Security considerations for Galois non-dual RLWE families

We explore further the hardness of the non-dual discrete variant of the Ring-LWE problem for various number rings, give improved attacks for certain rings satisfying some additional assumptions, construct a new family of vulnerable Galois number fields, and apply some number theoretic results on Gauss sums to deduce the likely failure of these attacks for 2-power cyclotomic rings and unramified moduli

Provably weak instances of ring-LWE revisited

In CRYPTO 2015, Elias, Lauter, Ozman and Stange described an attack on the non-dual decision version of the ring learning with errors problem (RLWE) for two special families of defining polynomials, whose construction depends on the modulus q that is being used. For particularly chosen error parameters, they managed to solve non-dual decision RLWE given 20 samples, with a success rate ranging from 10% to 80%. In this paper we show how to solve the search version for the same families and error parameters, using only 7 samples with a success rate of 100%. Moreover our attack works for every modulus q instead of the q that was used to construct the defining polynomial. The attack is based on the observation that the RLWE error distribution for these families of polynomials is very skewed in the directions of the polynomial basis. For the parameters chosen by Elias et al. the smallest errors are negligible and simple linear algebra suffices to recover the secret. But enlarging the error paremeters makes the largest errors wrap around, thereby turning the RLWE problem unsuitable for cryptographic applications. These observations also apply to dual RLWE, but do not contradict the seminal work by Lyubashevsky, Peikert and Regev

Fast Fourier Orthogonalization

The classical fast Fourier transform (FFT) allows to compute in quasi-linear time the product of two polynomials, in the {\em circular convolution ring} R[x]/(x^d−1) --- a task that naively requires quadratic time. Equivalently, it allows to accelerate matrix-vector products when the matrix is *circulant*. In this work, we discover that the ideas of the FFT can be applied to speed up the orthogonalization process of matrices with circulant blocks of size d×d . We show that, when d is composite, it is possible to proceed to the orthogonalization in an inductive way ---up to an appropriate re-indexation of rows and columns. This leads to a structured Gram-Schmidt decomposition. In turn, this structured Gram-Schmidt decomposition accelerates a cornerstone lattice algorithm: the nearest plane algorithm. The complexity of both algorithms may be brought down to Θ(dlogd). Our results easily extend to *cyclotomic rings*, and can be adapted to Gaussian samplers. This finds applications in lattice-based cryptography, improving the performances of trapdoor functions

Cystic Echinococcosis: Late Rupture and Complication of a Stable Pulmonary Cyst

Cystic echinococcosis is observed worldwide. Traditional management includes an invasive surgical approach with adjunctive chemotherapy. It has been suggested that observation alone may be appropriate in asymptomatic individuals with stable cysts. A case involving a 38-year-old Peruvian man with an asymptomatic bronchogenic cyst (suspected to be due to echinococcus, but never definitely diagnosed) is presented. The cyst was first noted in 1998, and was followed for 10 years during which time he remained asymptomatic with minimal radiographic change. One year later, in 2009, he presented with acute rupture of the cyst causing empyema. The patient required thoracotomy, decortication and resection of the ruptured cyst. Final pathology showed Echinococcus organisms. The patient responded well to treatment with albendazole and praziquantel, and became completely asymptomatic within six months. The present case demonstrates that echinococcal cysts may be at risk of spontaneous rupture, even after many years of clinical stability, thus supporting the case for resection of asymptomatic cysts suspected of being echinococcal at the time of diagnosis. In addition, the case illustrates that medical therapy with albendazole and praziquantel, in conjunction with surgical drainage, can be successful in the treatment of echinococcal empyema

Saber:module-LWR based key exchange, CPA-secure encryption and CCA-secure KEM

© Springer International Publishing AG, part of Springer Nature 2018. In this paper, we introduce Saber, a package of cryptographic primitives whose security relies on the hardness of the Module Learning With Rounding problem (Mod-LWR). We first describe a secure Diffie-Hellman type key exchangeprotocol, which is then transformed into an IND-CPA encryption scheme and finally into an IND-CCA secure key encapsulation mechanism using a post-quantum version of the Fujisaki-Okamoto transform. The design goals of this package were simplicity, efficiency and flexibility resulting in the following choices: all integer moduli are powers of 2 avoiding modular reduction and rejection sampling entirely; the use of LWR halves the amount of randomness required compared to LWE-based schemes and reduces bandwidth; the module structure provides flexibility by reusing one core component for multiple security levels. A constant-time AVX2 optimized software implementation of the KEM with parameters providing more than 128 bits of post-quantum security, requires only 101K, 125K and 129K cycles for key generation, encapsulation and decapsulation respectively on a Dell laptop with an Intel i7-Haswell processor

