Multi-key fully homomorphic encryption (MKFHE), a generalization of fully homomorphic encryption (FHE), enables a computation over encrypted data under multiple keys. The first MKFHE schemes were based on the NTRU primitive, however these early NTRU based FHE schemes were found to be insecure due to the problem of over-stretched parameters. Recently, in the case of standard (non-multi key) FHE a secure version, called FINAL, of NTRU has been found. In this work we extend FINAL to an MKFHE scheme, this allows us to benefit from some of the performance advantages provided by NTRU based primitives. Thus, our scheme provides competitive performance against current state-of-the-art multi-key TFHE, in particular reducing the computational complexity from quadratic to linear in the number of keys.

The security of lattice-based crytography (LWE, NTRU, and FHE) depends on the hardness of the shortest-vector problem (SVP). Sieving algorithms give the lowest asymptotic runtime to solve SVP, but depend on exponential memory. Memory access costs much more in reality than in the RAM model, so we consider a computational model where processors, memory, and meters of wire are in constant proportions to each other. While this adds substantial costs to route data during lattice sieving, we modify existing algorithms to amortize these costs and find that, asymptotically, a classical computer can achieve the previous RAM model cost of $2^{0.2925d+o(d)}$ to sieve a $d$-dimensional lattice for a computer existing in 3 or more spatial dimensions, and can reach $2^{0.3113d+o(d)}$ in 2 spatial dimensions, where “spatial dimensions” are the dimensions of the physical geometry in which the computer exists.

Since this result implies an increased cost in 2 spatial dimensions, we make several assumptions about the constant terms of memory access and lattice attacks so that we can give bit security estimates for Kyber and Dilithium. These estimates support but do not increase the security categories claimed in the Kyber and Dilithium specifications, at least with respect to known attacks.

The Learning with Errors (LWE) problem has become one of the most prominent candidates of post-quantum cryptography, offering promising potential to meet the challenge of quantum computing. From a theoretical perspective, optimizing algorithms to solve LWE is a vital task for the analysis of this cryptographic primitive. In this paper, we propose a fine-grained time/memory trade-off method to analyze c-sum BKW variants for LWE in both classical and quantum models, then offer new complexity bounds for multiple BKW variants determined by modulus q, dimension k, error rate alpha, and stripe size b. Through our analysis, optimal parameters can be efficiently found for different settings, and the minimized complexities are lower than existing results. Furthermore, we enhance the performance of c-sum BKW in the quantum computing model by adopting the quantum Meet-in-the-Middle technique as c-sum solver instead of the naive c-sum technique. Our complexity trade-off formula also applies to the quantum version of BKW, and optimizes the theoretical quantum time and memory costs, which are exponentially lower than existing quantum c-sum BKW variants.

Almost pairwise independence (API) is a quantitative property of a class of functions that is desirable in many cryptographic applications. This property is satisfied by Learning with errors (LWE)-mappings and by special Substitution-Permutation Networks (SPN). API block ciphers are known to be resilient to differential and linear cryptanalysis attacks. Recently, security of protocols against neural network-based attacks became a major trend in cryptographic studies. Therefore, it is relevant to study the hardness of learning a target function from an API class of functions by gradient-based methods.

We propose a theoretical analysis based on the study of the variance of the gradient of a general machine learning objective with respect to a random choice of target function from a class. We prove an upper bound and verify that, indeed, such a variance is extremely small for API classes of functions. This implies the resilience of actual LWE-based primitives against deep learning attacks, and to some extent, the security of SPNs. The hardness of learning reveals itself in the form of the barren plateau phenomenon during the training process, or in other words, in a low information content of the gradient about the target function. Yet, we emphasize that our bounds hold for the case of a regular parameterization of a neural network and the gradient may become informative if a class is mildly pairwise independent and a parameterization is non-regular. We demonstrate our theory in experiments on the learnability of LWE mappings.

Watermarking pseudorandom functions (PRF) allow an authority to embed an unforgeable and unremovable watermark into a PRF while preserving its functionality. In this work, we extend the work of Kim and Wu [Crypto'19] who gave a simple two-step construction of watermarking PRFs from a class of extractable PRFs satisfying several other properties – first construct a mark-embedding scheme, and then upgrade it to a message-embedding scheme.

While the message-embedding scheme of Kim and Wu is based on complex homomorphic evaluation techniques, we observe that much simpler constructions can be obtained and from a wider range of assumptions, if we forego the strong requirement of security against the watermarking authority. Concretely, we introduce a new notion called extractable PRGs (xPRGs), from which extractable PRFs (without security against authorities) suitable for the Kim-Wu transformations can be simply obtained via the Goldreich-Goldwasser-Micali (GGM) construction. We provide simple constructions of xPRGs from a wide range of assumptions such as hardness of computational Diffie-Hellman (CDH) in the random oracle model, as well as LWE and RSA in the standard model.

This paper develops Central Limit arguments for analysing the noise in ciphertexts in two homomorphic encryption schemes that are based on Ring-LWE. The first main contribution of this paper is to present and evaluate an average-case noise analysis for the BGV scheme. Our approach relies on the recent work of Costache et al.(SAC 2023) that gives the approximation of a polynomial product as a multivariate Normal distribution. We show how this result can be applied in the BGV context and evaluate its efficacy. We find this average-case approach can much more closely model the noise growth in BGV implementations than prior approaches, but in some cases it can also underestimate the practical noise growth. Our second main contribution is to develop a Central Limit framework to analyse the noise growth in the homomorphic Ring-LWE cryptosystem of Lyubashevsky, Peikert and Regev (Eurocrypt 2013, full version). Our approach is very general: apart from finite variance, no assumption on the distribution of the noise is required (in particular, the noise need not be subgaussian). We show that our approach leads to tighter bounds for the probability of decryption failure than those of prior work.