International Association
for Cryptologic Research

Masking is one of the most popular countermeasures to side- channel attacks, because it can offer provable security. However, depend- ing on the adversary’s model, useful security guarantees can be hard to provide. At first, masking has been shown secure against t-threshold probing adversaries by Ishai et al. at Crypto’03. It has then been shown secure in the more generic random probing model by Duc et al. at Euro- crypt’14. Prouff and Rivain have introduced the noisy leakage model to capture more realistic leakage at Eurocrypt’13. Reduction from noisy leakage to random probing has been introduced by Duc et al. at Eu- rocrypt’14, and security guarantees were improved for both models by Prest et al. at Crypto’19, Duc et al. in Eurocrypt’15/J. Cryptol’19, and Masure and Standaert at Crypto’23. Unfortunately, as it turns out, we found that previous proofs in either random probing or noisy leakage models are flawed, and such flaws do not appear easy to fix. In this work, we show that the Doeblin coefficient allows one to overcome these flaws. In fact, it yields optimal reductions from noisy leakage to random probing, thereby providing a correct and usable metric to prop- erly ground security proofs. This shows the inherent inevitable cost of a reduction from the noisy leakages to the random probing model. We show that it can also be used to derive direct formal security proofs using the subsequence decomposition of Prouff and Rivain.

The implementation of cryptographic algorithms must be protected against physical attacks. Side-channel and fault injection analyses are two prominent such implementation-level attacks. Protections against either do exist. Against sidechannel attacks, they are characterized by SNI security orders: the higher the order, the more difficult the attack.In this paper, we leverage fast discrete Fourier transform to reduce the complexity of high-order masking. The security paradigm is that of code-based masking. Coding theory is amenable both to mask material at a prescribed order, by mixing the information, and to detect and/or correct errors purposely injected by an attacker. For the first time, we show that quasi-linear masking (pioneered by Goudarzi, Joux and Rivain at ASIACRYPT 2018) can be achieved alongside with cost amortisation. This technique consists in masking several symbols/bytes with the same masking material, therefore improving the efficiency of the masking. We provide a security proof, leveraging both coding and probing security arguments. Regarding fault detection, our masking is capable of detecting up to d faults, where 2d + 1 is the length of the code, at any place of the algorithm, including within gadgets. In addition to the theory, that makes use of the Frobenius Additive Fast Fourier Transform, we show performance results, in a C language implementation, which confirms in practice that the complexity is quasi-linear in the code length.

Code-based masking is a highly generalized type of masking schemes, which can be instantiated into specific cases by assigning different encoders. It captivates by its side-channel resistance against higher-order attacks and the potential to withstand fault injection attacks. However, similar to other algebraically-involved masking schemes, code-based masking is also burdened with expensive computational overhead. To mitigate such cost and make it efficient, we contribute to several improvements to the original scheme proposed by Wang et al. in TCHES 2020. Specifically, we devise a computationally friendly encoder and accordingly accelerate masked gadgets to leverage efficient implementations. In addition, we highlight that the amortization technique introduced by Wang et al. does not always lead to efficient implementations as expected, but actually decreases the efficiency in some cases.From the perspective of practical security, we carry out an extensive evaluation of the concrete security of code-based masking in the real world. On one hand, we select three representative variations of code-based masking as targets for an extensive evaluation. On the other hand, we aim at security assessment of both encoding and computations to investigate whether the state-of-the-art computational framework for code-based masking reaches the security of the corresponding encoding. By leveraging both leakage assessment tool and side-channel attacks, we verify the existence of “security order amplification” in practice and validate the reliability of the leakage quantification method proposed by Cheng et al. in TCHES 2021. In addition, we also study the security decrease caused by the “cost amortization” technique and redundancy of code-based masking. We identify a security bottleneck in the gadgets computations which limits the whole masked implementation. To the best of our knowledge, this is the first time that allows us to narrow down the gap between the theoretical security order under the probing model (sometimes with simulation experiments) and the concrete side-channel security level of protected implementations by code-based masking in practice.

