| 1 | # -*- coding: utf-8 -*- |
| 2 | """ |
| 3 | (Ring-)LWE oracle generators |
| 4 | |
| 5 | The Learning with Errors problem (LWE) is solving linear systems of equations |
| 6 | where the right hand side has been disturbed 'slightly' where 'slightly' is made |
| 7 | precise by a noise distribution - typically a discrete Gaussian |
| 8 | distribution. See [Reg09]_ for details. |
| 9 | |
| 10 | The Ring Learning with Errors problem (LWE) is solving a set of univariate |
| 11 | polynomial equations - typically in a cyclotomic field - where the right hand |
| 12 | side was disturbed 'slightly'. See [LPR10]_ for details. |
| 13 | |
| 14 | This module implements generators of LWE samples where parameters are chosen |
| 15 | following proposals in the cryptographic literature. |
| 16 | |
| 17 | EXAMPLES: |
| 18 | |
| 19 | We get 30 samples from an LWE oracle parameterised by security parameter |
| 20 | ``n=20`` and where the modulus and the standard deviation of the noise are |
| 21 | chosen as in [Reg09]_:: |
| 22 | |
| 23 | sage: from sage.crypto.lwe import samples |
| 24 | sage: samples(30, 20, 'Regev') |
| 25 | [((360, 264, 123, 368, 398, 392, 41, 84, 25, 389, 311, 68, 322, 41, 161, 372, 222, 153, 243, 381), 126), |
| 26 | ... |
| 27 | ((138, 198, 204, 235, 339, 168, 269, 276, 392, 243, 86, 18, 378, 20, 369, 141, 108, 151, 336, 141), 102)] |
| 28 | |
| 29 | |
| 30 | We may also pass classes to the samples function, which is useful for users |
| 31 | implementing their own oracles:: |
| 32 | |
| 33 | sage: from sage.crypto.lwe import samples, LindnerPeikert |
| 34 | sage: samples(30, 20, LindnerPeikert) |
| 35 | [((350, 835, 2023, 1785, 1958, 1818, 1130, 1285, 1331, 284, 2048, 441, 1581, 1406, 1185, 1724, 1397, 258, 994, 1056), 1902), |
| 36 | ... |
| 37 | ((1918, 1823, 1598, 18, 588, 1093, 744, 1934, 689, 1327, 1632, 1867, 228, 378, 798, 511, 274, 1001, 1709, 154), 184)] |
| 38 | |
| 39 | |
| 40 | Finally, :func:`samples` also accepts instances of classes:: |
| 41 | |
| 42 | sage: from sage.crypto.lwe import LindnerPeikert |
| 43 | sage: lwe = LindnerPeikert(20) |
| 44 | sage: samples(30, 20, lwe) |
| 45 | [((1817, 1322, 818, 1232, 354, 639, 1770, 754, 1366, 1731, 649, 162, 483, 1741, 1942, 1232, 1424, 1034, 50, 448), 1316), |
| 46 | ... |
| 47 | ((2021, 829, 572, 1698, 1025, 170, 598, 1193, 1268, 607, 1502, 1984, 1655, 206, 958, 334, 1213, 1413, 827, 1423), 546)] |
| 48 | |
| 49 | Note that Ring-LWE samples are returned as vectors:: |
| 50 | |
| 51 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection, RingLWE |
| 52 | sage: D = DiscreteGaussianPolynomialSamplerRejection(euler_phi(16), 5) |
| 53 | sage: ringlwe = RingLWE(16, 257, D, secret_dist='uniform') |
| 54 | sage: samples(30, euler_phi(16), ringlwe) |
| 55 | [((158, 49, 174, 179, 109, 92, 234, 41), (200, 159, 131, 197, 241, 172, 1, 107)), |
| 56 | ... |
| 57 | ((80, 227, 249, 205, 149, 92, 46, 68), (69, 256, 29, 219, 218, 34, 182, 178))] |
| 58 | |
| 59 | One technical issue when working with these generators is that by default they |
| 60 | return vectors and scalars over/in rings modulo some `q`. These are represented |
| 61 | as elements in `(0,q-1)` by Sage. However, it usually is more natural to think |
| 62 | of these entries as integers in `(-q//2,q//2)`. To allow for this, this module |
| 63 | provides the option to balance the representation. In this case vectors and |
| 64 | scalars over/in the integers are returned:: |
| 65 | |
| 66 | sage: from sage.crypto.lwe import samples |
| 67 | sage: samples(30, 20, 'Regev', balanced=True) |
| 68 | [((-38, 59, -33, -80, 165, -55, -46, -49, -113, 135, -32, 185, -80, -184, 127, 153, 162, -31, 115, 178), 14), |
| 69 | ... |
| 70 | ((-165, -187, -87, 188, 160, -118, -7, 107, -77, -107, -109, 77, 63, -66, -55, -75, -12, 90, 58, -185), 6)] |
| 71 | |
| 72 | AUTHORS: |
| 73 | |
| 74 | - Martin Albrecht |
| 75 | - Robert Fitzpatrick |
| 76 | - Daniel Cabracas |
| 77 | - Florian Göpfert |
| 78 | - Michael Schneider |
| 79 | |
| 80 | REFERENCES: |
| 81 | |
| 82 | .. [Reg09] Oded Regev. On Lattices, Learning with Errors, Random Linear Codes, |
| 83 | and Cryptography. in Journal of the ACM 56(6). ACM 2009, |
| 84 | http://dx.doi.org/10.1145/1060590.1060603 |
| 85 | |
| 86 | .. [LP11] Richard Lindner and Chris Peikert. Better key sizes (and attacks) for |
| 87 | LWE-based encryption. in Proceeding of the 11th international conference on |
| 88 | Topics in cryptology: CT-RSA 2011. Springer 2011, |
| 89 | http://dx.doi.org/10.1007/978-3-642-19074-2_21 |
| 90 | |
| 91 | .. [LPR10] Vadim Lyubashevsky, Chris Peikert, and Oded Regev. On Ideal Lattices |
| 92 | and Learning with Errors over Rings. in Advances in Cryptology – EUROCRYPT |
| 93 | 2010. Springer 2010. http://dx.doi.org/10.1007/978-3-642-13190-5_1 |
| 94 | |
| 95 | .. [CGW13] Daniel Cabarcas, Florian Göpfert, and Patrick Weiden. Provably Secure |
| 96 | LWE-Encryption with Uniform Secret. Cryptology ePrint Archive, Report |
