Martin R. Albrecht

On the concrete hardness of Learning with Errors

Abstract The learning with errors (LWE) problem has become a central building block of modern cryptographic constructions. This work collects and presents hardness results for concrete instances of LWE. In particular, we discuss algorithms proposed in the literature and give the expected resources required to run them. We consider both generic instances of LWE as well as small secret variants. Since for several methods of solving LWE we require a lattice reduction step, we also review lattice reduction algorithms and use a refined model for estimating their running times. We also give concrete estimates for various families of LWE instances, provide a Sage module for computing these estimates and highlight gaps in the knowledge about algorithms for solving the LWE problem.

Plaintext Recovery Attacks against SSH

This paper presents a variety of plaintext-recovering attacks against SSH. We implemented a proof of concept of our attacks against OpenSSH, where we can verifiably recover 14 bits of plaintext from an arbitrary block of ciphertext with probability

Algebraic Techniques in Differential Cryptanalysis

2^{-14}

Block Ciphers – Focus on the Linear Layer (feat. PRIDE )

and 32 bits of plaintext from an arbitrary block of ciphertext with probability

Ciphers for MPC and FHE

2^{-18}

On the complexity of the BKW algorithm on LWE

. These attacks assume the default configuration of a 128-bit block cipher operating in CBC mode. The paper explains why a combination of flaws in the basic design of SSH leads implementations such as OpenSSH to be open to our attacks, why current provable security results for SSH do not cover our attacks, and how the attacks can be prevented in practice.

On the Efficacy of Solving LWE by Reduction to Unique-SVP

In this paper we propose a new cryptanalytic method against block ciphers, which combines both algebraic and statistical techniques. More specifically, we show how to use algebraic relations arising from differential characteristics to speed up and improve key-recovery differential attacks against block ciphers. To illustrate the new technique, we apply algebraic techniques to mount differential attacks against round reduced variants of Present-128.

Cold boot key recovery by solving polynomial systems with noise

The linear layer is a core component in any substitution-permutation network block cipher. Its design significantly influences both the security and the efficiency of the resulting block cipher. Surprisingly, not many general constructions are known that allow to choose trade-offs between security and efficiency. Especially, when compared to Sboxes, it seems that the linear layer is crucially understudied. In this paper, we propose a general methodology to construct good, sometimes optimal, linear layers allowing for a large variety of trade-offs. We give several instances of our construction and on top underline its value by presenting a new block cipher. PRIDE is optimized for 8-bit micro-controllers and significantly outperforms all academic solutions both in terms of code size and cycle count.

On Dual Lattice Attacks Against Small-Secret LWE and Parameter Choices in HElib and SEAL

Designing an efficient cipher was always a delicate balance between linear and non-linear operations. This goes back to the design of DES, and in fact all the way back to the seminal work of Shannon.

Algorithm 898: Efficient multiplication of dense matrices over GF(2)

This work presents a study of the complexity of the Blum–Kalai–Wasserman (BKW) algorithm when applied to the Learning with Errors (LWE) problem, by providing refined estimates for the data and computational effort requirements for solving concrete instances of the LWE problem. We apply this refined analysis to suggested parameters for various LWE-based cryptographic schemes from the literature and compare with alternative approaches based on lattice reduction. As a result, we provide new upper bounds for the concrete hardness of these LWE-based schemes. Rather surprisingly, it appears that BKW algorithm outperforms known estimates for lattice reduction algorithms starting in dimension