Ring-LWE:applications to cryptography and their efficient realization

© Springer International Publishing AG 2016. The persistent progress of quantum computing with algorithms of Shor and Proos and Zalka has put our present RSA and ECC based public key cryptosystems at peril. There is a flurry of activity in cryptographic research community to replace classical cryptography schemes with their post-quantum counterparts. The learning with errors problem introduced by Oded Regev offers a way to design secure cryptography schemes in the post-quantum world. Later for efficiency LWE was adapted for ring polynomials known as Ring-LWE. In this paper we discuss some of these ring-LWE based schemes that have been designed. We have also drawn comparisons of different implementations of those schemes to illustrate their evolution from theoretical proposals to practically feasible schemes

An Improved BKW Algorithm for LWE with Applications to Cryptography and Lattices

In this paper, we study the Learning With Errors problem and its binary variant, where secrets and errors are binary or taken in a small interval. We introduce a new variant of the Blum, Kalai and Wasserman algorithm, relying on a quantization step that generalizes and fine-tunes modulus switching. In general this new technique yields a significant gain in the constant in front of the exponent in the overall complexity. We illustrate this by solving p within half a day a LWE instance with dimension n = 128, modulus $q = n^2$, Gaussian noise $\alpha = 1/(\sqrt{n/\pi} \log^2 n)$ and binary secret, using $2^{28}$ samples, while the previous best result based on BKW claims a time complexity of $2^{74}$ with $2^{60}$ samples for the same parameters. We then introduce variants of BDD, GapSVP and UniqueSVP, where the target point is required to lie in the fundamental parallelepiped, and show how the previous algorithm is able to solve these variants in subexponential time. Moreover, we also show how the previous algorithm can be used to solve the BinaryLWE problem with n samples in subexponential time $2^{(\ln 2/2+o(1))n/\log \log n}$. This analysis does not require any heuristic assumption, contrary to other algebraic approaches; instead, it uses a variant of an idea by Lyubashevsky to generate many samples from a small number of samples. This makes it possible to asymptotically and heuristically break the NTRU cryptosystem in subexponential time (without contradicting its security assumption). We are also able to solve subset sum problems in subexponential time for density $o(1)$, which is of independent interest: for such density, the previous best algorithm requires exponential time. As a direct application, we can solve in subexponential time the parameters of a cryptosystem based on this problem proposed at TCC 2010.Comment: CRYPTO 201

Large FHE Gates from tensored homomorphic accumulator

The main bottleneck of all known Fully Homomorphic Encryption schemes lies in the bootstrapping procedure invented by Gentry (STOC’09). The cost of this procedure can be mitigated either using Homomorphic SIMD techniques, or by performing larger computation per bootstrapping procedure.In this work, we propose new techniques allowing to perform more operations per bootstrapping in FHEW-type schemes (EUROCRYPT’13). While maintaining the quasi-quadratic Õ(n2) complexity of the whole cycle, our new scheme allows to evaluate gates with Ω(log n) input bits, which constitutes a quasi-linear speed-up. Our scheme is also very well adapted to large threshold gates, natively admitting up to Ω(n) inputs. This could be helpful for homomorphic evaluation of neural networks.Our theoretical contribution is backed by a preliminary prototype implementation, which can perform 6-to-6 bit gates in less than 10s on a single core, as well as threshold gates over 63 input bits even faster.<p

Provably secure NTRU instances over prime cyclotomic rings

Due to its remarkable performance and potential resistance to quantum attacks, NTRUEncrypt has drawn much attention recently; it also has been standardized by IEEE. However, classical NTRUEncrypt lacks a strong security guarantee and its security still relies on heuristic arguments. At Eurocrypt 2011, Stehlé and Steinfeld first proposed a variant of NTRUEncrypt with a security reduction from standard problems on ideal lattices. This variant is restricted to the family of rings ℤ[X]/(Xn + 1) with n a power of 2 and its private keys are sampled by rejection from certain discrete Gaussian so that the public key is shown to be almost uniform. Despite the fact that partial operations, especially for RLWE, over ℤ[X]/(Xn + 1) are simple and efficient, these rings are quite scarce and different from the classical NTRU setting. In this work, we consider a variant of NTRUEncrypt over prime cyclotomic rings, i.e. ℤ[X]/(Xn-1 +…+ X + 1) with n an odd prime, and obtain IND-CPA secure results in the standard model assuming the hardness of worst-case problems on ideal lattices. In our setting, the choice of the rings is much more flexible and the scheme is closer to the original NTRU, as ℤ[X]/(Xn-1+…+X+1) is a large subring of the NTRU ring ℤ[X]/(Xn-1). Some tools for prime cyclotomic rings are also developed

Hard Instances of the Constrained Discrete Logarithm Problem

The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent $x$ belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the set. Motivated by cryptographic applications, we study sets with succinct representation for which the constrained DLP is hard. We draw on earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such as generalized Menelaus' theorem for proving lower bounds on the complexity of the constrained DLP, and construct sets with succinct representation with provable non-trivial lower bounds