What a lovely hat

Every cryptographic implementation on embedded device is vulnerable to side-channel attacks. To prevent these attacks, the main countermeasure consists in splitting each sensitive variable in shares and processing them independently. With the upcoming of new algorithms designed to resist quantum computers and the complexity of their operations, this protection represents a real challenge. In this article, we present an attack on an earlier attempt to protect the decoder of BIKE...

We introduce the use of machine learning in the cryptanalysis of code-based cryptography. Our focus is on distinguishing problems related to the security of NIST round-4 McEliece-like cryptosystems, particularly for Goppa codes used in ClassicMcEliece and Quasi-Cyclic Moderate Density Parity-Check (QC-MDPC) codes used in BIKE. We present DeepDistinguisher, a new algorithm for distinguishing structured codes from random linear codes that uses a transformer. The results show that the new...

The computation of the inverse of a polynomial over a quotient ring or a finite field plays a very important role during the key generation of post-quantum cryptosystems like NTRU, BIKE, and LEDACrypt. It is therefore important that there exist an efficient algorithm capable of running in constant time, to prevent timing side-channel attacks. In this article, we study both constant-time algorithms based on Fermat's Little Theorem and the Extended $GCD$ Algorithm, and provide a detailed...

Quasi-cyclic moderate-density parity check (QC-MDPC) code-based encryption schemes under iterative decoders offer highly-competitive performance in the quantum-resistant space of cryptography, but the decoding-failure rate (DFR) of these algorithms are not well-understood. The DFR decreases extremely rapidly as the ratio of code-length to error-bits increases, then decreases much more slowly in regimes known as the waterfall and error-floor, respectively. This work establishes three,...

Cryptography based on the presumed hardness of decoding codes -- i.e., code-based cryptography -- has recently seen increased interest due to its plausible security against quantum attackers. Notably, of the four proposals for the NIST post-quantum standardization process that were advanced to their fourth round for further review, two were code-based. The most efficient proposals -- including HQC and BIKE, the NIST submissions alluded to above -- in fact rely on the presumed hardness of...

We present experimental findings on the decoding failure rate (DFR) of BIKE, a fourth-round candidate in the NIST Post-Quantum Standardization process, at the 20-bit security level using graph-theoretic approaches. We select parameters according to BIKE design principles and conduct a series of experiments using Rust to generate significantly more decoding failure instances than in prior work using SageMath. For each decoding failure, we study the internal state of the decoder at each...

The Multi-Party Computation in the Head (MPCitH) paradigm has proven to be a versatile tool to design proofs of knowledge (PoK) based on variety of computationally hard problems. For instance, many post-quantum signatures have been designed from MPC based proofs combined with the Fiat-Shamir transformation. Over the years, MPCitH has evolved significantly with developments based on techniques such as threshold computing and other optimizations. Recently, Vector Oblivious Linear Evaluation...

In this work, we continue the analysis of the binding properties of implicitly-rejecting key-encapsulation mechanisms (KEMs) obtained via the Fujisaki-Okamoto (FO) transform. These binding properties, in earlier literature known under the term robustness, thwart attacks that can arise when using KEMs in larger protocols. Recently, Cremers et al. (ePrint'24) introduced a framework for binding notions, encompassing previously existing but also new ones. While implicitly-rejecting KEMs have...

Anticipating the advent of large quantum computers, NIST started a worldwide competition in 2016 aiming to define the next cryptographic standards. HQC is one of these post-quantum schemes still in contention, with three others already standardized. In 2022, Guo et al. introduced a timing attack that exploited an inconsistency in HQC rejection sampling function to recover its secret key in 866,000 calls to an oracle. The authors of HQC updated its specification by applying an algorithm to...

BIKE is a post-quantum key encapsulation mechanism (KEM) selected for the 4th round of the NIST’s standardization campaign. It relies on the hardness of the syndrome decoding problem for quasi-cyclic codes and on the indistinguishability of the public key from a random element, and provides the most competitive performance among round 4 candidates, which makes it relevant for future real-world use cases. Analyzing its side-channel resistance has been highly encouraged by the community and...

In this paper, we initiate the study of the Rank Decoding (RD) problem and LRPC codes with blockwise structures in rank-based cryptosystems. First, we introduce the blockwise errors ($\ell$-errors) where each error consists of $\ell$ blocks of coordinates with disjoint supports, and define the blockwise RD ($\ell$-RD) problem as a natural generalization of the RD problem whose solutions are $\ell$-errors (note that the standard RD problem is actually a special $\ell$-RD problem with...

