# (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 [LPR2010] 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), 122),
...
((155, 22, 357, 312, 87, 298, 182, 163, 296, 181, 219, 135, 164, 308, 248, 320, 64, 166, 214, 104), 152)]


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)
[((1275, 168, 1529, 2024, 1874, 1309, 16, 1869, 1114, 1696, 1645, 618, 1372, 1273, 683, 237, 1526, 879, 1305, 1355), 950),
...
((1787, 2033, 1677, 331, 1562, 49, 796, 1002, 627, 98, 91, 711, 1712, 418, 2024, 163, 1773, 184, 1548, 3), 1815)]


Finally, samples() also accepts instances of classes:

sage: from sage.crypto.lwe import LindnerPeikert
sage: lwe = LindnerPeikert(20)
sage: samples(30, 20, lwe)
[((465, 180, 440, 706, 1367, 106, 1380, 614, 1162, 1354, 1098, 2036, 1974, 1417, 1502, 1431, 863, 1894, 1368, 1771), 618),
...
((1050, 1017, 1314, 1310, 1941, 2041, 484, 104, 1199, 1744, 161, 1905, 679, 1663, 531, 1630, 168, 1559, 1040, 1719), 1006)]


Note that Ring-LWE samples are returned as vectors:

sage: from sage.crypto.lwe import RingLWE
sage: from sage.stats.distributions.discrete_gaussian_polynomial import DiscreteGaussianDistributionPolynomialSampler
sage: D = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], euler_phi(16), 5)
sage: ringlwe = RingLWE(16, 257, D, secret_dist='uniform')
sage: samples(30, euler_phi(16), ringlwe)
[((232, 79, 223, 85, 26, 68, 60, 72), (72, 158, 117, 166, 140, 103, 142, 223)),
...
((27, 191, 241, 179, 246, 204, 36, 72), (207, 158, 127, 240, 225, 141, 156, 201))]


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)
[((-46, -84, 21, -72, -47, -162, -40, -31, -9, -131, 74, 183, 62, -83, -135, 164, -33, -109, -127, -124), 96),
...
((-48, 185, 118, 69, 57, 109, 109, 138, -42, -45, -16, 180, 34, 178, 20, -119, -58, -136, -46, 169), -72)]


AUTHORS:

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

REFERENCES:

class sage.crypto.lwe.LWE(n, q, D, secret_dist='uniform', m=None)

Learning with Errors (LWE) oracle.

__init__(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 DiscreteGaussianDistributionIntegerSampler or 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)

EXAMPLES:

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

sage: from sage.stats.distributions.discrete_gaussian_integer import DiscreteGaussianDistributionIntegerSampler
sage: D = DiscreteGaussianDistributionIntegerSampler(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, Discrete Gaussian sampler over the Integers with sigma = 3.000000 and c = 0, '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.

__call__()

EXAMPLES:

sage: from sage.crypto.lwe import DiscreteGaussianDistributionIntegerSampler, LWE
sage: LWE(10, 401, DiscreteGaussianDistributionIntegerSampler(3))()
((309, 347, 198, 194, 336, 360, 264, 123, 368, 398), 198)

class sage.crypto.lwe.LindnerPeikert(n, delta=0.01, m=None)

LWE oracle with parameters as in [LP2011].

__init__(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 [LP2011].

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 [LP2011] (default: None)

EXAMPLES:

sage: from sage.crypto.lwe import LindnerPeikert
sage: LindnerPeikert(n=20)
LWE(20, 2053, Discrete Gaussian sampler over the Integers with sigma = 3.600954 and c = 0, 'noise', 168)

class sage.crypto.lwe.Regev(n, secret_dist='uniform', m=None)

LWE oracle with parameters as in [Reg09].

__init__(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 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, Discrete Gaussian sampler over the Integers with sigma = 1.915069 and c = 401, 'uniform', None)

class sage.crypto.lwe.RingLWE(N, q, D, poly=None, secret_dist='uniform', m=None)

Ring Learning with Errors oracle.

__init__(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 DiscreteGaussianDistributionPolynomialSampler or 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 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 RingLWE
sage: from sage.stats.distributions.discrete_gaussian_polynomial import DiscreteGaussianDistributionPolynomialSampler
sage: D = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], n=euler_phi(20), sigma=3.0)
sage: RingLWE(N=20, q=next_prime(800), D=D)
RingLWE(20, 809, Discrete Gaussian sampler for polynomials of degree < 8 with σ=3.000000 in each component, x^8 - x^6 + x^4 - x^2 + 1, 'uniform', None)

__call__()

EXAMPLES:

sage: from sage.crypto.lwe import DiscreteGaussianDistributionPolynomialSampler, RingLWE
sage: N = 16
sage: n = euler_phi(N)
sage: D = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], n, 5)
sage: ringlwe = RingLWE(N, 257, D, secret_dist='uniform')
sage: ringlwe()
((226, 198, 38, 222, 222, 127, 194, 124), (11, 191, 177, 59, 105, 203, 108, 42))

class sage.crypto.lwe.RingLWEConverter(ringlwe)

Wrapper callable to convert Ring-LWE oracles into LWE oracles by disregarding the additional structure.

