International Association
for Cryptologic Research

A recent work from Eurocrypt 2023 suggests that prime-field masking has excellent potential to improve the efficiency vs. security tradeoff of masked implementations against side-channel attacks, especially in contexts where physical leakages show low noise. We pick up on the main open challenge that this seed result leads to, namely the design of an optimized prime cipher able to take advantage of this potential. Given the interest of tweakable block ciphers with cheap inverses in many leakage-resistant designs, we start by describing the FPM (Feistel for Prime Masking) family of tweakable block ciphers based on a generalized Feistel structure. We then propose a first instantiation of FPM, which we denote as small-pSquare. It builds on the recent observation that the square operation (which is non-linear in Fp) can lead to masked gadgets that are more efficient than those for multiplication, and is tailored for efficient masked implementations in hardware. We analyze the mathematical security of the FPM family of ciphers and the small-pSquare instance, trying to isolate the parts of our study that can be re-used for other instances. We additionally evaluate the implementation features of small-pSquare by comparing the efficiency vs. security tradeoff of masked FPGA circuits against those of a state-of-the art binary cipher, namely SKINNY, confirming significant gains in relevant contexts.

<p> Fully Homomorphic Encryption (FHE) is a powerful tool to achieve non-interactive privacy preserving protocols with optimal computation/communication complexity. However, the main disadvantage is that the actual communication cost (bandwidth) is high due to the large size of FHE ciphertexts. As a solution, a technique called transciphering (also known as Hybrid Homomorphic Encryption) was introduced to achieve almost optimal bandwidth for such protocols. However, all existing works require clients to fix a precision for the messages or a mathematical structure for the message space beforehand. It results in unwanted constraints on the plaintext size or underlying structure of FHE based applications.</p><p> In this article, we introduce a new approach for transciphering which does not require fixed message precision decided by the client, for the first time. In more detail, a client uses any kind of FHE-friendly symmetric cipher for $\{0,1\}$ to send its input data encrypted bit-by-bit, then the server can choose a precision $p$ depending on the application and homomorphically transforms the encrypted bits into FHE ciphertexts encrypting integers in $\mathbb{Z}_p$. To illustrate our new technique, we evaluate a transciphering using FiLIP cipher and adapt the most practical homomorphic evaluation technique [CCS'22] to keep the practical latency. As a result, our proof-of-concept implementation for $p$ from $2^2$ to $2^8$ takes only from $13$ ms to $137$ ms. </p>

Embedded devices used in security applications are natural targets for physical attacks. Thus, enhancing their side-channel resistance is an important research challenge. A standard solution for this purpose is the use of Boolean masking schemes, as they are well adapted to current block ciphers with efficient bitslice representations. Boolean masking guarantees that the security of an implementation grows exponentially in the number of shares under the assumption that leakages are sufficiently noisy (and independent). Unfortunately, it has been shown that this noise assumption is hardly met on low-end devices. In this paper, we therefore investigate techniques to mask cryptographic algorithms in such a way that their resistance can survive an almost complete lack of noise. Building on seed theoretical results of Dziembowski et al., we put forward that arithmetic encodings in prime fields can reach this goal. We first exhibit the gains that such encodings lead to thanks to a simulated information theoretic analysis of their leakage (with up to six shares). We then provide figures showing that on platforms where optimized arithmetic adders and multipliers are readily available (i.e., most MCUs and FPGAs), performing masked operations in small to medium Mersenne-prime fields as opposed to binary extension fields will not lead to notable implementation overheads. We compile these observations into a new AES-like block cipher, called AES-prime, which is well-suited to illustrate the remarkable advantages of masking in prime fields. We also confirm the practical relevance of our findings by evaluating concrete software (ARM Cortex-M3) and hardware (Xilinx Spartan-6) implementations. Our experimental results show that security gains over Boolean masking (and, more generally, binary encodings) can reach orders of magnitude despite the same amount of information being leaked per share.

