of LWE samples where parameters are chosen
following proposals in the cryptographic literature.

EXAMPLES:

We get 30 samples from an LWE oracle parameterised by security parameter
``n=20`` and where the modulus and the standard deviation of the noise are
chosen as in [Reg09]_::

    sage: from sage.crypto.lwe import samples
    sage: samples(30, 20, 'Regev')
    [((360, 264, 123, 368, 398, 392, 41, 84, 25, 389, 311, 68, 322, 41, 161, 372, 222, 153, 243, 381), 126),
    ...
    ((138, 198, 204, 235, 339, 168, 269, 276, 392, 243, 86, 18, 378, 20, 369, 141, 108, 151, 336, 141), 102)]


We may also pass classes to the samples function, which is useful for users
implementing their own oracles::

    sage: from sage.crypto.lwe import samples, LindnerPeikert
    sage: samples(30, 20, LindnerPeikert)
    [((350, 835, 2023, 1785, 1958, 1818, 1130, 1285, 1331, 284, 2048, 441, 1581, 1406, 1185, 1724, 1397, 258, 994, 1056), 1902),
    ...
    ((1918, 1823, 1598, 18, 588, 1093, 744, 1934, 689, 1327, 1632, 1867, 228, 378, 798, 511, 274, 1001, 1709, 154), 184)]


Finally, :func:`samples` also accepts instances of classes::

    sage: from sage.crypto.lwe import LindnerPeikert
    sage: lwe = LindnerPeikert(20)
    sage: samples(30, 20, lwe)
    [((1817, 1322, 818, 1232, 354, 639, 1770, 754, 1366, 1731, 649, 162, 483, 1741, 1942, 1232, 1424, 1034, 50, 448), 1316),
    ...
    ((2021, 829, 572, 1698, 1025, 170, 598, 1193, 1268, 607, 1502, 1984, 1655, 206, 958, 334, 1213, 1413, 827, 1423), 546)]

Note that Ring-LWE samples are returned as vectors::

    sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection, RingLWE
    sage: D = DiscreteGaussianPolynomialSamplerRejection(euler_phi(16), 5)
    sage: ringlwe = RingLWE(16, 257, D, secret_dist='uniform')
    sage: samples(30, euler_phi(16), ringlwe)
    [((158, 49, 174, 179, 109, 92, 234, 41), (200, 159, 131, 197, 241, 172, 1, 107)),
    ...
    ((80, 227, 249, 205, 149, 92, 46, 68), (69, 256, 29, 219, 218, 34, 182, 178))]

One technical issue when working with these generators is that by default they
return vectors and scalars over/in rings modulo some `q`. These are represented
as elements in `(0,q-1)` by Sage. However, it usually is more natural to think
of these entries as integers in `(-q//2,q//2)`. To allow for this, this module
provides the option to balance the representation. In this case vectors and
scalars over/in the integers are returned::

    sage: from sage.crypto.lwe import samples
    sage: samples(30, 20, 'Regev', balanced=True)
    [((-38, 59, -33, -80, 165, -55, -46, -49, -113, 135, -32, 185, -80, -184, 127, 153, 162, -31, 115, 178), 14),
    ...
    ((-165, -187, -87, 188, 160, -118, -7, 107, -77, -107, -109, 77, 63, -66, -55, -75, -12, 90, 58, -185), 6)]

AUTHORS:

- Martin Albrecht
- Robert Fitzpatrick
- Daniel Cabracas
- Florian Göpfert
- Michael Schneider

REFERENCES:

.. [Reg09] Oded Regev. On Lattices, Learning with Errors, Random Linear Codes,
   and Cryptography. in Journal of the ACM 56(6). ACM 2009,
   http://dx.doi.org/10.1145/1060590.1060603

.. [LP11] Richard Lindner and Chris Peikert. Better key sizes (and attacks) for
   LWE-based encryption. in Proceeding of the 11th international conference on
   Topics in cryptology: CT-RSA 2011. Springer 2011,
   http://dx.doi.org/10.1007/978-3-642-19074-2_21

.. [LPR10] Vadim Lyubashevsky, Chris Peikert, and Oded Regev. On Ideal Lattices
   and Learning with Errors over Rings. in Advances in Cryptology – EUROCRYPT
   2010. Springer 2010. http://dx.doi.org/10.1007/978-3-642-13190-5_1

.. [CGW13] Daniel Cabarcas, Florian Göpfert, and Patrick Weiden. Provably Secure
   LWE-Encryption with Uniform Secret. Cryptology ePrint Archive, Report
   2013/164. 2013.  2013/164. http://eprint.iacr.org/2013/164
"""

from sage.calculus.var import var
from sage.functions.log import exp, log
from sage.functions.other import sqrt, floor, ceil
from sage.misc.functional import cyclotomic_polynomial
from sage.misc.randstate import set_random_seed
from sage.misc.prandom import randint
from sage.misc.misc import get_verbose
from sage.modules.free_module import FreeModule
from sage.modules.free_module_element import random_vector, vector
from sage.numerical.optimize import find_root
from sage.rings.all import ZZ, RealField, IntegerModRing, RR
from sage.rings.arith import next_prime, euler_phi
from sage.structure.element import parent
from sage.structure.sage_object import SageObject
from sage.symbolic.constants import pi

class DiscreteGaussianSamplerRejection(SageObject):
    """
    Discrete Gaussian sampler using rejection sampling.

