International Association
for Cryptologic Research

As for any cryptographic algorithm, the deployment of post-quantum CCA-secure public key encryption schemes may come with the need to be protected against side-channel attacks. For existing post-quantum schemes that have not been developed with leakage in mind, recent results showed that the cost of these protections can make their implementations more expensive by orders of magnitude. In this paper, we describe a new design, coined POLKA, that is specifically tailored for this purpose. It leverages various ingredients in order to enable efficient side-channel protected implementations such as: (i) the rigidity property (which intuitively means that de-randomized encryption and decryption are injective functions) to avoid the very leaky re-encryption step of the Fujisaki-Okamoto transform, (ii) the randomization of the decryption thanks to the incorporation of a dummy ciphertext, removing the adversary's control of its intermediate computations and making these computations ephemeral, (iii) key-homomorphic computations that can be masked against side-channel attacks with overheads that scale linearly in the number of shares, (iv) hard physical learning problem to argue about the security of some critical unmasked operations. Furthermore, we use an explicit rejection mechanism (returning an error symbol for invalid ciphertexts) to avoid the additional leakage caused by implicit rejection. As a result, all the operations of POLKA can be protected against leakage in a much cheaper way than state-of-the-art designs, opening the way towards schemes that are both quantum-safe and leakage-resistant.

GIMPS and PrimeGrid are large-scale distributed projects dedicated to searching giant prime numbers, usually of special forms like Mersenne and Proth primes. The numbers in the current search-space are millions of digits large and the participating volunteers need to run resource-consuming primality tests. Once a candidate prime $N$ has been found, the only way for another party to independently verify the primality of $N$ used to be by repeating the expensive primality test. To avoid the need for second recomputation of each primality test, these projects have recently adopted certifying mechanisms that enable efficient verification of performed tests. However, the mechanisms presently in place only detect benign errors and there is no guarantee against adversarial behavior: a malicious volunteer can mislead the project to reject a giant prime as being non-prime. In this paper, we propose a practical, cryptographically-sound mechanism for certifying the non-primality of Proth numbers. That is, a volunteer can -- parallel to running the primality test for $N$ -- generate an efficiently verifiable proof at a little extra cost certifying that $N$ is not prime. The interactive protocol has statistical soundness and can be made non-interactive using the Fiat-Shamir heuristic. Our approach is based on a cryptographic primitive called Proof of Exponentiation (PoE) which, for a group $\G$, certifies that a tuple $(x,y,T)\in\G^2\times\mathbb{N}$ satisfies $x^{2^T}=y$ (Pietrzak, ITCS 2019 and Wesolowski, J. Cryptol. 2020). In particular, we show how to adapt Pietrzak's PoE at a moderate additional cost to make it a cryptographically-sound certificate of non-primality.

A proof of exponentiation (PoE) in a group G of unknown order allows a prover to convince a verifier that a tuple (x, q, T, y) ∈G × N × N × G satisfies x^q^T= y. This primitive has recently found exciting applications in the constructions of verifiable delay functions and succinct arguments of knowledge. The most practical PoEs only achieve soundness either under computational assumptions, i.e., they are arguments (Wesolowski, Journal of Cryptology 2020), or in groups that come with the promise of not having any small subgroups (Pietrzak, ITCS 2019). The only statistically-sound PoE in general groups of unknown order is due to Block et al. (CRYPTO 2021), and can be seen as an elaborate parallel repetition of Pietrzak’s PoE: to achieve λ bits of security, say λ = 80, the number of repetitions required (and thus the blow-up in communication) is as large as λ. In this work we propose a statistically-sound PoE for the case where the exponent q is the product of all primes up to some bound B. We show that, in this case, it suffices to run only λ/ log(B) parallel instances of Pietrzak’s PoE, which reduces the concrete proof-size compared to Block et al. by an order of magnitude. Furthermore, we show that in the known applications where PoEs are used as a building block such structured exponents are viable. Finally, we also discuss batching of our PoE, showing that many proofs (for the same G and q but different x and T) can be batched by adding only a single element to the proof per additional statement.

