A New Algorithm for Solving Ring-LPN with a Reducible Polynomial

The LPN (Learning Parity with Noise) problem has recently proved to be of great importance in cryptology. A special and very useful case is the RING-LPN problem, which typically provides improved efficiency in the constructed cryptographic primitive. We present a new algorithm for solving the RING-LPN problem in the case when the polynomial used is reducible. It greatly outperforms previous algorithms for solving this problem. Using the algorithm, we can break the Lapin authentication protocol for the proposed instance using a reducible polynomial, in about 2^70 bit operations

Some Notes on Code-Based Cryptography

This thesis presents new cryptanalytic results in several areas of coding-based cryptography. In addition, we also investigate the possibility of using convolutional codes in code-based public-key cryptography. The first algorithm that we present is an information-set decoding algorithm, aiming towards the problem of decoding random linear codes. We apply the generalized birthday technique to information-set decoding, improving the computational complexity over previous approaches. Next, we present a new version of the McEliece public-key cryptosystem based on convolutional codes. The original construction uses Goppa codes, which is an algebraic code family admitting a well-defined code structure. In the two constructions proposed, large parts of randomly generated parity checks are used. By increasing the entropy of the generator matrix, this presumably makes structured attacks more difficult. Following this, we analyze a McEliece variant based on quasi-cylic MDPC codes. We show that when the underlying code construction has an even dimension, the system is susceptible to, what we call, a squaring attack. Our results show that the new squaring attack allows for great complexity improvements over previous attacks on this particular McEliece construction. Then, we introduce two new techniques for finding low-weight polynomial multiples. Firstly, we propose a general technique based on a reduction to the minimum-distance problem in coding, which increases the multiplicity of the low-weight codeword by extending the code. We use this algorithm to break some of the instances used by the TCHo cryptosystem. Secondly, we propose an algorithm for finding weight-4 polynomials. By using the generalized birthday technique in conjunction with increasing the multiplicity of the low-weight polynomial multiple, we obtain a much better complexity than previously known algorithms. Lastly, two new algorithms for the learning parities with noise (LPN) problem are proposed. The first one is a general algorithm, applicable to any instance of LPN. The algorithm performs favorably compared to previously known algorithms, breaking the 80-bit security of the widely used (512,1/8) instance. The second one focuses on LPN instances over a polynomial ring, when the generator polynomial is reducible. Using the algorithm, we break an 80-bit security instance of the Lapin cryptosystem

A New Birthday-Type Algorithm for Attacking the Fresh Re-Keying Countermeasure

The fresh re-keying scheme is a countermeasure designed to protect low-cost devices against side-channel attacks. In this paper, we present a new birthday-type attack based on a refined reduction to Ring-LPN with a reducible polynomial. Compared with the previous research, our algorithm significantly reduces the time complexity in the 128-bit leakage model—with an SNR equal to 8 and at most $2^{20}$ traces, for instance, the key can be recovered using $2^{41.99}$ bit-operations

Correlated Pseudorandomness from the Hardness of Quasi-Abelian Decoding

Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.Comment: This is a long version of a paper accepted at CRYPTO'2

Efficient Pseudorandom Correlation Generators from Ring-LPN

Secure multiparty computation can often utilize a trusted source of correlated randomness to achieve better efficiency. A recent line of work, initiated by Boyle et al. (CCS 2018, Crypto 2019), showed how useful forms of correlated randomness can be generated using a cheap, one-time interaction, followed by only silent local computation. This is achieved via a pseudorandom correlation generator (PCG), a deterministic function that stretches short correlated seeds into long instances of a target correlation. Previous works constructed concretely efficient PCGs for simple but useful correlations, including random oblivious transfer and vector-OLE, together with efficient protocols to distribute the PCG seed generation. Most of these constructions were based on variants of the Learning Parity with Noise (LPN) assumption. PCGs for other useful correlations had poor asymptotic and concrete efficiency. In this work, we design a new class of efficient PCGs based on different flavors of the ring-LPN assumption. Our new PCGs can generate OLE correlations, authenticated multiplication triples, matrix product correlations, and other types of useful correlations over large fields. These PCGs are more efficient by orders of magnitude than the previous constructions and can be used to improve the preprocessing phase of many existing MPC protocols

How Practical is Public-Key Encryption Based on LPN and Ring-LPN?

We conduct a study of public-key cryptosystems based on variants of the Learning Parity with Noise (LPN) problem. The main LPN variant in consideration was introduced by Alekhnovich (FOCS 2003), and we describe several improvements to the originally proposed scheme, inspired by similar existing variants of Regev\u27s LWE-based cryptosystem. To achieve further efficiency, we propose the first public-key cryptosystem based on the ring-LPN problem, which is a more recently introduced LPN variant that makes for substantial improvement in terms of both time and space. We also introduce a variant of this problem called the transposed Ring-LPN problem. Our public-key scheme based on this problem is even more efficient. For all cases, we compute the parameters required for various security levels in practice, given the best currently known attacks. Our conclusion is that the basic LPN-based scheme is in several respects not competitive with existing practical schemes, as the public key, ciphertexts and encryption time become very large already for 80-bit security. On the other hand, the scheme based on transposed Ring-LPN is far better in all these respects. Although the public key and ciphertexts are still larger than for, say, RSA at comparable security levels, they are not prohibitively large; moreover, for decryption, the scheme outperforms RSA for security levels of 112 bits or more. The Ring-LPN based scheme is less efficient, however. Thus, LPN-based public-key cryptography seems to be somewhat more promising for practical use than has been generally assumed so far

Discrete Exponential Equations and Noisy Systems

The history of equations dates back to thousands of years ago, though the equals sign = was only invented in 1557. We formalize the processes of decomposition and restoration in mathematics and physics by defining discrete exponential equations and noisy equation systems over an abstract structure called a land , which is more general than fields, rings, groups, and monoids. Our abstract equations and systems provide general languages for many famous computational problems such as integer factorization, ideal factorization, isogeny factorization, learning parity with noise, learning with errors, learning with rounding, etc. From the abstract equations and systems we deduce a list of new decomposition problems and noisy learning problems. We also give algorithms for discrete exponential equations and systems over algebraic integers. Our motivations are to develop a theory of decomposition and restoration; to unify the scattered studies of decomposition problems and noisy learning problems; and to further permeate the ideas of decomposition and restoration into all possible branches of mathematics. A direct application is a methodology for finding new hardness assumptions for cryptography

Cryptography based on the Hardness of Decoding

This thesis provides progress in the fields of for lattice and coding based cryptography. The first contribution consists of constructions of IND-CCA2 secure public key cryptosystems from both the McEliece and the low noise learning parity with noise assumption. The second contribution is a novel instantiation of the lattice-based learning with errors problem which uses uniform errors