    EXAMPLE::

        sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection
        sage: DiscreteGaussianSamplerRejection(3.0)()
        -1
        sage: gs = DiscreteGaussianSamplerRejection(3.0, precision=100, tailcut=1.0)
        sage: all(gs() <= 3.0 for _ in xrange(1000))
        True

    .. automethod:: __init__
    .. automethod:: __call__
    """
    def __init__(self, stddev, precision=53, tailcut=4):
        """
        Construct a new discrete Gaussian sampler.

        INPUT:

        - ``stddev`` - standard deviation
        - ``precision`` - precision used for internal computations (default: ``53``)
        - ``tailcut`` - cut the tail at ``tailcut`` standard deviations (default: ``4``)

        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection
            sage: gs = DiscreteGaussianSamplerRejection(3.0)
            sage: sqrt(variance([gs() for _ in xrange(1000)])).n()
            2.965...
        """
        self.stddev = stddev
        self.precision = precision
        self.tailcut = tailcut
        self.max_precs = 2**precision
        self.upper_bound = ZZ(round(tailcut*stddev))

    def __call__(self):
        """
        Return a new sample.

        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection
            sage: sampler = DiscreteGaussianSamplerRejection(12.0)
            sage: sampler()
            -5
        """
        x = 0
        y = 0
        z = 0
        while y >= z :
            x = randint(0, self.upper_bound-1)
            y = randint(0, self.max_precs-1)
            z = self.rho[x]
        return (2*randint(0,1)-1)*x

    def _repr_(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection
            sage: DiscreteGaussianSamplerRejection(3.0)
            DiscreteGaussianSamplerRejection(3.000000, 53, 4)
        """
        return "DiscreteGaussianSamplerRejection(%f, %d, %d)"%(self.stddev, self.precision, self.tailcut)


    def __getattr__(self, name):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection
            sage: DiscreteGaussianSamplerRejection(3.0).foo
            Traceback (most recent call last):
            ...
            AttributeError: 'DiscreteGaussianSamplerRejection' object has no attribute 'foo'
        """
        if name == "rho":
            # we delay the creation of rho until we actually need it
            R = RealField(self.precision)
            self.rho = [round(self.max_precs * exp((-(R(x) / R(self.stddev))**2)/R(2))) for x in range(0,self.upper_bound)]
            self.rho[0] = self.rho[0] / 2
            return self.rho
        else:
            raise AttributeError("'%s' object has no attribute '%s'"%(self.__class__.__name__, name))

# By default we use rejection sampling
DiscreteGaussianSampler = DiscreteGaussianSamplerRejection

class DiscreteGaussianPolynomialSamplerRejection(SageObject):
    """
    Discrete Gaussian sampler for polynomials.

    EXAMPLE::

        sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection
        sage: DiscreteGaussianPolynomialSamplerRejection(8, 3.0)()
        x^7 - x^6 - 2*x^4 + 2*x^3 - x^2 + x - 1
        sage: gs = DiscreteGaussianPolynomialSamplerRejection(8, 3.0, precision=100, tailcut=1.0)
        sage: [gs() for _ in xrange(3)]
        [-x^7 + x^6 + 2*x^5 + 2*x^4 - x^3 - x^2 - 1,
         x^7 - 2*x^6 + 2*x^5 + x^4 - x^3 + 2*x^2 - x + 2,
         x^5 + 2*x^3 + 2*x + 1]

    .. automethod:: __init__
    .. automethod:: __call__
    """
    def __init__(self, n, stddev, precision=53, tailcut=4, D=DiscreteGaussianSampler):
        """
        Construct a sampler for univariate polynomials of degree ``n-1``
        where coefficients are drawn independently with standard deviation
        ``stddev`` using ``D``.

