Safe-Error Attacks on SIKE and CSIDH

The isogeny-based post-quantum schemes SIKE (NIST PQC round 3 alternate candidate) and CSIDH (Asiacrypt 2018) have received only little attention with respect to their fault attack resilience so far. We aim to fill this gap and provide a better understanding of their vulnerability by analyzing their resistance towards safe-error attacks. We present four safe-error attacks, two against SIKE and two against a constant-time implementation of CSIDH that uses dummy isogenies. The attacks use targeted bitflips during the respective isogeny-graph traversals. All four attacks lead to full key recovery. By using voltage and clock glitching, we physically carried out two of the attacks - one against each scheme -, thus demonstrate that full key recovery is also possible in practice

First-Order Masked Kyber on ARM Cortex-M4

In this work, we present a fast and first-order secure Kyber implementation optimized for ARM Cortex-M4. Most notably, to our knowledge this is the first liberally-licensed open-source Cortex-M4 implementation of masked Kyber. The ongoing NIST standardization process for post-quantum cryptography and newly proposed side-channel attacks have increased the demand for side-channel analysis and countermeasures for the finalists. On the foundation of the commonly used PQM4 project, we make use of the previously presented optimizations for Kyber on a Cortex-M4 and further combine different ideas from various recent works to achieve a better performance and improve the security in comparison to the original implementations. We show our performance results for first-order secure implementations. Our masked Kyber768 decapsulation on the ARM Cortex-M4 requires only 2 978 441 cycles, including randomness generation from the internal RNG. We then practically verify our implementation by using the t-test methodology with 100 000 traces

To Infect Or Not To Infect: A Critical Analysis Of Infective Countermeasures In Fault Attacks

As fault based cryptanalysis is becoming more and more of a practical threat, it is imperative to make efforts to devise suitable countermeasures. In this regard, the so-called ``infective countermeasures\u27\u27 have garnered particular attention from the community due to its ability in inhibiting differential fault attacks without explicitly detecting the fault. We observe that despite being adopted over a decade ago, a systematic study of infective countermeasures is missing from the literature. Moreover, there seems to be a lack of proper security analysis of the schemes proposed, as quite a few of them have been broken promptly. Our first contribution comes in the form of a generalization of infective schemes which aids us with a better insight into the vulnerabilities, scopes for cost reduction and possible improvements. This way, we are able to propose lightweight alternatives of two existing schemes. Further we analyze shortcomings of LatinCrypt\u2712 and CHES\u2714 schemes and propose a simple patch for the former

Analogue of Vélu\u27s Formulas for Computing Isogenies over Hessian Model of Elliptic Curves

Vélu\u27s formulas for computing isogenies over Weierstrass model of elliptic curves has been extended to other models of elliptic curves such as the Huff model, the Edwards model and the Jacobi model of elliptic curves. This work continues this line of research by providing efficient formulas for computing isogenies over elliptic curves of Hessian form. We provide explicit formulas for computing isogenies of degree 3 and isogenies of degree l not divisible by 3. The theoretical cost of computing these maps in this case is slightly faster than the case with other curves. We also extend the formulas to obtain isogenies over twisted and generalized Hessian forms of elliptic curves. The formulas in this work have been verified with the Sage software and are faster than previous results on the same curve

Effective Pairings in Isogeny-based Cryptography

Pairings are useful tools in isogeny-based cryptography and have been used in SIDH/SIKE and other protocols. As a general technique, pairings can be used to move problems about points on curves to elements in finite fields. However, until now, their applicability was limited to curves over fields with primes of a specific shape and pairings seemed too costly for the type of primes that are nowadays often used in isogeny-based cryptography. We remove this roadblock by optimizing pairings for highly-composite degrees such as those encountered in CSIDH and SQISign. This makes the general technique viable again: We apply our low-cost pairing to problems of general interest, such as supersingularity verification and finding full-torsion points, and show that we can outperform current methods, in some cases up to four times faster than the state-of-the-art. Furthermore, we analyze how pairings can be used to improve deterministic and dummy-free CSIDH. Finally, we provide a constant-time implementation (in Rust) that shows the practicality of these algorithms

A Statistical Verification Method of Random Permutations for Hiding Countermeasure Against Side-Channel Attacks

As NIST is putting the final touches on the standardization of PQC (Post Quantum Cryptography) public key algorithms, it is a racing certainty that peskier cryptographic attacks undeterred by those new PQC algorithms will surface. Such a trend in turn will prompt more follow-up studies of attacks and countermeasures. As things stand, from the attackers’ perspective, one viable form of attack that can be implemented thereupon is the so-called “side-channel attack”. Two best-known countermeasures heralded to be durable against side-channel attacks are: “masking” and “hiding”. In that dichotomous picture, of particular note are successful single-trace attacks on some of the NIST’s PQC then-candidates, which worked to the detriment of the former: “masking”. In this paper, we cast an eye over the latter: “hiding”. Hiding proves to be durable against both side-channel attacks and another equally robust type of attacks called “fault injection attacks”, and hence is deemed an auspicious countermeasure to be implemented. Mathematically, the hiding method is fundamentally based on random permutations. There has been a cornucopia of studies on generating random permutations. However, those are not tied to implementation of the hiding method. In this paper, we propose a reliable and efficient verification of permutation implementation, through employing Fisher–Yates’ shuffling method. We introduce the concept of an -th order permutation and explain how it can be used to verify that our implementation is more efficient than its previous-gen counterparts for hiding countermeasures

Belief Propagation Meets Lattice Reduction: Security Estimates for Error-Tolerant Key Recovery from Decryption Errors