| 97 | 2013/164. 2013. 2013/164. http://eprint.iacr.org/2013/164 |
| 98 | """ |
| 99 | |
| 100 | from sage.calculus.var import var |
| 101 | from sage.functions.log import exp, log |
| 102 | from sage.functions.other import sqrt, floor, ceil |
| 103 | from sage.misc.functional import cyclotomic_polynomial |
| 104 | from sage.misc.randstate import set_random_seed |
| 105 | from sage.misc.prandom import randint |
| 106 | from sage.misc.misc import get_verbose |
| 107 | from sage.modules.free_module import FreeModule |
| 108 | from sage.modules.free_module_element import random_vector, vector |
| 109 | from sage.numerical.optimize import find_root |
| 110 | from sage.rings.all import ZZ, RealField, IntegerModRing, RR |
| 111 | from sage.rings.arith import next_prime, euler_phi |
| 112 | from sage.structure.element import parent |
| 113 | from sage.structure.sage_object import SageObject |
| 114 | from sage.symbolic.constants import pi |
| 115 | |
| 116 | class DiscreteGaussianSamplerRejection(SageObject): |
| 117 | """ |
| 118 | Discrete Gaussian sampler using rejection sampling. |
| 119 | |
| 120 | EXAMPLE:: |
| 121 | |
| 122 | sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection |
| 123 | sage: DiscreteGaussianSamplerRejection(3.0)() |
| 124 | -1 |
| 125 | sage: gs = DiscreteGaussianSamplerRejection(3.0, precision=100, tailcut=1.0) |
| 126 | sage: all(gs() <= 3.0 for _ in xrange(1000)) |
| 127 | True |
| 128 | |
| 129 | .. automethod:: __init__ |
| 130 | .. automethod:: __call__ |
| 131 | """ |
| 132 | def __init__(self, stddev, precision=53, tailcut=4): |
| 133 | """ |
| 134 | Construct a new discrete Gaussian sampler. |
| 135 | |
| 136 | INPUT: |
| 137 | |
| 138 | - ``stddev`` - standard deviation |
| 139 | - ``precision`` - precision used for internal computations (default: ``53``) |
| 140 | - ``tailcut`` - cut the tail at ``tailcut`` standard deviations (default: ``4``) |
| 141 | |
| 142 | EXAMPLE:: |
| 143 | |
| 144 | sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection |
| 145 | sage: gs = DiscreteGaussianSamplerRejection(3.0) |
| 146 | sage: sqrt(variance([gs() for _ in xrange(1000)])).n() |
| 147 | 2.965... |
| 148 | """ |
| 149 | self.stddev = stddev |
| 150 | self.precision = precision |
| 151 | self.tailcut = tailcut |
| 152 | self.max_precs = 2**precision |
| 153 | self.upper_bound = ZZ(round(tailcut*stddev)) |
| 154 | |
| 155 | def __call__(self): |
| 156 | """ |
| 157 | Return a new sample. |
| 158 | |
| 159 | EXAMPLE:: |
| 160 | |
| 161 | sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection |
| 162 | sage: sampler = DiscreteGaussianSamplerRejection(12.0) |
| 163 | sage: sampler() |
| 164 | -5 |
| 165 | """ |
| 166 | x = 0 |
| 167 | y = 0 |
| 168 | z = 0 |
| 169 | while y >= z : |
| 170 | x = randint(0, self.upper_bound-1) |
| 171 | y = randint(0, self.max_precs-1) |
| 172 | z = self.rho[x] |
| 173 | return (2*randint(0,1)-1)*x |
| 174 | |
| 175 | def _repr_(self): |
| 176 | """ |
| 177 | EXAMPLE:: |
| 178 | |
| 179 | sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection |
| 180 | sage: DiscreteGaussianSamplerRejection(3.0) |
| 181 | DiscreteGaussianSamplerRejection(3.000000, 53, 4) |
| 182 | """ |
| 183 | return "DiscreteGaussianSamplerRejection(%f, %d, %d)"%(self.stddev, self.precision, self.tailcut) |
| 184 | |
| 185 | |
| 186 | def __getattr__(self, name): |
| 187 | """ |
| 188 | EXAMPLE:: |
| 189 | |
| 190 | sage: from sage.crypto.lwe import DiscreteGaussianSamplerRejection |
| 191 | sage: DiscreteGaussianSamplerRejection(3.0).foo |
| 192 | Traceback (most recent call last): |
| 193 | ... |
| 194 | AttributeError: 'DiscreteGaussianSamplerRejection' object has no attribute 'foo' |
| 195 | """ |
| 196 | if name == "rho": |
| 197 | # we delay the creation of rho until we actually need it |
| 198 | R = RealField(self.precision) |
| 199 | self.rho = [round(self.max_precs * exp((-(R(x) / R(self.stddev))**2)/R(2))) for x in range(0,self.upper_bound)] |
| 200 | self.rho[0] = self.rho[0] / 2 |
| 201 | return self.rho |
| 202 | else: |
| 203 | raise AttributeError("'%s' object has no attribute '%s'"%(self.__class__.__name__, name)) |
| 204 | |
| 205 | # By default we use rejection sampling |
| 206 | DiscreteGaussianSampler = DiscreteGaussianSamplerRejection |
| 207 | |
| 208 | class DiscreteGaussianPolynomialSamplerRejection(SageObject): |
| 209 | """ |
| 210 | Discrete Gaussian sampler for polynomials. |
| 211 | |
| 212 | EXAMPLE:: |
| 213 | |
| 214 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection |
| 215 | sage: DiscreteGaussianPolynomialSamplerRejection(8, 3.0)() |
| 216 | x^7 - x^6 - 2*x^4 + 2*x^3 - x^2 + x - 1 |
| 217 | sage: gs = DiscreteGaussianPolynomialSamplerRejection(8, 3.0, precision=100, tailcut=1.0) |
| 218 | sage: [gs() for _ in xrange(3)] |
| 219 | [-x^7 + x^6 + 2*x^5 + 2*x^4 - x^3 - x^2 - 1, |
| 220 | x^7 - 2*x^6 + 2*x^5 + x^4 - x^3 + 2*x^2 - x + 2, |
| 221 | x^5 + 2*x^3 + 2*x + 1] |
| 222 | |
| 223 | .. automethod:: __init__ |
| 224 | .. automethod:: __call__ |