__init__(ringlwe)

INPUT:

EXAMPLES:

sage: from sage.crypto.lwe import DiscreteGaussianDistributionPolynomialSampler, RingLWE, RingLWEConverter
sage: D = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], euler_phi(16), 5)
sage: lwe = RingLWEConverter(RingLWE(16, 257, D, secret_dist='uniform'))
sage: set_random_seed(1337)
sage: lwe()
((32, 216, 3, 125, 58, 197, 171, 43), 81)

__call__()

EXAMPLES:

sage: from sage.crypto.lwe import DiscreteGaussianDistributionPolynomialSampler, RingLWE, RingLWEConverter
sage: D = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], euler_phi(16), 5)
sage: lwe = RingLWEConverter(RingLWE(16, 257, D, secret_dist='uniform'))
sage: set_random_seed(1337)
sage: lwe()
((32, 216, 3, 125, 58, 197, 171, 43), 81)

class sage.crypto.lwe.RingLindnerPeikert(N, delta=0.01, m=None)

Ring-LWE oracle with parameters as in [LP2011].

__init__(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 [LP2011].

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, Discrete Gaussian sampler for polynomials of degree < 8 with σ=2.803372 in each component, x^8 + 1, 'noise', 24)

class sage.crypto.lwe.UniformNoiseLWE(n, instance='key', m=None)

LWE oracle with uniform secret with parameters as in [CGW2013].

__init__(n, instance='key', m=None)

Construct LWE instance parameterised by security parameter n where all other parameters are chosen as in [CGW2013].

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 [CGW2013]. (default: None)

EXAMPLES:

sage: from sage.crypto.lwe import UniformNoiseLWE
sage: UniformNoiseLWE(89)
LWE(89, 64311834871, UniformSampler(0, 6577), 'noise', 131)

sage: UniformNoiseLWE(89, instance='encrypt')
LWE(131, 64311834871, UniformSampler(0, 11109), 'noise', 181)

class sage.crypto.lwe.UniformPolynomialSampler(P, n, lower_bound, upper_bound)

Uniform sampler for polynomials.

EXAMPLES:

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

__init__(P, 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:

• P - a univariate polynomial ring over the Integers
• n - number of coefficients to be sampled
• lower_bound - integer
• upper_bound - integer

EXAMPLES:

sage: from sage.crypto.lwe import UniformPolynomialSampler
sage: UniformPolynomialSampler(ZZ['x'], 10, -10, 10)
UniformPolynomialSampler(10, -10, 10)

__call__()

Return a new sample.

EXAMPLES:

sage: from sage.crypto.lwe import UniformPolynomialSampler
sage: sampler = UniformPolynomialSampler(ZZ['x'], 8, -12, 12)
sage: sampler()
-10*x^7 + 5*x^6 - 8*x^5 + x^4 - 4*x^3 - 11*x^2 - 10

class sage.crypto.lwe.UniformSampler(lower_bound, upper_bound)

Uniform sampling in a range of integers.

EXAMPLES:

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

__init__(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

EXAMPLES:

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

__call__()

Return a new sample.

EXAMPLES:

sage: from sage.crypto.lwe import UniformSampler
sage: sampler = UniformSampler(-12, 12)
sage: sampler()
-10

sage.crypto.lwe.balance_sample(s, q=None)

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)

EXAMPLES:

sage: from sage.crypto.lwe import balance_sample, samples, Regev
sage: [balance_sample(s) for s in samples(10, 5, Regev)]
[((-9, -4, -4, 4, -4), 4), ((-8, 11, 12, -11, -11), -7),
...
((-11, 12, 0, -6, -3), 7), ((-7, 14, 8, 11, -8), -12)]

sage: from sage.crypto.lwe import balance_sample, DiscreteGaussianDistributionPolynomialSampler, RingLWE, samples
sage: D = DiscreteGaussianDistributionPolynomialSampler(ZZ['x'], 8, 5)
sage: rlwe = RingLWE(20, 257, D)
sage: [balance_sample(s) for s in samples(10, 8, rlwe)]
[((-64, 107, -91, -24, 120, 54, 38, -35), (-84, 121, 28, -99, 91, 54, -60, 11)),
...
((-40, -117, 35, -69, -11, 10, 122, 48), (-80, -2, 119, -91, 27, 66, 121, -1))]


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.

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

EXAMPLES:

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), 15),
((365, 227, 333, 165, 76, 328, 288, 206, 286, 42, 175, 155, 190, 275, 114, 280, 45, 218, 304, 386), 143)]

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), 15),
((-36, -174, -68, 165, 76, -73, -113, -195, -115, 42, 175, 155, 190, -126, 114, -121, 45, -183, -97, -15), 143)]

sage: from sage.crypto.lwe import samples
sage: samples(2, 20, 'LindnerPeikert')
[((506, 1205, 398, 0, 337, 106, 836, 75, 1242, 642, 840, 262, 1823, 1798, 1831, 1658, 1084, 915, 1994, 163), 1447),
((463, 250, 1226, 1906, 330, 933, 1014, 1061, 1322, 2035, 1849, 285, 1993, 1975, 864, 1341, 41, 1955, 1818, 1357), 312)]