In LWE-based KEMs, observed decryption errors leak information about the secret key in the form of equations or inequalities. Several practical fault attacks have already exploited such leakage by either directly applying a fault or enabling a chosen-ciphertext attack using a fault. When the leaked information is in the form of inequalities, the recovery of the secret key is not trivial. Recent methods use either statistical or algebraic methods (but not both), with some being able to handle incorrect information. Having in mind that integration of the side-channel information is a crucial part of several classes of implementation attacks on LWE-based schemes, it is an important question whether statistically processed information can be successfully integrated in lattice reduction algorithms. We answer this question positively by proposing an error-tolerant combination of statistical and algebraic methods that make use of the advantages of both approaches. The combination enables us to improve upon existing methods -- we use both fewer inequalities and are more resistant to errors. We further provide precise security estimates based on the number of available inequalities. Our recovery method applies to several types of implementation attacks in which decryption errors are used in a chosen-ciphertext attack. We practically demonstrate the improved performance of our approach in a key-recovery attack against Kyber with fault-induced decryption errors

Adapting Belief Propagation to Counter Shuffling of NTTs

The Number Theoretic Transform (NTT) is a major building block in recently introduced lattice based post-quantum (PQ) cryptography. The NTT was target of a number of recently proposed Belief Propagation (BP)-based Side Channel Attacks (SCAs). Ravi et al. have recently proposed a number of countermeasures mitigating these attacks. In 2021, Hamburg et al. presented a chosen-ciphertext enabled SCA improving noise-resistance, which we use as a starting point to state our findings. We introduce a pre-processing step as well as a new factor node which we call shuffle node. Shuffle nodes allow for a modified version of BP when included into a factor graph. The node iteratively learns the shuffling permutation of fine shuffling within a BP run. We further expand our attacker model and describe several matching algorithms to find inter-layer connections based on shuffled measurements. Our matching algorithm allows for either mixing prior distributions according to a doubly stochastic mix matrix or to extract permutations and perform an exact un-matching of layers. We additionally discuss the usage of sub-graph inference to reduce uncertainty and improve un-shuffling of butterflies. Based on our results, we conclude that the proposed countermeasures of Ravi et al. are powerful and counter Hamburg et al., yet could lead to a false security perception – a powerful adversary could still launch successful attacks. We discuss on the capabilities needed to defeat shuffling in the setting of Hamburg et al. using our expanded attacker model. Our methods are not limited to the presented case but provide a toolkit to analyze and evaluate shuffling countermeasures in BP-based attack scenarios

Carry Your Fault: A Fault Propagation Attack on Side-Channel Protected LWE-based KEM

Post-quantum cryptographic (PQC) algorithms, especially those based on the learning with errors (LWE) problem, have been subjected to several physical attacks in the recent past. Although the attacks broadly belong to two classes -- passive side-channel attacks and active fault attacks, the attack strategies vary significantly due to the inherent complexities of such algorithms. Exploring further attack surfaces is, therefore, an important step for eventually securing the deployment of these algorithms. Also, it is important to test the robustness of the already proposed countermeasures in this regard. In this work, we propose a new fault attack on side-channel secure masked implementation of LWE-based key-encapsulation mechanisms (KEMs) exploiting fault propagation. The attack typically originates due to an algorithmic modification widely used to enable masking, namely the Arithmetic-to-Boolean ($\mathtt{A2B}$) conversion. We exploit the data dependency of the adder carry chain in $\mathtt{A2B}$ and extract sensitive information, albeit masking (of arbitrary order) being present. As a practical demonstration of the exploitability of this information leakage, we show key recovery attacks of Kyber, although the leakage also exists for other schemes like Saber. The attack on Kyber targets the decapsulation module and utilizes Belief Propagation (BP) for key recovery. To the best of our knowledge, it is the first attack exploiting an algorithmic component introduced to ease masking rather than only exploiting the randomness introduced by masking to obtain desired faults (as done by Delvaux). Finally, we performed both simulated and electromagnetic (EM) fault-based practical validation of the attack for an open-source first-order secure Kyber implementation running on an STM32 platform

Breaking DPA-protected Kyber via the pair-pointwise multiplication

We present a new template attack that allows us to recover the secret key in Kyber directly from the polynomial multiplication in the decapsulation process. This multiplication corresponds to pair-pointwise multiplications between the NTT representations of the secret key and an input ciphertext. For each pair-point multiplication, a pair of secret coefficients are multiplied in isolation with a pair of ciphertext coefficients, leading to side-channel information which depends solely on these two pairs of values. Hence, we propose to exploit leakage coming from each pair-point multiplication and use it for identifying the values of all secret coefficients. Interestingly, the same leakage is present in DPA-protected implementations. Namely, masked implementations of Kyber simply compute the pair-pointwise multiplication process sequentially on secret shares, allowing us to apply the same strategy for recovering the secret coefficients of each share of the key. Moreover, as we show, our attack can be easily extended to target designs implementing shuffling of the polynomial multiplication. We also show that our attacks can be generalised to work with a known ciphertext rather than a chosen one. To evaluate the effectiveness of our attack, we target the open source implementation of masked Kyber from the mkm4 repository. We conduct extensive simulations which confirm high success rates in the Hamming weight model, even when running the simplest versions of our attack with a minimal number of templates. We show that the success probabilities of our attacks can be increased exponentially only by a linear (in the modulus q) increase in the number of templates. Additionally, we provide partial experimental evidence of our attack’s success. In fact, we show via power traces that, if we build templates for pairs of coefficients used within a pair-point multiplication, we can perform a key extraction by simply calculating the difference between the target trace and the templates. Our attack is simple, straightforward and should not require any deep learning or heavy machinery means for template building or matching. Our work shows that countermeasures such as masking and shuffling may not be enough for protecting the polynomial multiplication in lattice-based schemes against very basic side-channel attacks