| 225 | """ |
| 226 | def __init__(self, n, stddev, precision=53, tailcut=4, D=DiscreteGaussianSampler): |
| 227 | """ |
| 228 | Construct a sampler for univariate polynomials of degree ``n-1`` |
| 229 | where coefficients are drawn independently with standard deviation |
| 230 | ``stddev`` using ``D``. |
| 231 | |
| 232 | INPUT: |
| 233 | |
| 234 | - ``n`` - number of coeficients to be sampled |
| 235 | - ``stddev`` - standard deviation |
| 236 | - ``precision`` - precision used for internal computations (default: ``53``) |
| 237 | - ``tailcut`` - cut the tail at ``tailcut`` standard deviations |
| 238 | (default: ``4``) |
| 239 | - ``D`` - a discrete Gaussian sampler (default: |
| 240 | :class:`DiscreteGaussianSampler`) |
| 241 | |
| 242 | EXAMPLE:: |
| 243 | |
| 244 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection |
| 245 | sage: DiscreteGaussianPolynomialSamplerRejection(8, 3.0)() |
| 246 | x^7 - x^6 - 2*x^4 + 2*x^3 - x^2 + x - 1 |
| 247 | sage: gs = DiscreteGaussianPolynomialSamplerRejection(8, 3.0, precision=100, tailcut=1.0) |
| 248 | sage: [gs() for _ in xrange(3)] |
| 249 | [-x^7 + x^6 + 2*x^5 + 2*x^4 - x^3 - x^2 - 1, |
| 250 | x^7 - 2*x^6 + 2*x^5 + x^4 - x^3 + 2*x^2 - x + 2, |
| 251 | x^5 + 2*x^3 + 2*x + 1] |
| 252 | """ |
| 253 | self.stddev = stddev |
| 254 | self.precision = precision |
| 255 | self.tailcut = tailcut |
| 256 | self.D = D(stddev, precision, tailcut) |
| 257 | self.n = ZZ(n) |
| 258 | self.P = ZZ['x'] |
| 259 | |
| 260 | def __call__(self): |
| 261 | """ |
| 262 | Return a new sample. |
| 263 | |
| 264 | EXAMPLE:: |
| 265 | |
| 266 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection |
| 267 | sage: sampler = DiscreteGaussianPolynomialSamplerRejection(8, 12.0) |
| 268 | sage: sampler() |
| 269 | x^7 - 9*x^5 + 2*x^4 + 8*x^3 - 5*x^2 + 7*x - 5 |
| 270 | """ |
| 271 | coeff = [self.D() for _ in range(self.n)] |
| 272 | f = self.P(coeff) |
| 273 | return f |
| 274 | |
| 275 | def _repr_(self): |
| 276 | """ |
| 277 | EXAMPLE:: |
| 278 | |
| 279 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSamplerRejection |
| 280 | sage: DiscreteGaussianPolynomialSamplerRejection(8, 3.0) |
| 281 | DiscreteGaussianPolynomialSamplerRejection(8, 3.000000, 53, 4) |
| 282 | """ |
| 283 | return "DiscreteGaussianPolynomialSamplerRejection(%d, %f, %d, %d)"%(self.n, self.stddev, self.precision, self.tailcut) |
| 284 | |
| 285 | |
| 286 | # By default we use rejection sampling |
| 287 | DiscreteGaussianPolynomialSampler = DiscreteGaussianPolynomialSamplerRejection |
| 288 | |
| 289 | class UniformSampler(SageObject): |
| 290 | """ |
| 291 | Uniform sampling in a range of integers. |
| 292 | |
| 293 | EXAMPLE:: |
| 294 | |
| 295 | sage: from sage.crypto.lwe import UniformSampler |
| 296 | sage: sampler = UniformSampler(-2, 2); sampler |
| 297 | UniformSampler(-2, 2) |
| 298 | sage: sampler() |
| 299 | -2 |
| 300 | |
| 301 | .. automethod:: __init__ |
| 302 | .. automethod:: __call__ |
| 303 | """ |
| 304 | def __init__(self, lower_bound, upper_bound): |
| 305 | """ |
| 306 | Construct a uniform sampler with bounds ``lower_bound`` and |
| 307 | ``upper_bound`` (both endpoints inclusive). |
| 308 | |
| 309 | INPUT: |
| 310 | |
| 311 | - ``lower_bound`` - integer |
| 312 | - ``upper_bound`` - integer |
| 313 | |
| 314 | EXAMPLE:: |
| 315 | |
| 316 | sage: from sage.crypto.lwe import UniformSampler |
| 317 | sage: UniformSampler(-2, 2) |
| 318 | UniformSampler(-2, 2) |
| 319 | """ |
| 320 | if lower_bound > upper_bound: |
| 321 | raise TypeError("lower bound must be <= than upper bound.") |
| 322 | self.lower_bound = ZZ(lower_bound) |
| 323 | self.upper_bound = ZZ(upper_bound) |
| 324 | |
| 325 | def __call__(self): |
| 326 | """ |
| 327 | Return a new sample. |
| 328 | |
| 329 | EXAMPLE:: |
| 330 | |
| 331 | sage: from sage.crypto.lwe import UniformSampler |
| 332 | sage: sampler = UniformSampler(-12, 12) |
| 333 | sage: sampler() |
| 334 | -10 |
| 335 | """ |
| 336 | return randint(self.lower_bound, self.upper_bound) |
| 337 | |
| 338 | def _repr_(self): |
| 339 | """ |
| 340 | EXAMPLE:: |
| 341 | |
| 342 | sage: from sage.crypto.lwe import UniformSampler |
| 343 | sage: UniformSampler(-2, 2) |
| 344 | UniformSampler(-2, 2) |
| 345 | """ |
| 346 | return "UniformSampler(%d, %d)"%(self.lower_bound, self.upper_bound) |
| 347 | |
| 348 | |
| 349 | class UniformPolynomialSampler(SageObject): |
| 350 | """ |
| 351 | uniform sampler for polynomials. |
| 352 | |
| 353 | EXAMPLE:: |
| 354 | |
| 355 | sage: from sage.crypto.lwe import UniformPolynomialSampler |
| 356 | sage: UniformPolynomialSampler(8, -2, 2)() |
| 357 | -2*x^7 + x^6 - 2*x^5 - x^3 - 2*x^2 - 2 |
| 358 | |
| 359 | .. automethod:: __init__ |
| 360 | .. automethod:: __call__ |
| 361 | """ |
| 362 | def __init__(self, n, lower_bound, upper_bound): |
| 363 | """ |
| 364 | Construct a sampler for univariate polynomials of degree ``n-1`` where |
| 365 | coefficients are drawn uniformly at random between ``lower_bound`` and |