        INPUT:

        - ``n`` - number of coefficients to be sampled
        - ``stddev`` - standard deviation
        - ``precision`` - precision used for internal computations (default: ``53``)
        - ``tailcut`` - cut the tail at ``tailcut`` standard deviations
          (default: ``4``)
        - ``D`` - a discrete Gaussian sampler (default:
          :class:`DiscreteGaussianSampler`)

        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection
            sage: DiscreteGaussianPolynomialSamplerRejection(8, 3.0)()
            x^7 - x^6 - 2*x^4 + 2*x^3 - x^2 + x - 1
            sage: gs = DiscreteGaussianPolynomialSamplerRejection(8, 3.0, precision=100, tailcut=1.0)
            sage: [gs() for _ in xrange(3)]
            [-x^7 + x^6 + 2*x^5 + 2*x^4 - x^3 - x^2 - 1,
             x^7 - 2*x^6 + 2*x^5 + x^4 - x^3 + 2*x^2 - x + 2,
             x^5 + 2*x^3 + 2*x + 1]
        """
        self.stddev = stddev
        self.precision = precision
        self.tailcut = tailcut
        self.D = D(stddev, precision, tailcut)
        self.n = ZZ(n)
        self.P = ZZ['x']

    def __call__(self):
        """
        Return a new sample.

        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection
            sage: sampler = DiscreteGaussianPolynomialSamplerRejection(8, 12.0)
            sage: sampler()
            x^7 - 9*x^5 + 2*x^4 + 8*x^3 - 5*x^2 + 7*x - 5
        """
        coeff = [self.D() for _ in range(self.n)]
        f = self.P(coeff)
        return f

    def _repr_(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection
            sage: DiscreteGaussianPolynomialSamplerRejection(8, 3.0)
            DiscreteGaussianPolynomialSamplerRejection(8, 3.000000, 53, 4)
        """
        return "DiscreteGaussianPolynomialSamplerRejection(%d, %f, %d, %d)"%(self.n, self.stddev, self.precision, self.tailcut)


# By default we use rejection sampling
DiscreteGaussianPolynomialSampler = DiscreteGaussianPolynomialSamplerRejection

class UniformSampler(SageObject):
    """
    Uniform sampling in a range of integers.

    EXAMPLE::

        sage: from sage.crypto.lwe import UniformSampler
        sage: sampler = UniformSampler(-2, 2); sampler
        UniformSampler(-2, 2)
        sage: sampler()
        -2

    .. automethod:: __init__
    .. automethod:: __call__
    """
    def __init__(self, lower_bound, upper_bound):
        """
        Construct a uniform sampler with bounds ``lower_bound`` and
        ``upper_bound`` (both endpoints inclusive).

        INPUT:

        - ``lower_bound`` - integer
        - ``upper_bound`` - integer

        EXAMPLE::

            sage: from sage.crypto.lwe import UniformSampler
            sage: UniformSampler(-2, 2)
            UniformSampler(-2, 2)
        """
        if lower_bound > upper_bound:
            raise TypeError("lower bound must be <= upper bound.")
        self.lower_bound = ZZ(lower_bound)
        self.upper_bound = ZZ(upper_bound)

    def __call__(self):
        """
        Return a new sample.

        EXAMPLE::

            sage: from sage.crypto.lwe import UniformSampler
            sage: sampler = UniformSampler(-12, 12)
            sage: sampler()
            -10
        """
        return randint(self.lower_bound, self.upper_bound)

    def _repr_(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import UniformSampler
            sage: UniformSampler(-2, 2)
            UniformSampler(-2, 2)
        """
        return "UniformSampler(%d, %d)"%(self.lower_bound, self.upper_bound)


class UniformPolynomialSampler(SageObject):
    """
    Uniform sampler for polynomials.

    EXAMPLE::

        sage: from sage.crypto.lwe import UniformPolynomialSampler
        sage: UniformPolynomialSampler(8, -2, 2)()
        -2*x^7 + x^6 - 2*x^5 - x^3 - 2*x^2 - 2

    .. automethod:: __init__
    .. automethod:: __call__
    """
    def __init__(self, n, lower_bound, upper_bound):
        """
        Construct a sampler for univariate polynomials of degree ``n-1`` where
        coefficients are drawn uniformly at random between ``lower_bound`` and
        ``upper_bound`` (both endpoints inclusive).

        INPUT:

        - ``n`` - number of coefficients to be sampled
        - ``lower_bound`` - integer
        - ``upper_bound`` - integer

        EXAMPLE::

            sage: from sage.crypto.lwe import UniformPolynomialSampler
            sage: UniformPolynomialSampler(10, -10, 10)
            UniformPolynomialSampler(10, -10, 10)
        """
        self.n = ZZ(n)
        self.P = ZZ['x']
        if lower_bound > upper_bound:
            raise TypeError("lower bound must be <= upper bound.")
        self.lower_bound = ZZ(lower_bound)
        self.upper_bound = ZZ(upper_bound)
        self.D = UniformSampler(self.lower_bound, self.upper_bound)

    def __call__(self):
        """
        Return a new sample.

