Removing the Field Size Loss from Duc et al.\u27s Conjectured Bound for Masked Encodings

At Eurocrypt 2015, Duc et al. conjectured that the success rate of a side-channel attack targeting an intermediate computation encoded in a linear secret-sharing, a.k.a masking with $d+1$ shares, could be inferred by measuring the mutual information between the leakage and each share separately. This way, security bounds can be derived without having to mount the complete attack. So far, the best proven bounds for masked encodings were nearly tight with the conjecture, up to a constant factor overhead equal to the field size, which may still give loose security guarantees compared to actual attacks. In this paper, we improve upon the state-of-the-art bounds by removing the field size loss, in the cases of Boolean masking and arithmetic masking modulo a power of two. As an example, when masking in the AES field, our new bound outperforms the former ones by a factor $256$. Moreover, we provide theoretical hints that similar results could hold for masking in other fields as well

Verification of the security in Boolean masked circuits

We introduce a novel method for reducing an arbitrary $\delta$-noisy leakage function to a collection of $\epsilon$-random probing leakages. These reductions combined with linear algebra tools are utilized to study the security of linear Boolean masked circuits in a practical and concrete setting. The secret recovery probability (SRP) that measures an adversary\u27s ability to obtain secrets of a masked circuit is used to quantify the security. Leakage data and the parity-check relations imposed by the algorithm\u27s structure are employed to estimate the SRP Both the reduction method and the SRP metric were used in the previous works. Here, as our main contribution, the SRP evaluation task is decomposed from the given $\mathbb{F}_q$ field to $q-1$ different binary systems indexed with $i$. Where for the $i$th system, the equivalent $\delta_i$-noisy leakage is reduced optimally to a $\epsilon_i$-random probing leakage with $\epsilon_i=2\delta_i$. Each binary system is targeting a particular bit-composition of the secret. The $q-1$ derived $\delta_i\leq \delta$ values are shown to be a good measure for the informativeness of the given $\delta$-noisy leakage function. Our works here can be considered as an extension of the work of TCC 2016. There, only $\delta$-noisy leakage from the shares of a secret was considered. Here, we also incorporate the leakages that are introduced by the computations over the shares

Information Leakage in Code-based Masking: A Systematic Evaluation by Higher-Order Attacks

Code-based masking is a recent line of research on masking schemes aiming at provably counteracting side-channel attacks. It generalizes and unifies many masking schemes within a coding-theoretic formalization. In code-based masking schemes, the tuning parameters are the underlying linear codes, whose choice significantly affects the side-channel resilience. In this paper, we investigate the exploitability of the information leakage in code-based masking and present attack-based evaluation results of higher-order optimal distinguisher (HOOD). Particularly, we consider two representative instances of code-based masking, namely inner product masking (IPM) and Shamir\u27s secret sharing (SSS) based masking. Our results do confirm the state-of-the-art theoretical derivatives in an empirical manner with numerically simulated measurements. Specifically, theoretical results are based on quantifying information leakage; we further complete the panorama with attack-based evaluations by investigating the exploitability of the leakage. Moreover, we classify all possible candidates of linear codes in IPM with 2 and 3 shares and (3,1)-SSS based masking, and highlight both optimal and worst codes for them. Relying on our empirical evaluations, we therefore recommend investigating the coding-theoretic properties to find the best linear codes in strengthening instances of code-based masking. As for applications, our attack-based evaluation directly empowers designers, by employing optimal linear codes, to enhance the protection of code-based masking. Our framework leverages simulated leakage traces, hence allowing for source code validation or patching in case it is found to be attackable

Towards Tight Random Probing Security

Proving the security of masked implementations in theoretical models that are relevant to practice and match the best known attacks of the side-channel literature is a notoriously hard problem. The random probing model is a good candidate to contribute to this challenge, due to its ability to capture the continuous nature of physical leakage (contrary to the threshold probing model), while also being convenient to manipulate in proofs and to automate with verification tools. Yet, despite recent progresses in the design of masked circuits with good asymptotic security guarantees in this model, existing results still fall short when it comes to analyze the security of concretely useful circuits under realistic noise levels and with low number of shares. In this paper, we contribute to this issue by introducing a new composability notion, the Probe Distribution Table (PDT), and a new tool (called STRAPS, for the Sampled Testing of the RAndom Probing Security). Their combination allows us to significantly improve the tightness of existing analyses in the most practical (low noise, low number of shares) region of the design space. We illustrate these improvements by quantifying the random probing security of an AES S-box circuit, masked with the popular multiplication gadget of Ishai, Sahai and Wagner from Crypto 2003, with up to six shares