| 366 | ``upper_bound`` (both endpoints inclusive). |
| 367 | |
| 368 | INPUT: |
| 369 | |
| 370 | - ``n`` - number of coeficients to be sampled |
| 371 | - ``lower_bound`` - integer |
| 372 | - ``upper_bound`` - integer |
| 373 | |
| 374 | EXAMPLE:: |
| 375 | |
| 376 | sage: from sage.crypto.lwe import UniformPolynomialSampler |
| 377 | sage: UniformPolynomialSampler(10, -10, 10) |
| 378 | UniformPolynomialSampler(10, -10, 10) |
| 379 | """ |
| 380 | self.n = ZZ(n) |
| 381 | self.P = ZZ['x'] |
| 382 | if lower_bound > upper_bound: |
| 383 | raise TypeError("lower bound must be <= than upper bound.") |
| 384 | self.lower_bound = ZZ(lower_bound) |
| 385 | self.upper_bound = ZZ(upper_bound) |
| 386 | self.D = UniformSampler(self.lower_bound, self.upper_bound) |
| 387 | |
| 388 | def __call__(self): |
| 389 | """ |
| 390 | Return a new sample. |
| 391 | |
| 392 | EXAMPLE:: |
| 393 | |
| 394 | sage: from sage.crypto.lwe import UniformPolynomialSampler |
| 395 | sage: sampler = UniformPolynomialSampler(8, -12, 12) |
| 396 | sage: sampler() |
| 397 | -10*x^7 + 5*x^6 - 8*x^5 + x^4 - 4*x^3 - 11*x^2 - 10 |
| 398 | """ |
| 399 | coeff = [self.D() for _ in range(self.n)] |
| 400 | f = self.P(coeff) |
| 401 | return f |
| 402 | |
| 403 | def _repr_(self): |
| 404 | """ |
| 405 | EXAMPLE:: |
| 406 | |
| 407 | sage: from sage.crypto.lwe import UniformPolynomialSampler |
| 408 | sage: UniformPolynomialSampler(8, -3, 3) |
| 409 | UniformPolynomialSampler(8, -3, 3) |
| 410 | """ |
| 411 | return "UniformPolynomialSampler(%d, %d, %d)"%(self.n, self.lower_bound, self.upper_bound) |
| 412 | |
| 413 | |
| 414 | class LWE(SageObject): |
| 415 | """ |
| 416 | Learning with Errors (LWE) oracle. |
| 417 | |
| 418 | .. automethod:: __init__ |
| 419 | .. automethod:: __call__ |
| 420 | """ |
| 421 | def __init__(self, n, q, D, secret_dist='uniform', m=None): |
| 422 | """ |
| 423 | Construct an LWE oracle in dimension ``n`` over a ring of order |
| 424 | ``q`` with noise distribution ``D``. |
| 425 | |
| 426 | INPUT: |
| 427 | |
| 428 | - ``n`` - dimension (integer > 0) |
| 429 | - ``q`` - modulus typically > n (integer > 0) |
| 430 | - ``D`` - an error distribution such as an instance of |
| 431 | :class:`DiscreteGaussianSamplerRejection` or :class:`UniformSampler` |
| 432 | - ``secret_dist`` - distribution of the secret; one of |
| 433 | |
| 434 | - "uniform" - secret follows the uniform distribution in `\Zmod{q}` |
| 435 | - "noise" - secret follows the noise distrbution |
| 436 | - ``(lb,ub)`` - the secret is chosen uniformly from ``[lb,...,ub]`` including both endpoints |
| 437 | |
| 438 | - ``m`` - number of allowed samples or ``None`` if no such limit exists |
| 439 | (default: ``None``) |
| 440 | |
| 441 | EXAMPLE: |
| 442 | |
| 443 | First, we construct a noise distribution with standard deviation 3.0:: |
| 444 | |
| 445 | sage: from sage.crypto.lwe import DiscreteGaussianSampler |
| 446 | sage: D = DiscreteGaussianSampler(3.0) |
| 447 | |
| 448 | Next, we construct our oracle:: |
| 449 | |
| 450 | sage: from sage.crypto.lwe import LWE |
| 451 | sage: lwe = LWE(n=20, q=next_prime(400), D=D); lwe |
| 452 | LWE(20, 401, DiscreteGaussianSamplerRejection(3.000000, 53, 4), 'uniform', None) |
| 453 | |
| 454 | and sample 1000 samples:: |
| 455 | |
| 456 | sage: L = [lwe() for _ in range(1000)] |
| 457 | |
| 458 | To test the oracle, we use the internal secret to evaluate the samples |
| 459 | in the secret:: |
| 460 | |
| 461 | sage: S = [ZZ(a.dot_product(lwe._LWE__s) - c) for (a,c) in L] |
| 462 | |
| 463 | However, while Sage represents finite field elements between 0 and q-1 |
| 464 | we rely on a balanced representation of those elements here. Hence, we |
| 465 | fix the representation and recover the correct standard deviation of the |
| 466 | noise:: |
| 467 | |
| 468 | sage: sqrt(variance([e if e <= 200 else e-401 for e in S]).n()) |
| 469 | 3.0... |
| 470 | |
| 471 | If ``m`` is not ``None`` the number of available samples is restricted:: |
| 472 | |
| 473 | sage: from sage.crypto.lwe import LWE |
| 474 | sage: lwe = LWE(n=20, q=next_prime(400), D=D, m=30) |
| 475 | sage: _ = [lwe() for _ in range(30)] |
| 476 | sage: lwe() # 31 |
| 477 | Traceback (most recent call last): |
| 478 | ... |
| 479 | IndexError: Number of available samples exhausted. |
| 480 | """ |
| 481 | self.n = ZZ(n) |
| 482 | self.m = m |
| 483 | self.__i = 0 |
| 484 | self.K = IntegerModRing(q) |
| 485 | self.FM = FreeModule(self.K, n) |
| 486 | self.D = D |
| 487 | |
| 488 | self.secret_dist = secret_dist |
| 489 | if secret_dist == 'uniform': |
| 490 | self.__s = random_vector(self.K, self.n) |
| 491 | elif secret_dist == 'noise': |
| 492 | self.__s = vector(self.K, self.n, [self.D() for _ in range(n)]) |
| 493 | else: |
| 494 | try: |
| 495 | lb, ub = map(ZZ,secret_dist) |
| 496 | self.__s = vector(self.K, self.n, [randint(lb,ub) for _ in range(n)]) |
| 497 | except (IndexError, TypeError): |
| 498 | raise TypeError("Parameter secret_dist=%s not understood."%(secret_dist)) |