        EXAMPLE::

            sage: from sage.crypto.lwe import UniformPolynomialSampler
            sage: sampler = UniformPolynomialSampler(8, -12, 12)
            sage: sampler()
            -10*x^7 + 5*x^6 - 8*x^5 + x^4 - 4*x^3 - 11*x^2 - 10
        """
        coeff = [self.D() for _ in range(self.n)]
        f = self.P(coeff)
        return f

    def _repr_(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import UniformPolynomialSampler
            sage: UniformPolynomialSampler(8, -3, 3)
            UniformPolynomialSampler(8, -3, 3)
        """
        return "UniformPolynomialSampler(%d, %d, %d)"%(self.n, self.lower_bound, self.upper_bound)


class LWE(SageObject):
    """
    Learning with Errors (LWE) oracle.

    .. automethod:: __init__
    .. automethod:: __call__
    """
    def __init__(self, n, q, D, secret_dist='uniform', m=None):
        """
        Construct an LWE oracle in dimension ``n`` over a ring of order
        ``q`` with noise distribution ``D``.

        INPUT:

        - ``n`` - dimension (integer > 0)
        - ``q`` - modulus typically > n (integer > 0)
        - ``D`` - an error distribution such as an instance of
          :class:`DiscreteGaussianSamplerRejection` or :class:`UniformSampler`
        - ``secret_dist`` - distribution of the secret (default: 'uniform'); one of

          - "uniform" - secret follows the uniform distribution in `\Zmod{q}`
          - "noise" - secret follows the noise distribution
          - ``(lb,ub)`` - the secret is chosen uniformly from ``[lb,...,ub]`` including both endpoints

        - ``m`` - number of allowed samples or ``None`` if no such limit exists
          (default: ``None``)

        EXAMPLE:

        First, we construct a noise distribution with standard deviation 3.0::

            sage: from sage.crypto.lwe import DiscreteGaussianSampler
            sage: D = DiscreteGaussianSampler(3.0)

        Next, we construct our oracle::

            sage: from sage.crypto.lwe import LWE
            sage: lwe = LWE(n=20, q=next_prime(400), D=D); lwe
            LWE(20, 401, DiscreteGaussianSamplerRejection(3.000000, 53, 4), 'uniform', None)

        and sample 1000 samples::

            sage: L = [lwe() for _ in range(1000)]

        To test the oracle, we use the internal secret to evaluate the samples
        in the secret::

            sage: S = [ZZ(a.dot_product(lwe._LWE__s) - c) for (a,c) in L]

        However, while Sage represents finite field elements between 0 and q-1
        we rely on a balanced representation of those elements here. Hence, we
        fix the representation and recover the correct standard deviation of the
        noise::

            sage: sqrt(variance([e if e <= 200 else e-401 for e in S]).n())
            3.0...

        If ``m`` is not ``None`` the number of available samples is restricted::

            sage: from sage.crypto.lwe import LWE
            sage: lwe = LWE(n=20, q=next_prime(400), D=D, m=30)
            sage: _ = [lwe() for _ in range(30)]
            sage: lwe() # 31
            Traceback (most recent call last):
            ...
            IndexError: Number of available samples exhausted.
        """
        self.n  = ZZ(n)
        self.m =  m
        self.__i = 0
        self.K  = IntegerModRing(q)
        self.FM = FreeModule(self.K, n)
        self.D = D

        self.secret_dist = secret_dist
        if secret_dist == 'uniform':
            self.__s = random_vector(self.K, self.n)
        elif secret_dist == 'noise':
            self.__s = vector(self.K, self.n, [self.D() for _ in range(n)])
        else:
            try:
                lb, ub = map(ZZ,secret_dist)
                self.__s = vector(self.K, self.n, [randint(lb,ub) for _ in range(n)])
            except (IndexError, TypeError):
                raise TypeError("Parameter secret_dist=%s not understood."%(secret_dist))

    def _repr_(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianSampler, LWE
            sage: D = DiscreteGaussianSampler(3.0)
            sage: lwe = LWE(n=20, q=next_prime(400), D=D); lwe
            LWE(20, 401, DiscreteGaussianSamplerRejection(3.000000, 53, 4), 'uniform', None)

            sage: lwe = LWE(n=20, q=next_prime(400), D=D, secret_dist=(-3, 3)); lwe
            LWE(20, 401, DiscreteGaussianSamplerRejection(3.000000, 53, 4), (-3, 3), None)
        """
        if type(self.secret_dist) == str:
            return "LWE(%d, %d, %s, '%s', %s)"%(self.n,self.K.order(),self.D,self.secret_dist, self.m)
        else:
            return "LWE(%d, %d, %s, %s, %s)"%(self.n,self.K.order(),self.D,self.secret_dist, self.m)


    def __call__(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianSampler, LWE
            sage: LWE(10, 401, DiscreteGaussianSampler(3))()
            ((309, 347, 198, 194, 336, 360, 264, 123, 368, 398), 198)
        """
        if self.m is not None:
            if self.__i >= self.m:
                raise IndexError("Number of available samples exhausted.")
        self.__i+=1
        a = self.FM.random_element()
        return a, a.dot_product(self.__s) + self.K(self.D())


class Regev(LWE):
    """
    LWE oracle with parameters as in [Reg09]_.