Fresh re-keying is a countermeasure against side-channel analysis where an ephemeral key is derived from a long-term key using a public random value. Popular instances of such schemes rely on key-homomorphic primitives, so that the re-keying process is easy to mask and the rest of the (e.g., block cipher) computations can run with cheaper countermeasures. The main requirement for these schemes to be secure is that the leakages of the ephemeral keys do not allow recovering the long-term key. The Learning with Physical Rounding (LWPR) problem formalizes this security in a practically-relevant model where the adversary can observe noise-free leakages. It can be viewed as a physical version of the Learning With Rounding (LWR) problem, where the rounding is performed by a leakage function and therefore does not have to be computed explicitly. In this paper, we first consolidate the intuition that LWPR cannot be secure in a serial implementation context without additional countermeasures (like shuffling), due to attacks exploiting worst-case leakages that can be mounted with practical data complexity. We then extend the understanding of LWPR in a parallel implementation setting. On the one hand, we generalize its robustness against cryptanalysis taking advantage of any (i.e., not only worst-case) leakage. A previous work claimed security in the specific context of a Hamming weight leakage function. We clarify necessary conditions to maintain this guarantee, based on the degree of the leakage function and the accuracy of its coefficients. On the other hand, we show that parallelism inherently provides good security against attacks exploiting worst-case leakages. We finally confirm the practical relevance of these findings by validating our assumptions experimentally for an exemplary implementation.

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.

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.

Code-based masking is a very general type of masking scheme that covers Boolean masking, inner product masking, direct sum masking, and so on. The merits of the generalization are twofold. Firstly, the higher algebraic complexity of the sharing function decreases the information leakage in “low noise conditions” and may increase the “statistical security order” of an implementation (with linear leakages). Secondly, the underlying error-correction codes can offer improved fault resistance for the encoded variables. Nevertheless, this higher algebraic complexity also implies additional challenges. On the one hand, a generic multiplication algorithm applicable to any linear code is still unknown. On the other hand, masking schemes with higher algebraic complexity usually come with implementation overheads, as for example witnessed by inner-product masking. In this paper, we contribute to these challenges in two directions. Firstly, we propose a generic algorithm that allows us (to the best of our knowledge for the first time) to compute on data shared with linear codes. Secondly, we introduce a new amortization technique that can significantly mitigate the implementation overheads of code-based masking, and illustrate this claim with a case study. Precisely, we show that, although performing every single code-based masked operation is relatively complex, processing multiple secrets in parallel leads to much better performances. This property enables code-based masked implementations of the AES to compete with the state-of-the-art in randomness complexity. Since our masked operations can be instantiated with various linear codes, we hope that these investigations open new avenues for the study of code-based masking schemes, by specializing the codes for improved performances, better side-channel security or improved fault tolerance.

State-of-the-art re-keying schemes can be viewed as a tradeoff between efficient but heuristic solutions based on binary field multiplications, that are only secure if implemented with a sufficient amount of noise, and formal but more expensive solutions based on weak pseudorandom functions, that remain secure if the adversary accesses their output in full. Recent results on “crypto dark matter” (TCC 2018) suggest that low-complexity pseudorandom functions can be obtained by mixing linear functions over different small moduli. In this paper, we conjecture that by mixing some matrix multiplications in a prime field with a physical mapping similar to the leakage functions exploited in side-channel analysis, we can build efficient re-keying schemes based on “crypto-physical dark matter”, that remain secure against an adversary who can access noise-free measurements. We provide first analyzes of the security and implementation properties that such schemes provide. Precisely, we first show that they are more secure than the initial (heuristic) proposal by Medwed et al. (AFRICACRYPT 2010). For example, they can resist attacks put forward by Belaid et al. (ASIACRYPT 2014), satisfy some relevant cryptographic properties and can be connected to a “Learning with Physical Rounding” problem that shares some similarities with standard learning problems. We next show that they are significantly more efficient than the weak pseudorandom function proposed by Dziembowski et al. (CRYPTO 2016), by exhibiting hardware implementation results.

Local pseudorandom generators allow to expand a short random string into a long pseudo-random string, such that each output bit depends on a constant number d of input bits. Due to its extreme efficiency features, this intriguing primitive enjoys a wide variety of applications in cryptography and complexity. In the polynomial regime, where the seed is of size n and the output of size $$n^{\textsf {s}}$$ for $$\textsf {s}> 1$$ , the only known solution, commonly known as Goldreich’s PRG, proceeds by applying a simple d-ary predicate to public random size-d subsets of the bits of the seed.While the security of Goldreich’s PRG has been thoroughly investigated, with a variety of results deriving provable security guarantees against class of attacks in some parameter regimes and necessary criteria to be satisfied by the underlying predicate, little is known about its concrete security and efficiency. Motivated by its numerous theoretical applications and the hope of getting practical instantiations for some of them, we initiate a study of the concrete security of Goldreich’s PRG, and evaluate its resistance to cryptanalytic attacks. Along the way, we develop a new guess-and-determine-style attack, and identify new criteria which refine existing criteria and capture the security guarantees of candidate local PRGs in a more fine-grained way.

We study the main cryptographic features of Boolean functions (balancedness, nonlinearity, algebraic immunity) when, for a given number n of variables, the input to these functions is restricted to some subset E of