| 499 | |
| 500 | def _repr_(self): |
| 501 | """ |
| 502 | EXAMPLE:: |
| 503 | |
| 504 | sage: from sage.crypto.lwe import DiscreteGaussianSampler, LWE |
| 505 | sage: D = DiscreteGaussianSampler(3.0) |
| 506 | sage: lwe = LWE(n=20, q=next_prime(400), D=D); lwe |
| 507 | LWE(20, 401, DiscreteGaussianSamplerRejection(3.000000, 53, 4), 'uniform', None) |
| 508 | |
| 509 | sage: lwe = LWE(n=20, q=next_prime(400), D=D, secret_dist=(-3, 3)); lwe |
| 510 | LWE(20, 401, DiscreteGaussianSamplerRejection(3.000000, 53, 4), (-3, 3), None) |
| 511 | """ |
| 512 | if type(self.secret_dist) == str: |
| 513 | return "LWE(%d, %d, %s, '%s', %s)"%(self.n,self.K.order(),self.D,self.secret_dist, self.m) |
| 514 | else: |
| 515 | return "LWE(%d, %d, %s, %s, %s)"%(self.n,self.K.order(),self.D,self.secret_dist, self.m) |
| 516 | |
| 517 | |
| 518 | def __call__(self): |
| 519 | """ |
| 520 | EXAMPLE:: |
| 521 | |
| 522 | sage: from sage.crypto.lwe import DiscreteGaussianSampler, LWE |
| 523 | sage: LWE(10, 401, DiscreteGaussianSampler(3))() |
| 524 | ((309, 347, 198, 194, 336, 360, 264, 123, 368, 398), 198) |
| 525 | """ |
| 526 | if self.m is not None: |
| 527 | if self.__i >= self.m: |
| 528 | raise IndexError("Number of available samples exhausted.") |
| 529 | self.__i+=1 |
| 530 | a = self.FM.random_element() |
| 531 | return a, a.dot_product(self.__s) + self.K(self.D()) |
| 532 | |
| 533 | |
| 534 | class Regev(LWE): |
| 535 | """ |
| 536 | LWE oracle with parameters as in [Reg09]_. |
| 537 | |
| 538 | .. automethod:: __init__ |
| 539 | """ |
| 540 | def __init__(self, n, secret_dist='uniform', m=None): |
| 541 | """ |
| 542 | Construct LWE instance parameterised by security paramter ``n`` where |
| 543 | the modulus ``q`` and the ``stddev`` of the noise are chosen as in |
| 544 | [Reg09]_. |
| 545 | |
| 546 | INPUT: |
| 547 | |
| 548 | - ``n`` - security paramter (integer > 0) |
| 549 | - ``secret_dist`` - distribution of the secret. See documentation of :class:`LWE` |
| 550 | for details (default='uniform') |
| 551 | - ``m`` - number of allowed samples or ``None`` if no such limit exists |
| 552 | (default: ``None``) |
| 553 | |
| 554 | EXAMPLES:: |
| 555 | |
| 556 | sage: from sage.crypto.lwe import Regev |
| 557 | sage: Regev(n=20) |
| 558 | LWE(20, 401, DiscreteGaussianSamplerRejection(1.915069, 401, 4), 'uniform', None) |
| 559 | """ |
| 560 | q = ZZ(next_prime(n**2)) |
| 561 | s = RR(1/(RR(n).sqrt() * log(n, 2)**2) * q) |
| 562 | D = DiscreteGaussianSampler(s/sqrt(2*pi.n()), q) |
| 563 | LWE.__init__(self, n=n, q=q, D=D, secret_dist=secret_dist, m=m) |
| 564 | |
| 565 | class LindnerPeikert(LWE): |
| 566 | """ |
| 567 | LWE oracle with parameters as in [LP11]_. |
| 568 | |
| 569 | .. automethod:: __init__ |
| 570 | """ |
| 571 | def __init__(self, n, delta=0.01, m=None): |
| 572 | """ |
| 573 | Construct LWE instance parameterised by security paramter ``n`` where |
| 574 | the modulus ``q`` and the ``stddev`` of the noise is chosen as in |
| 575 | [LP11]_. |
| 576 | |
| 577 | INPUT: |
| 578 | |
| 579 | - ``n`` - security paramter (integer > 0) |
| 580 | - ``delta`` - error probability per symbol (default: 0.01) |
| 581 | - ``m`` - number of allowed samples or ``None`` in which case ``m=2*n + |
| 582 | 128`` as in [LP11]_ (default: ``None``) |
| 583 | |
| 584 | EXAMPLES:: |
| 585 | |
| 586 | sage: from sage.crypto.lwe import LindnerPeikert |
| 587 | sage: LindnerPeikert(n=20) |
| 588 | LWE(20, 2053, DiscreteGaussianSamplerRejection(3.600954, 53, 4), 'noise', 168) |
| 589 | """ |
| 590 | if m is None: |
| 591 | m = 2*n + 128 |
| 592 | # Find c>=1 such that c*exp((1-c**2)/2))**(2*n) == 2**-40 |
| 593 | # (c*exp((1-c**2)/2))**(2*n) == 2**-40 |
| 594 | # log((c*exp((1-c**2)/2))**(2*n)) == -40*log(2) |
| 595 | # (2*n)*log(c*exp((1-c**2)/2)) == -40*log(2) |
| 596 | # 2*n*(log(c)+log(exp((1-c**2)/2))) == -40*log(2) |
| 597 | # 2*n*(log(c)+(1-c**2)/2) == -40*log(2) |
| 598 | # 2*n*log(c)+n*(1-c**2) == -40*log(2) |
| 599 | # 2*n*log(c)+n*(1-c**2) + 40*log(2) == 0 |
| 600 | c = var('c') |
| 601 | c = find_root(2*n*log(c)+n*(1-c**2) + 40*log(2) == 0, 1, 10) |
| 602 | # Upper bound on s**2/t |
| 603 | s_t_bound = (sqrt(2) * pi / c / sqrt(2*n*log(2/delta))).n() |
| 604 | # Interpretation of "choose q just large enough to allow for a Gaussian parameter s>=8" in [LP11]_ |
| 605 | q = next_prime(floor(2**round(log(256 / s_t_bound, 2)))) |
| 606 | # Gaussian parameter as defined in [LP11]_ |
| 607 | s = sqrt(s_t_bound*floor(q/4)) |
| 608 | # Transform s into stddev |
| 609 | stddev = s/sqrt(2*pi.n()) |
| 610 | D = DiscreteGaussianSampler(stddev) |
| 611 | LWE.__init__(self, n=n, q=q, D=D, secret_dist='noise', m=m) |
| 612 | |
| 613 | |
| 614 | class UniformNoiseLWE(LWE): |
| 615 | """ |
| 616 | LWE oracle with uniform sectet with parameters as in [CGW13]_. |
| 617 | |
| 618 | .. automethod:: __init__ |
| 619 | """ |
| 620 | def __init__(self, n, instance='key', m=None): |
| 621 | """ |