    .. automethod:: __init__
    """
    def __init__(self, n, secret_dist='uniform', m=None):
        """
        Construct LWE instance parameterised by security parameter ``n`` where
        the modulus ``q`` and the ``stddev`` of the noise are chosen as in
        [Reg09]_.

        INPUT:

        - ``n`` - security parameter (integer > 0)
        - ``secret_dist`` - distribution of the secret. See documentation of :class:`LWE`
          for details (default='uniform')
        - ``m`` - number of allowed samples or ``None`` if no such limit exists
          (default: ``None``)

        EXAMPLES::

            sage: from sage.crypto.lwe import Regev
            sage: Regev(n=20)
            LWE(20, 401, DiscreteGaussianSamplerRejection(1.915069, 401, 4), 'uniform', None)
        """
        q = ZZ(next_prime(n**2))
        s = RR(1/(RR(n).sqrt() * log(n, 2)**2) * q)
        D = DiscreteGaussianSampler(s/sqrt(2*pi.n()), q)
        LWE.__init__(self, n=n, q=q, D=D, secret_dist=secret_dist, m=m)

class LindnerPeikert(LWE):
    """
    LWE oracle with parameters as in [LP11]_.

    .. automethod:: __init__
    """
    def __init__(self, n, delta=0.01, m=None):
        """
        Construct LWE instance parameterised by security parameter ``n`` where
        the modulus ``q`` and the ``stddev`` of the noise is chosen as in
        [LP11]_.

        INPUT:

        - ``n`` - security parameter (integer > 0)
        - ``delta`` - error probability per symbol (default: 0.01)
        - ``m`` - number of allowed samples or ``None`` in which case ``m=2*n +
          128`` as in [LP11]_ (default: ``None``)

        EXAMPLES::

            sage: from sage.crypto.lwe import LindnerPeikert
            sage: LindnerPeikert(n=20)
            LWE(20, 2053, DiscreteGaussianSamplerRejection(3.600954, 53, 4), 'noise', 168)
        """
        if m is None:
            m = 2*n + 128
        # Find c>=1 such that c*exp((1-c**2)/2))**(2*n) == 2**-40
        #         (c*exp((1-c**2)/2))**(2*n) == 2**-40
        #    log((c*exp((1-c**2)/2))**(2*n)) == -40*log(2)
        #       (2*n)*log(c*exp((1-c**2)/2)) == -40*log(2)
        #  2*n*(log(c)+log(exp((1-c**2)/2))) == -40*log(2)
        #            2*n*(log(c)+(1-c**2)/2) == -40*log(2)
        #              2*n*log(c)+n*(1-c**2) == -40*log(2)
        #  2*n*log(c)+n*(1-c**2) + 40*log(2) == 0
        c = var('c')
        c = find_root(2*n*log(c)+n*(1-c**2) + 40*log(2) == 0, 1, 10)
        # Upper bound on s**2/t
        s_t_bound = (sqrt(2) * pi / c / sqrt(2*n*log(2/delta))).n()
        # Interpretation of "choose q just large enough to allow for a Gaussian parameter s>=8" in [LP11]_
        q = next_prime(floor(2**round(log(256 / s_t_bound, 2))))
        # Gaussian parameter as defined in [LP11]_
        s = sqrt(s_t_bound*floor(q/4))
        # Transform s into stddev
        stddev = s/sqrt(2*pi.n())
        D   = DiscreteGaussianSampler(stddev)
        LWE.__init__(self, n=n, q=q, D=D, secret_dist='noise', m=m)


class UniformNoiseLWE(LWE):
    """
    LWE oracle with uniform secret with parameters as in [CGW13]_.

    .. automethod:: __init__
    """
    def __init__(self, n, instance='key', m=None):
        """
        Construct LWE instance parameterised by security parameter ``n`` where
        all other parameters are chosen as in [CGW13]_.

        INPUT:

        - ``n`` - security parameter (integer >= 89)
        - ``instance`` - one of

          - "key" - the LWE-instance that hides the secret key is generated
          - "encrypt" - the LWE-instance that hides the message is generated
            (default: ``key``)

        - ``m`` - number of allowed samples or ``None`` in which case ``m`` is
          chosen as in [CGW13_].  (default: ``None``)