Unconditionally Secure Computation Against Low-Complexity Leakage

We consider the problem of constructing leakage-resilient circuit compilers that are secure against global leakage functions with bounded output length. By global, we mean that the leakage can depend on all circuit wires and output a low-complexity function (represented as a multi-output Boolean circuit) applied on these wires. In this work, we design compilers both in the stateless (a.k.a. single-shot leakage) setting and the stateful (a.k.a. continuous leakage) setting that are unconditionally secure against AC0 leakage and similar low-complexity classes. In the stateless case, we show that the original private circuits construction of Ishai, Sahai, and Wagner (Crypto 2003) is actually secure against AC0 leakage. In the stateful case, we modify the construction of Rothblum (Crypto 2012), obtaining a simple construction with unconditional security. Prior works that designed leakage-resilient circuit compilers against AC0 leakage had to rely either on secure hardware components (Faust et al., Eurocrypt 2010, Miles-Viola, STOC 2013) or on (unproven) complexity-theoretic assumptions (Rothblum, Crypto 2012)

Towards Achieving Provable Side-Channel Security in Practice

Physical side-channel attacks are powerful attacks that exploit a device\u27s physical emanations to break the security of cryptographic implementations. Many countermeasures have been proposed against these attacks, especially the widely-used and efficient masking countermeasure. Nevertheless, proving the security of masked implementations is challenging. Current techniques rely on empirical approaches to validate the security of such implementations. On the other hand, the theoretical community introduced leakage models to provide formal proofs of the security of masked implementations. Meanwhile, these leakage models rely on physical assumptions that are difficult to satisfy in practice, and the literature lacks a clear framework to implement proven secure constructions on a physical device while preserving the proven security. In this paper, we present a complete methodology describing the steps to turn an abstract masking scheme proven secure in a theoretical leakage model into a physical implementation satisfying provable security against side-channel attacks in practice. We propose new tools to enforce or relax the physical assumptions the indeal noisy leakage model rely on and provide novel ways of including them in a physical implementation. We also highlight the design goals for an embedded device to reach high levels of proven security, discussing the limitations and open problems of the practical usability of the leakage models. Our goal is to show that it is possible to bridge theory and practice and to motivate further research to fully close the gap and get practical implementations proven secure against side-channel attacks on a physical device without any ideal assumption about the leakage

Concrete Evaluation of the Random Probing Security

We study masked implementation\u27s security when an adversary randomly probes each of its internal variables, intending to recover non-trivial knowledge about its secrets. We introduce a novel metric called Secret Recovery Probability (SRP) for assessing the informativeness of the probing leakages about the masked secrets. To evaluate SRP, our starting point is to describe the relations of the intermediate variables with a parity equation system where the target secret is an unknown of this system ..

Long Paper: Provable Secure Parallel Gadgets

Side-channel attacks are a fundamental threat to the security of cryptographic implementations. One of the most prominent countermeasures against side-channel attacks is masking, where each intermediate value of the computation is secret shared, thereby concealing the computation\u27s sensitive information. An important security model to study the security of masking schemes is the random probing model, in which the adversary obtains each intermediate value of the computation with some probability $p$. To construct secure masking schemes, an important building block is the refreshing gadget, which updates the randomness of the secret shared intermediate values. Recently, Dziembowski, Faust, and Zebrowski (ASIACRYPT\u2719) analyzed the security of a simple refreshing gadget by using a new technique called the leakage diagram. In this work, we follow the approach of Dziembowski et al. and significantly improve its methodology. Concretely, we refine the notion of a leakage diagram via so-called dependency graphs, and show how to use this technique for arbitrary complex circuits via composition results and approximation techniques. To illustrate the power of our new techniques, as a case study, we designed provably secure parallel gadgets for the random probing model, and adapted the ISW multiplication such that all gadgets can be parallelized. Finally, we evaluate concrete security levels, and show how our new methodology can further improve the concrete security level of masking schemes. This results in a compiler provable secure up to a noise level of $O({1})$ for affine circuits and $O({1}/{\sqrt{n}})$ in general