| 622 | Construct LWE instance parameterised by security paramter ``n`` where |
| 623 | all other parameters are chosen as in [CGW13]_. |
| 624 | |
| 625 | INPUT: |
| 626 | |
| 627 | - ``n`` - security paramter (integer >= 80) |
| 628 | - ``instance`` - one of |
| 629 | |
| 630 | - "key" - the LWE-instance that hides the secret key is generated |
| 631 | - "encrypt" - the LWE-instance that hides the message is generated |
| 632 | (default: ``key``) |
| 633 | - ``m`` - number of allowed samples or ``None`` in which case ``m`` is |
| 634 | chosen as in [CGW13_]. (default: ``None``) |
| 635 | |
| 636 | EXAMPLES:: |
| 637 | |
| 638 | sage: from sage.crypto.lwe import UniformNoiseLWE |
| 639 | sage: UniformNoiseLWE(89) |
| 640 | LWE(89, 154262477, UniformSampler(0, 351), 'noise', 131) |
| 641 | |
| 642 | sage: UniformNoiseLWE(89, instance='encrypt') |
| 643 | LWE(131, 154262477, UniformSampler(0, 497), 'noise', 181) |
| 644 | """ |
| 645 | |
| 646 | if n<89: |
| 647 | raise TypeError("Parameter too small") |
| 648 | |
| 649 | n2 = n |
| 650 | C = 4/sqrt(2*pi) |
| 651 | kk = floor((n2-2*log(n2, 2)**2)/5) |
| 652 | n1 = floor((3*n2-5*kk)/2) |
| 653 | ke = floor((n1-2*log(n1, 2)**2)/5) |
| 654 | l = floor((3*n1-5*ke)/2)-n2 |
| 655 | sk = ceil((C*(n1+n2))**(3/2)) |
| 656 | se = ceil((C*(n1+n2+l))**(3/2)) |
| 657 | 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))) |
| 658 | |
| 659 | if kk<=0: |
| 660 | raise TypeError("Parameter too small") |
| 661 | |
| 662 | if instance == 'key': |
| 663 | D = UniformSampler(0, sk-1) |
| 664 | if m is None: |
| 665 | m = n1 |
| 666 | LWE.__init__(self, n=n2, q=q, D=D, secret_dist='noise', m=m) |
| 667 | elif instance == 'encrypt': |
| 668 | D = UniformSampler(0, se-1) |
| 669 | if m is None: |
| 670 | m = n2+l |
| 671 | LWE.__init__(self, n=n1, q=q, D=D, secret_dist='noise', m=m) |
| 672 | else: |
| 673 | raise TypeError("Parameter instance=%s not understood."%(instance)) |
| 674 | |
| 675 | class RingLWE(SageObject): |
| 676 | """ |
| 677 | Ring Learning with Errors oracle. |
| 678 | |
| 679 | .. automethod:: __init__ |
| 680 | .. automethod:: __call__ |
| 681 | """ |
| 682 | def __init__(self, N, q, D, poly=None, secret_dist='uniform', m=None): |
| 683 | """ |
| 684 | Construct a Ring-LWE oracle in dimension ``n=phi(N)`` over a ring of order |
| 685 | ``q`` with noise distribution ``D``. |
| 686 | |
| 687 | INPUT: |
| 688 | |
| 689 | - ``N`` - index of cyclotomic polynomial (integer > 0, must be power of 2) |
| 690 | - ``q`` - modulus typically > N (integer > 0) |
| 691 | - ``D`` - an error distribution such as an instance of |
| 692 | :class:`DiscreteGaussianPolynomialSamplerRejection` or :class:`UniformSampler` |
| 693 | - ``poly`` - a polynomial of degree ``phi(N)``. If ``None`` the |
| 694 | cyclotomic polynomial used (default: ``None``). |
| 695 | - ``secret_dist`` - distribution of the secret. See documentation of |
| 696 | :class:`LWE` for details (default='uniform') |
| 697 | - ``m`` - number of allowed samples or ``None`` if no such limit exists |
| 698 | (default: ``None``) |
| 699 | |
| 700 | EXAMPLE:: |
| 701 | |
| 702 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE |
| 703 | sage: D = DiscreteGaussianPolynomialSampler(n=euler_phi(20), stddev=3.0) |
| 704 | sage: RingLWE(N=20, q=next_prime(800), D=D); |
| 705 | RingLWE(20, 809, DiscreteGaussianPolynomialSamplerRejection(8, 3.000000, 53, 4), x^8 - x^6 + x^4 - x^2 + 1, 'uniform', None) |
| 706 | """ |
| 707 | self.N = ZZ(N) |
| 708 | self.n = euler_phi(N) |
| 709 | self.m = m |
| 710 | self.__i = 0 |
| 711 | self.K = IntegerModRing(q) |
| 712 | |
| 713 | if self.n != D.n: |
| 714 | raise ValueError("Noise distribution has dimensions %d != %d"%(D.n, self.n)) |
| 715 | |
| 716 | self.D = D |
| 717 | self.q = q |
| 718 | if poly is not None: |
| 719 | self.poly = poly |
| 720 | else: |
| 721 | self.poly = cyclotomic_polynomial(self.N, 'x') |
| 722 | |
| 723 | self.R_q = self.K['x'].quotient(self.poly, 'x') |
| 724 | |
| 725 | self.secret_dist = secret_dist |
| 726 | if secret_dist == 'uniform': |
| 727 | self.__s = self.R_q.random_element() # uniform sampling of secret |
| 728 | elif secret_dist == 'noise': |
| 729 | self.__s = self.D() |
| 730 | else: |
| 731 | raise TypeError("Parameter secret_dist=%s not understood."%(secret_dist)) |
| 732 | |
| 733 | def _repr_(self): |
| 734 | """ |
| 735 | EXAMPLE:: |
| 736 | |
| 737 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE |
| 738 | sage: D = DiscreteGaussianPolynomialSampler(n=8, stddev=3.0) |
| 739 | sage: RingLWE(N=16, q=next_prime(400), D=D); |
| 740 | RingLWE(16, 401, DiscreteGaussianPolynomialSamplerRejection(8, 3.000000, 53, 4), x^8 + 1, 'uniform', None) |
| 741 | """ |
| 742 | if type(self.secret_dist) == str: |
| 743 | return "RingLWE(%d, %d, %s, %s, '%s', %s)"%(self.N, self.K.order(), self.D, self.poly, self.secret_dist, self.m) |
| 744 | else: |
| 745 | return "RingLWE(%d, %d, %s, %s, %s, %s)"%(self.N, self.K.order(), self.D, self.poly, self.secret_dist, self.m) |
| 746 | |