        EXAMPLES::

            sage: from sage.crypto.lwe import UniformNoiseLWE
            sage: UniformNoiseLWE(89)
            LWE(89, 154262477, UniformSampler(0, 351), 'noise', 131)

            sage: UniformNoiseLWE(89, instance='encrypt')
            LWE(131, 154262477, UniformSampler(0, 497), 'noise', 181)
        """

        if n<89:
            raise TypeError("Parameter too small")

        n2 = n
        C  = 4/sqrt(2*pi)
        kk = floor((n2-2*log(n2, 2)**2)/5)
        n1 = floor((3*n2-5*kk)/2)
        ke = floor((n1-2*log(n1, 2)**2)/5)
        l  = floor((3*n1-5*ke)/2)-n2
        sk = ceil((C*(n1+n2))**(3/2))
        se = ceil((C*(n1+n2+l))**(3/2))
        q = next_prime(max(ceil((4*sk)**((n1+n2)/n1)), ceil((4*se)**((n1+n2+l)/(n2+l))), ceil(4*(n1+n2)*se*sk+4*se+1)))

        if kk<=0:
            raise TypeError("Parameter too small")

        if instance == 'key':
            D  = UniformSampler(0, sk-1)
            if m is None:
                m = n1
            LWE.__init__(self, n=n2, q=q, D=D, secret_dist='noise', m=m)
        elif instance == 'encrypt':
            D   = UniformSampler(0, se-1)
            if m is None:
                m = n2+l
            LWE.__init__(self, n=n1, q=q, D=D, secret_dist='noise', m=m)
        else:
            raise TypeError("Parameter instance=%s not understood."%(instance))

class RingLWE(SageObject):
    """
    Ring Learning with Errors oracle.

    .. automethod:: __init__
    .. automethod:: __call__
    """
    def __init__(self, N, q, D, poly=None, secret_dist='uniform', m=None):
        """
        Construct a Ring-LWE oracle in dimension ``n=phi(N)`` over a ring of order
        ``q`` with noise distribution ``D``.

        INPUT:

        - ``N`` - index of cyclotomic polynomial (integer > 0, must be power of 2)
        - ``q`` - modulus typically > N (integer > 0)
        - ``D`` - an error distribution such as an instance of
          :class:`DiscreteGaussianPolynomialSamplerRejection` or :class:`UniformSampler`
        - ``poly`` - a polynomial of degree ``phi(N)``. If ``None`` the
          cyclotomic polynomial used (default: ``None``).
        - ``secret_dist`` - distribution of the secret. See documentation of
          :class:`LWE` for details (default='uniform')
        - ``m`` - number of allowed samples or ``None`` if no such limit exists
          (default: ``None``)

        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE
            sage: D = DiscreteGaussianPolynomialSampler(n=euler_phi(20), stddev=3.0)
            sage: RingLWE(N=20, q=next_prime(800), D=D);
            RingLWE(20, 809, DiscreteGaussianPolynomialSamplerRejection(8, 3.000000, 53, 4), x^8 - x^6 + x^4 - x^2 + 1, 'uniform', None)
        """
        self.N  = ZZ(N)
        self.n = euler_phi(N)
        self.m =  m
        self.__i = 0
        self.K  = IntegerModRing(q)

        if self.n != D.n:
            raise ValueError("Noise distribution has dimensions %d != %d"%(D.n, self.n))

        self.D = D
        self.q = q
        if poly is not None:
            self.poly = poly
        else:
            self.poly = cyclotomic_polynomial(self.N, 'x')

        self.R_q = self.K['x'].quotient(self.poly, 'x')

        self.secret_dist = secret_dist
        if secret_dist == 'uniform':
            self.__s = self.R_q.random_element()  # uniform sampling of secret
        elif secret_dist == 'noise':
            self.__s = self.D()
        else:
            raise TypeError("Parameter secret_dist=%s not understood."%(secret_dist))

    def _repr_(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE
            sage: D = DiscreteGaussianPolynomialSampler(n=8, stddev=3.0)
            sage: RingLWE(N=16, q=next_prime(400), D=D);
            RingLWE(16, 401, DiscreteGaussianPolynomialSamplerRejection(8, 3.000000, 53, 4), x^8 + 1, 'uniform', None)
        """
        if type(self.secret_dist) == str:
            return "RingLWE(%d, %d, %s, %s, '%s', %s)"%(self.N, self.K.order(), self.D, self.poly, self.secret_dist, self.m)
        else:
            return "RingLWE(%d, %d, %s, %s, %s, %s)"%(self.N, self.K.order(), self.D, self.poly, self.secret_dist, self.m)


    def __call__(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE
            sage: N = 16
            sage: n = euler_phi(N)
            sage: D = DiscreteGaussianPolynomialSampler(n, 5)
            sage: ringlwe = RingLWE(N, 257, D, secret_dist='uniform')
            sage: ringlwe()
            ((228, 149, 226, 198, 38, 222, 222, 127), (177, 138, 68, 134, 74, 162, 203, 243))
        """
        if self.m is not None:
            if self.__i >= self.m:
                raise IndexError("Number of available samples exhausted.")
        self.__i+=1
        a = self.R_q.random_element()
        return vector(a), vector(a * (self.__s) + self.D())

class RingLindnerPeikert(RingLWE):
    """
    Ring-LWE oracle with parameters as in [LP11]_.