Hard physical learning problems have been introduced as an alternative option to implement cryptosystems based on hard learning problems. Their high-level idea is to use inexact computing to generate erroneous computations directly, rather than to first compute correctly and add errors afterwards. Previous works focused on the applicability of this idea to the Learning Parity with Noise (LPN) problem as a first step, and formalized it as Learning Parity with Physical Noise (LPPN). In this work, we generalize it to the Learning With Errors (LWE) problem, formalized as Learning With Physical Errors (LWPE). We first show that the direct application of the design ideas used for LPPN prototypes leads to a new source of (mathematical) data dependencies in the error distributions that can reduce the security of the underlying problem. We then show that design tweaks can be used to avoid this issue, making LWPE samples natively robust against such data dependencies. We additionally put forward that these ideas open a quite wide design space that could make hard physical learning problems relevant in various applications. And we conclude by presenting a first prototype FPGA design confirming our claims.

Hard physical learning problems have been introduced as an alternative option to implement cryptosystems based on hard learning problems. Their high-level idea is to use inexact computing to generate erroneous computations directly, rather than to first compute correctly and add errors afterwards. Previous works focused on the applicability of this idea to the Learning Parity with Noise (LPN) problem as a first step, and formalized it as Learning Parity with Physical Noise (LPPN). In this work, we generalize it to the Learning With Errors (LWE) problem, formalized as Learning With Physical Errors (LWPE). We first show that the direct application of the design ideas used for LPPN prototypes leads to a new source of (mathematical) data dependencies in the error distributions that can reduce the security of the underlying problem. We then show that design tweaks can be used to avoid this issue, making LWPE samples natively robust against such data dependencies. We additionally put forward that these ideas open a quite wide design space that could make hard physical learning problems relevant in various applications. And we conclude by presenting a first prototype FPGA design confirming our claims.

The homogeneous continuous LWE (hCLWE) problem is to distinguish samples of a specific high-dimensional Gaussian mixture from standard normal samples. It was shown to be at least as hard as Learning with Errors, but no reduction in the other direction is currently known. We present four new public-key encryption schemes based on the hardness of hCLWE, with varying tradeoffs between decryption and security errors, and different discretization techniques. Our schemes yield a polynomial-time algorithm for solving hCLWE using a Statistical Zero-Knowledge oracle.

Hybrid Homomorphic Encryption (HHE) reduces the amount of computation client-side and bandwidth usage in a Fully Homomorphic Encryption (FHE) framework. HHE requires the usage of specific symmetric schemes that can be evaluated homomorphically efficiently. In this paper, we introduce the paradigm of Group Filter Permutator (GFP) as a generalization of the Improved Filter Permutator paradigm introduced by M ́eaux et al. From this paradigm, we specify Elisabeth , a family of stream cipher and give an instance: Elisabeth-4. After proving the security of this scheme, we provide a Rust implementation of it and ensure its performance is comparable to state-of-the-art HHE. The true strength of Elisabeth lies in the available operations server-side: while the best HHE applications were limited to a few multiplications server-side, we used data sent through Elisabeth-4 to homomorphically evaluate a neural network inference. Finally, we discuss the improvement and loss between the HHE and the FHE framework and give ideas to build more efficient schemes from the Elisabeth family.

Hard learning problems are important building blocks for the design of various cryptographic functionalities such as authentication protocols and post-quantum public key encryption. The standard implementations of such schemes add some controlled errors to simple (e.g., inner product) computations involving a public challenge and a secret key. Hard physical learning problems formalize the potential gains that could be obtained by leveraging inexact computing to directly generate erroneous samples. While they have good potential for improving the performances and physical security of more conventional samplers when implemented in specialized integrated circuits, it remains unknown whether physical defaults that inevitably occur in their instantiation can lead to security losses, nor whether their implementation can be viable on standard platforms such as FPGAs. We contribute to these questions in the context of the Learning Parity with Physical Noise (LPPN) problem by: (1) exhibiting new (output) data dependencies of the error probabilities that LPPN samples may suffer from; (2) formally showing that LPPN instances with such dependencies are as hard as the standard LPN problem; (3) analyzing an FPGA prototype of LPPN processor that satisfies basic security and performance requirements.