| 747 | |
| 748 | def __call__(self): |
| 749 | """ |
| 750 | EXAMPLE:: |
| 751 | |
| 752 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE |
| 753 | sage: N = 16 |
| 754 | sage: n = euler_phi(N) |
| 755 | sage: D = DiscreteGaussianPolynomialSampler(n, 5) |
| 756 | sage: ringlwe = RingLWE(N, 257, D, secret_dist='uniform') |
| 757 | sage: ringlwe() |
| 758 | ((228, 149, 226, 198, 38, 222, 222, 127), (177, 138, 68, 134, 74, 162, 203, 243)) |
| 759 | """ |
| 760 | if self.m is not None: |
| 761 | if self.__i >= self.m: |
| 762 | raise IndexError("Number of available samples exhausted.") |
| 763 | self.__i+=1 |
| 764 | a = self.R_q.random_element() |
| 765 | return vector(a), vector(a * (self.__s) + self.D()) |
| 766 | |
| 767 | class RingLindnerPeikert(RingLWE): |
| 768 | """ |
| 769 | Ring-LWE oracle with parameters as in [LP11]_. |
| 770 | |
| 771 | .. automethod:: __init__ |
| 772 | """ |
| 773 | def __init__(self, N, delta=0.01, m=None): |
| 774 | """ |
| 775 | Construct a Ring-LWE oracle in dimension ``n=phi(N)`` where |
| 776 | the modulus ``q`` and the ``stddev`` of the noise is chosen as in |
| 777 | [LP11]_. |
| 778 | |
| 779 | INPUT: |
| 780 | |
| 781 | - ``N`` - index of cyclotomic polynomial (integer > 0, must be power of 2) |
| 782 | - ``delta`` - error probability per symbol (default: 0.01) |
| 783 | - ``m`` - number of allowed samples or ``None`` in which case ``3*n`` is |
| 784 | used (default: ``None``) |
| 785 | |
| 786 | EXAMPLES:: |
| 787 | |
| 788 | sage: from sage.crypto.lwe import RingLindnerPeikert |
| 789 | sage: RingLindnerPeikert(N=16) |
| 790 | RingLWE(16, 1031, DiscreteGaussianPolynomialSamplerRejection(8, 2.803372, 53, 4), x^8 + 1, 'noise', 24) |
| 791 | """ |
| 792 | n = euler_phi(N) |
| 793 | if m is None: |
| 794 | m = 3*n |
| 795 | # Find c>=1 such that c*exp((1-c**2)/2))**(2*n) == 2**-40 |
| 796 | # i.e c>=1 such that 2*n*log(c)+n*(1-c**2) + 40*log(2) == 0 |
| 797 | c = var('c') |
| 798 | c = find_root(2*n*log(c)+n*(1-c**2) + 40*log(2) == 0, 1, 10) |
| 799 | # Upper bound on s**2/t |
| 800 | s_t_bound = (sqrt(2) * pi / c / sqrt(2*n*log(2/delta))).n() |
| 801 | # Interpretation of "choose q just large enough to allow for a Gaussian parameter s>=8" in [LP11]_ |
| 802 | q = next_prime(floor(2**round(log(256 / s_t_bound, 2)))) |
| 803 | # Gaussian parameter as defined in [LP11]_ |
| 804 | s = sqrt(s_t_bound*floor(q/4)) |
| 805 | # Transform s into stddev |
| 806 | stddev = s/sqrt(2*pi.n()) |
| 807 | D = DiscreteGaussianPolynomialSampler(n, stddev) |
| 808 | RingLWE.__init__(self, N=N, q=q, D=D, poly=None, secret_dist='noise', m=m) |
| 809 | |
| 810 | class RingLWEConverter(SageObject): |
| 811 | """ |
| 812 | Wrapper callable to convert Ring-LWE oracles into LWE oracles by |
| 813 | disregarding the additional structure. |
| 814 | |
| 815 | .. automethod:: __init__ |
| 816 | .. automethod:: __call__ |
| 817 | """ |
| 818 | def __init__(self, ringlwe): |
| 819 | """ |
| 820 | INPUT: |
| 821 | |
| 822 | - ``ringlwe`` - an instance of a :class:`RingLWE` |
| 823 | |
| 824 | EXAMPLE:: |
| 825 | |
| 826 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE, RingLWEConverter |
| 827 | sage: D = DiscreteGaussianPolynomialSampler(euler_phi(16), 5) |
| 828 | sage: lwe = RingLWEConverter(RingLWE(16, 257, D, secret_dist='uniform')) |
| 829 | sage: set_random_seed(1337) |
| 830 | sage: lwe() |
| 831 | ((130, 32, 216, 3, 125, 58, 197, 171), 182) |
| 832 | """ |
| 833 | self.ringlwe = ringlwe |
| 834 | self._i = 0 |
| 835 | self._ac = None |
| 836 | self.n = self.ringlwe.n |
| 837 | |
| 838 | def __call__(self): |
| 839 | """ |
| 840 | EXAMPLE:: |
| 841 | |
| 842 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE, RingLWEConverter |
| 843 | sage: D = DiscreteGaussianPolynomialSampler(euler_phi(16), 5) |
| 844 | sage: lwe = RingLWEConverter(RingLWE(16, 257, D, secret_dist='uniform')) |
| 845 | sage: set_random_seed(1337) |
| 846 | sage: lwe() |
| 847 | ((130, 32, 216, 3, 125, 58, 197, 171), 182) |
| 848 | """ |
| 849 | R_q = self.ringlwe.R_q |
| 850 | |
| 851 | if (self._i % self.n) == 0: |
| 852 | self._ac = self.ringlwe() |
| 853 | a, c = self._ac |
| 854 | x = R_q.gen() |
| 855 | r = vector((x**(self._i % self.n) * R_q(a.list())).list()), c[self._i % self.n] |
| 856 | self._i += 1 |
| 857 | return r |
| 858 | |
| 859 | def _repr_(self): |
| 860 | """ |
| 861 | EXAMPLE:: |
| 862 | |
| 863 | sage: from sage.crypto.lwe import DiscreteGaussianPolynomialSampler, RingLWE, RingLWEConverter |
| 864 | sage: D = DiscreteGaussianPolynomialSampler(euler_phi(20), 5) |
| 865 | sage: rlwe = RingLWE(20, 257, D) |
| 866 | sage: lwe = RingLWEConverter(rlwe) |
| 867 | sage: lwe |
| 868 | RingLWEConverter(RingLWE(20, 257, DiscreteGaussianPolynomialSamplerRejection(8, 5.000000, 53, 4), x^8 - x^6 + x^4 - x^2 + 1, 'uniform', None)) |
| 869 | |
| 870 | """ |
| 871 | return "RingLWEConverter(%s)"%str(self.ringlwe) |
| 872 | |
| 873 | def samples(m, n, lwe, seed=None, balanced=False, **kwds): |
| 874 | """ |