    .. automethod:: __init__
    """
    def __init__(self, N, delta=0.01, m=None):
        """
        Construct a Ring-LWE oracle in dimension ``n=phi(N)`` where
        the modulus ``q`` and the ``stddev`` of the noise is chosen as in
        [LP11]_.

        INPUT:

        - ``N`` - index of cyclotomic polynomial (integer > 0, must be power of 2)
        - ``delta`` - error probability per symbol (default: 0.01)
        - ``m`` - number of allowed samples or ``None`` in which case ``3*n`` is
          used (default: ``None``)

        EXAMPLES::

            sage: from sage.crypto.lwe import RingLindnerPeikert
            sage: RingLindnerPeikert(N=16)
            RingLWE(16, 1031, DiscreteGaussianPolynomialSamplerRejection(8, 2.803372, 53, 4), x^8 + 1, 'noise', 24)
        """
        n = euler_phi(N)
        if m is None:
            m = 3*n
        # Find c>=1 such that c*exp((1-c**2)/2))**(2*n) == 2**-40
        #  i.e c>=1 such that 2*n*log(c)+n*(1-c**2) + 40*log(2) == 0
        c = var('c')
        c = find_root(2*n*log(c)+n*(1-c**2) + 40*log(2) == 0, 1, 10)
        # Upper bound on s**2/t
        s_t_bound = (sqrt(2) * pi / c / sqrt(2*n*log(2/delta))).n()
        # Interpretation of "choose q just large enough to allow for a Gaussian parameter s>=8" in [LP11]_
        q = next_prime(floor(2**round(log(256 / s_t_bound, 2))))
        # Gaussian parameter as defined in [LP11]_
        s = sqrt(s_t_bound*floor(q/4))
        # Transform s into stddev
        stddev = s/sqrt(2*pi.n())
        D = DiscreteGaussianPolynomialSampler(n, stddev)
        RingLWE.__init__(self, N=N, q=q, D=D, poly=None, secret_dist='noise', m=m)

class RingLWEConverter(SageObject):
    """
    Wrapper callable to convert Ring-LWE oracles into LWE oracles by
    disregarding the additional structure.

    .. automethod:: __init__
    .. automethod:: __call__
    """
    def __init__(self, ringlwe):
        """
        INPUT:

        - ``ringlwe`` - an instance of a :class:`RingLWE`

        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE, RingLWEConverter
            sage: D = DiscreteGaussianPolynomialSampler(euler_phi(16), 5)
            sage: lwe = RingLWEConverter(RingLWE(16, 257, D, secret_dist='uniform'))
            sage: set_random_seed(1337)
            sage: lwe()
            ((130, 32, 216, 3, 125, 58, 197, 171), 182)
        """
        self.ringlwe = ringlwe
        self._i = 0
        self._ac = None
        self.n = self.ringlwe.n

    def __call__(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE, RingLWEConverter
            sage: D = DiscreteGaussianPolynomialSampler(euler_phi(16), 5)
            sage: lwe = RingLWEConverter(RingLWE(16, 257, D, secret_dist='uniform'))
            sage: set_random_seed(1337)
            sage: lwe()
            ((130, 32, 216, 3, 125, 58, 197, 171), 182)
        """
        R_q = self.ringlwe.R_q

        if (self._i % self.n) == 0:
            self._ac = self.ringlwe()
        a, c = self._ac
        x = R_q.gen()
        r = vector((x**(self._i % self.n) * R_q(a.list())).list()), c[self._i % self.n]
        self._i += 1
        return r

    def _repr_(self):
        """
        EXAMPLE::

            sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE, RingLWEConverter
            sage: D = DiscreteGaussianPolynomialSampler(euler_phi(20), 5)
            sage: rlwe = RingLWE(20, 257, D)
            sage: lwe = RingLWEConverter(rlwe)
            sage: lwe
            RingLWEConverter(RingLWE(20, 257, DiscreteGaussianPolynomialSamplerRejection(8, 5.000000, 53, 4), x^8 - x^6 + x^4 - x^2 + 1, 'uniform', None))

        """
        return "RingLWEConverter(%s)"%str(self.ringlwe)

def samples(m, n, lwe, seed=None, balanced=False, **kwds):
    """
    Return ``m`` LWE samples.