Block ciphers are protected against side-channel attacks by masking. On one hand, when the leakage model is unknown, second-order correlation attacks are typically used. On the other hand, when the leakage model can be profiled, template attacks are prescribed. But what if the profiled model does not exactly match that of the attacked device?One solution consists in regressing on-the-fly the scaling parameters from the model. In this paper, we leverage an Expectation-Maximization (EM) algorithm to implement such an attack. The resulting unprofiled EM attack, termed U-EM, is shown to be both efficient (in terms of number of traces) and effective (computationally speaking). Based on synthetic and real traces, we introduce variants of our U-EM attack to optimize its performance, depending on trade-offs between model complexity and epistemic noise. We show that the approach is flexible, in that it can easily be adapted to refinements such as different points of interest and number of parameters in the leakage model.

This paper presents a unified approach to quantifying the information leakages in the most general code-based masking schemes. Specifically, by utilizing a uniform representation, we highlight first that all code-based masking schemes’ side-channel resistance can be quantified by an all-in-one framework consisting of two easy-tocompute parameters (the dual distance and the number of conditioned codewords) from a coding-theoretic perspective. In particular, we use signal-to-noise ratio (SNR) and mutual information (MI) as two complementary metrics, where a closed-form expression of SNR and an approximation of MI are proposed by connecting both metrics to the two coding-theoretic parameters. Secondly, considering the connection between Reed-Solomon code and SSS (Shamir’s Secret Sharing) scheme, the SSS-based masking is viewed as a particular case of generalized code-based masking. Hence as a straightforward application, we evaluate the impact of public points on the side-channel security of SSS-based masking schemes, namely the polynomial masking, and enhance the SSS-based masking by choosing optimal public points for it. Interestingly, we show that given a specific security order, more shares in SSS-based masking leak more information on secrets in an information-theoretic sense. Finally, our approach provides a systematic method for optimizing the side-channel resistance of every code-based masking. More precisely, this approach enables us to select optimal linear codes (parameters) for the generalized code-based masking by choosing appropriate codes according to the two coding-theoretic parameters. Summing up, we provide a best-practice guideline for the application of code-based masking to protect cryptographic implementations.

In some practical enciphering frameworks, operational constraints may require that a secret key be embedded into the cryptographic algorithm. Such implementations are referred to as White-Box Cryptography (WBC). One technique consists of the algorithm’s tabulation specialized for its key, followed by obfuscating the resulting tables. The obfuscation consists of the application of invertible diffusion and confusion layers at the interface between tables so that the analysis of input/output does not provide exploitable information about the concealed key material.Several such protections have been proposed in the past and already cryptanalyzed thanks to a complete WBC scheme analysis. In this article, we study a particular pattern for local protection (which can be leveraged for robust WBC); we formalize it as DIBO (for Diffused-Input-Blocked-Output). This notion has been explored (albeit without having been nicknamed DIBO) in previous works. However, we notice that guidelines to adequately select the invertible diffusion ∅and the blocked bijections B were missing. Therefore, all choices for ∅ and B were assumed as suitable. Actually, we show that most configurations can be attacked, and we even give mathematical proof for the attack. The cryptanalysis tool is the number of zeros in a Walsh-Hadamard spectrum. This “spectral distinguisher” improves on top of the previously known one (Sasdrich, Moradi, Güneysu, at FSE 2016). However, we show that such an attack does not work always (even if it works most of the time).Therefore, on the defense side, we give a straightforward rationale for the WBC implementations to be secure against such spectral attacks: the random diffusion part ∅ shall be selected such that the rank of each restriction to bytes is full. In AES’s case, this seldom happens if ∅ is selected at random as a linear bijection of F322. Thus, specific care shall be taken. Notice that the entropy of the resulting ∅ (suitable for WBC against spectral attacks) is still sufficient to design acceptable WBC schemes.

Using information-theoretic tools, this paper establishes a mathematical link between the probability of success of a side-channel attack and the minimum number of queries to reach a given success rate, valid for any possible distinguishing rule and with the best possible knowledge on the attacker’s side. This link is a lower bound on the number of queries highly depends on Shannon’s mutual information between the traces and the secret key. This leads us to derive upper bounds on the mutual information that are as tight as possible and can be easily calculated. It turns out that, in the case of an additive white Gaussian noise, the bound on the probability of success of any attack is directly related to the signal to noise ratio. This leads to very easy computations and predictions of the success rate in any leakage model.