| 875 | Return ``m`` LWE samples. |
| 876 | |
| 877 | INPUT: |
| 878 | |
| 879 | - ``m`` - the number of samples (integer > 0) |
| 880 | - ``n`` - the security parameter (integer > 0) |
| 881 | - ``lwe`` - either |
| 882 | |
| 883 | - a subclass of :class:`LWE` such as :class:`Regev` or :class:`LindnerPeikert` |
| 884 | - an instance of :class:`LWE` or any subclass |
| 885 | - the name of any such class (e.g., "Regev", "LindnerPeikert") |
| 886 | |
| 887 | - ``seed`` - seed to be used for generation or ``None`` if no specific seed |
| 888 | shall be set (default: ``None``) |
| 889 | - ``balanced`` - use function :func:`balance_sample` to return balanced |
| 890 | representations of finite field elements (default: ``False``) |
| 891 | - ``**kwds`` - passed through to LWE constructor |
| 892 | |
| 893 | EXAMPLE:: |
| 894 | |
| 895 | sage: from sage.crypto.lwe import samples, Regev |
| 896 | sage: samples(2, 20, Regev, seed=1337) |
| 897 | [((199, 388, 337, 53, 200, 284, 336, 215, 75, 14, 274, 234, 97, 255, 246, 153, 268, 218, 396, 351), 18), |
| 898 | ((286, 42, 175, 155, 190, 275, 114, 280, 45, 218, 304, 386, 98, 235, 77, 0, 65, 20, 163, 14), 334)] |
| 899 | |
| 900 | sage: from sage.crypto.lwe import samples, Regev |
| 901 | sage: samples(2, 20, Regev, balanced=True, seed=1337) |
| 902 | [((199, -13, -64, 53, 200, -117, -65, -186, 75, 14, -127, -167, 97, -146, -155, 153, -133, -183, -5, -50), 18), |
| 903 | ((-115, 42, 175, 155, 190, -126, 114, -121, 45, -183, -97, -15, 98, -166, 77, 0, 65, 20, 163, 14), -67)] |
| 904 | |
| 905 | sage: from sage.crypto.lwe import samples |
| 906 | sage: samples(2, 20, 'LindnerPeikert') |
| 907 | [((1302, 718, 1397, 147, 278, 979, 1185, 133, 902, 1180, 1264, 734, 2029, 314, 428, 18, 707, 2021, 1153, 173), 1127), |
| 908 | ((2015, 1278, 455, 429, 1391, 186, 149, 1199, 220, 1629, 843, 719, 1744, 1568, 674, 1462, 1549, 972, 248, 1066), 1422)] |
| 909 | |
| 910 | """ |
| 911 | if seed is not None: |
| 912 | set_random_seed(seed) |
| 913 | |
| 914 | if isinstance(lwe, str): |
| 915 | lwe = eval(lwe) |
| 916 | |
| 917 | if type(lwe) == type: |
| 918 | lwe = lwe(n, m=m, **kwds) |
| 919 | else: |
| 920 | lwe = lwe |
| 921 | if lwe.n != n: |
| 922 | raise ValueError("Passed LWE instance has n=%d, but n=%d was passed to this function."%(lwe.n, n)) |
| 923 | |
| 924 | if balanced is False: |
| 925 | f = lambda (a,c): (a,c) |
| 926 | else: |
| 927 | f = balance_sample |
| 928 | return [f(lwe()) for _ in xrange(m)] |
| 929 | |
| 930 | def balance_sample(s, q=None): |
| 931 | r""" |
| 932 | Given ``(a,c) = s`` return a tuple ``(a',c')`` where ``a'`` is a integer |
| 933 | vector with entries between -q//2 and q//2 and ``c`` is also within these |
| 934 | bounds. |
| 935 | |
| 936 | If ``q`` is given ``(a,c) = s`` may live in the integers. If ``q`` is not |
| 937 | given, then ``(a,c)`` are assumed to live in `\Zmod{q}`. |
| 938 | |
| 939 | INPUT: |
| 940 | |
| 941 | - ``s`` - sample of the form (a,c) where a is a vector and c is a scalar |
| 942 | - ``q`` - modulus (default: ``None``) |
| 943 | |
| 944 | EXAMPLE:: |
| 945 | |
| 946 | sage: from sage.crypto.lwe import balance_sample, samples, Regev |
| 947 | sage: map(balance_sample, samples(10, 5, Regev)) |
| 948 | [((-9, -4, -4, 4, -4), 6), ((-3, -10, 8, -3, -1), -10), ((-6, -12, -3, -2, -6), -6), |
| 949 | ... |
| 950 | ((-1, -8, -11, 13, 4), -6), ((10, 11, -3, -13, 0), 6), ((6, -1, 2, -11, 14), 2)] |
| 951 | |
| 952 | |
| 953 | sage: from sage.crypto.lwe import balance_sample, DiscreteGaussianPolynomialSampler, RingLWE, samples |
| 954 | sage: D = DiscreteGaussianPolynomialSampler(8, 5) |
| 955 | sage: rlwe = RingLWE(20, 257, D) |
| 956 | sage: map(balance_sample, samples(10, 8, rlwe)) |
| 957 | [((5, -55, -31, -90, 6, 100, -46, -107), (6, -64, -40, 117, 27, 54, -98, -56)), |
| 958 | ((109, -106, 28, 77, -14, -109, 115, 34), (82, 17, -89, 62, 1, -77, 128, 64)), |
| 959 | ... |
| 960 | ((-32, 51, -110, -106, 35, -82, 14, -113), (126, -120, 126, 119, 101, 3, -122, -75))] |
| 961 | |
| 962 | .. note:: |
| 963 | |
| 964 | This function is useful to convert between Sage's standard |
| 965 | representation of elements in `\Zmod{q}` as integers between 0 and q-1 |
| 966 | and the usual representation of such elements in lattice cryptography as |
| 967 | integers between -q//2 and q//2. |
| 968 | """ |
| 969 | a, c = s |
| 970 | |
| 971 | try: |
| 972 | c[0] |
| 973 | scalar = False |
| 974 | except TypeError: |
| 975 | c = vector(c.parent(),[c]) |
| 976 | scalar = True |
| 977 | |
| 978 | if q is None: |
| 979 | q = parent(c[0]).order() |
| 980 | a = a.change_ring(ZZ) |
| 981 | c = c.change_ring(ZZ) |
| 982 | else: |
| 983 | K = IntegerModRing(q) |
| 984 | a = a.change_ring(K).change_ring(ZZ) |
| 985 | c = c.change_ring(K).change_ring(ZZ) |
| 986 | |
| 987 | q2 = q//2 |
| 988 | |
| 989 | if scalar: |
| 990 | 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 |
| 991 | else: |
| 992 | 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]) |