    INPUT:

    - ``m`` - the number of samples (integer > 0)
    - ``n`` - the security parameter (integer > 0)
    - ``lwe`` - either

      - a subclass of :class:`LWE` such as :class:`Regev` or :class:`LindnerPeikert`
      - an instance of :class:`LWE` or any subclass
      - the name of any such class (e.g., "Regev", "LindnerPeikert")

    - ``seed`` - seed to be used for generation or ``None`` if no specific seed
      shall be set (default: ``None``)
    - ``balanced`` - use function :func:`balance_sample` to return balanced
      representations of finite field elements (default: ``False``)
    - ``**kwds`` - passed through to LWE constructor

    EXAMPLE::

        sage: from sage.crypto.lwe import samples, Regev
        sage: samples(2, 20, Regev, seed=1337)
        [((199, 388, 337, 53, 200, 284, 336, 215, 75, 14, 274, 234, 97, 255, 246, 153, 268, 218, 396, 351), 18),
        ((286, 42, 175, 155, 190, 275, 114, 280, 45, 218, 304, 386, 98, 235, 77, 0, 65, 20, 163, 14), 334)]

        sage: from sage.crypto.lwe import samples, Regev
        sage: samples(2, 20, Regev, balanced=True, seed=1337)
        [((199, -13, -64, 53, 200, -117, -65, -186, 75, 14, -127, -167, 97, -146, -155, 153, -133, -183, -5, -50), 18),
        ((-115, 42, 175, 155, 190, -126, 114, -121, 45, -183, -97, -15, 98, -166, 77, 0, 65, 20, 163, 14), -67)]

        sage: from sage.crypto.lwe import samples
        sage: samples(2, 20, 'LindnerPeikert')
        [((1302, 718, 1397, 147, 278, 979, 1185, 133, 902, 1180, 1264, 734, 2029, 314, 428, 18, 707, 2021, 1153, 173), 1127),
        ((2015, 1278, 455, 429, 1391, 186, 149, 1199, 220, 1629, 843, 719, 1744, 1568, 674, 1462, 1549, 972, 248, 1066), 1422)]

    """
    if seed is not None:
        set_random_seed(seed)

    if isinstance(lwe, str):
        lwe = eval(lwe)

    if type(lwe) == type:
        lwe = lwe(n, m=m, **kwds)
    else:
        lwe = lwe
        if lwe.n != n:
            raise ValueError("Passed LWE instance has n=%d, but n=%d was passed to this function."%(lwe.n, n))

    if balanced is False:
        f = lambda (a,c): (a,c)
    else:
        f = balance_sample
    return [f(lwe()) for _ in xrange(m)]

def balance_sample(s, q=None):
    r"""
    Given ``(a,c) = s`` return a tuple ``(a',c')`` where ``a'`` is an integer
    vector with entries between -q//2 and q//2 and ``c`` is also within these
    bounds.

    If ``q`` is given ``(a,c) = s`` may live in the integers. If ``q`` is not
    given, then ``(a,c)`` are assumed to live in `\Zmod{q}`.

    INPUT:

    - ``s`` - sample of the form (a,c) where a is a vector and c is a scalar
    - ``q`` - modulus (default: ``None``)

    EXAMPLE::

        sage: from sage.crypto.lwe import balance_sample, samples, Regev
        sage: map(balance_sample, samples(10, 5, Regev))
        [((-9, -4, -4, 4, -4), 6), ((-3, -10, 8, -3, -1), -10), ((-6, -12, -3, -2, -6), -6),
        ...
        ((-1, -8, -11, 13, 4), -6), ((10, 11, -3, -13, 0), 6), ((6, -1, 2, -11, 14), 2)]


        sage: from sage.crypto.lwe import balance_sample, DiscreteGaussianPolynomialSampler, RingLWE, samples
        sage: D = DiscreteGaussianPolynomialSampler(8, 5)
        sage: rlwe = RingLWE(20, 257, D)
        sage: map(balance_sample, samples(10, 8, rlwe))
        [((5, -55, -31, -90, 6, 100, -46, -107), (6, -64, -40, 117, 27, 54, -98, -56)),
         ((109, -106, 28, 77, -14, -109, 115, 34), (82, 17, -89, 62, 1, -77, 128, 64)),
         ...
         ((-32, 51, -110, -106, 35, -82, 14, -113), (126, -120, 126, 119, 101, 3, -122, -75))]

    .. note::

        This function is useful to convert between Sage's standard
        representation of elements in `\Zmod{q}` as integers between 0 and q-1
        and the usual representation of such elements in lattice cryptography as
        integers between -q//2 and q//2.
    """
    a, c = s

    try:
        c[0]
        scalar = False
    except TypeError:
        c = vector(c.parent(),[c])
        scalar = True

    if q is None:
        q = parent(c[0]).order()
        a = a.change_ring(ZZ)
        c = c.change_ring(ZZ)
    else:
        K = IntegerModRing(q)
        a = a.change_ring(K).change_ring(ZZ)
        c = c.change_ring(K).change_ring(ZZ)

    q2 = q//2

    if scalar:
        return vector(ZZ, len(a), [e if e <= q2 else e-q for e in a]), c[0] if c[0] <= q2 else c[0]-q
    else:
        return vector(ZZ, len(a), [e if e <= q2 else e-q for e in a]), vector(ZZ, len(c), [e if e <= q2 else e-q for e in c])