Splitting up sensitive data into multiple shares – termed masking – has proven an effective countermeasure against various types of Side-Channel Analysis (SCA) on cryptographic implementations. However, in software this approach not only leads to dramatic performance overheads for non-linear operations, but also suffers from microarchitectural leakage, which is hard to avoid. Both problems can be addressed with one solution: masked hardware accelerators. In this context, Gao et al. [GGM+...

Code-based cryptography has received a lot of attention recently because it is considered secure under quantum computing. Among them, the QC-MDPC based scheme is one of the most promising due to its excellent performance. QC-MDPC based scheme is usually subject to a small rate of decryption failure, which can leak information about the secret key. This raises two crucial problems: how to accurately estimate the decryption failure rate and how to use the failure information to recover the...

With the advancement of NIST PQC standardization, three of the four candidates in Round 4 are code-based schemes, namely Classic McEliece, HQC and BIKE. Currently, one of the most important tasks is to further analyze their security levels for the suggested parameter sets. At PKC 2022 Esser and Bellini restated the major information set decoding (ISD) algorithms by using nearest neighbor search and then applied these ISD algorithms to estimate the bit security of Classic McEliece, HQC and...

The problem of decoding random codes is a fundamental problem for code-based cryptography, including recent code-based candidates in the NIST post-quantum standardization process. In this paper, we present a novel sieving-style information-set decoding (ISD) algorithm, addressing the task of solving the syndrome decoding problem. Our approach involves maintaining a list of weight-$2p$ solution vectors to a partial syndrome decoding problem and then creating new vectors by identifying pairs...

Since 2016, NIST has been standardrizing Post-Quantum Cryptosystems, PQCs. Code-Based Cryptosystem, CBC, which is considered to be one of PQCs, uses the Syndrome Decoding Problem as the basis for its security. NIST's PQC standardization project is currently in its 4th round and some CBC encryption schemes remain there. In this paper, we consider the quantum security for these cryptosystems.

The sampling of polynomials with fixed weight is a procedure required by round-4 Key Encapsulation Mechanisms (KEMs) for Post-Quantum Cryptography (PQC) standardization (BIKE, HQC, McEliece) as well as NTRU, Streamlined NTRU Prime, and NTRU LPRrime. Recent attacks have shown in this context that side-channel leakage of sampling methods can be exploited for key recoveries. While countermeasures regarding such timing attacks have already been presented, still, there is no comprehensive work...

We propose new time-memory trade-offs for the random subset sum problem defined on $(a_1,\ldots,a_n,t)$ over $\mathbb{Z}_{2^n}$. Our trade-offs yield significant running time improvements for every fixed memory limit $M\geq2^{0.091n}$. Furthermore, we interpolate to the running times of the fastest known algorithms when memory is not limited. Technically, our design introduces a pruning strategy to the construction by Becker-Coron-Joux (BCJ) that allows for an exponentially small...

We present experimental findings on the decoding failure rate (DFR) of BIKE, a fourth-round candidate in the NIST Post-Quantum Standardization process, at the 20-bit security level. We select parameters according to BIKE design principles and conduct a series of experiments. We directly compute the average DFR on a range of BIKE block sizes and identify both the waterfall and error floor regions of the DFR curve. We then study the influence on the average DFR of three sets $\mathcal{C}$,...

Physical side-channel analysis poses a huge threat to post-quantum cryptographic schemes implemented on embedded devices. Still, secure implementations are missing for many schemes. In this paper, we present an efficient solution for masked polynomial inversion, a main component of the key generation of multiple post-quantum KEMs. For this, we introduce a polynomial-multiplicative masking scheme with efficient arbitrary order conversions from and to additive masking. Furthermore, we show how...

Code-based cryptography received attention after the NIST started the post-quantum cryptography standardization process in 2016. A central NP-hard problem is the binary syndrome decoding problem, on which the security of many code-based cryptosystems lies. The best known methods to solve this problem all stem from the information-set decoding strategy, first introduced by Prange in 1962. A recent line of work considers augmented versions of this strategy, with hints typically provided by...

In a so-called partial key exposure attack one obtains some information about the secret key, e.g. via some side-channel leakage. This information might be a certain fraction of the secret key bits (erasure model) or some erroneous version of the secret key (error model). The goal is to recover the secret key from the leaked information. There is a common belief that, as opposed to e.g. the RSA cryptosystem, most post-quantum cryptosystems are usually resistant against partial key...

Cache systems are crucial for reducing communication overhead on the Internet. The importance of communication privacy is being increasingly and widely recognized; therefore, we anticipate that nearly all end-to-end communication will be encrypted via secure sockets layer/transport layer security (SSL/TLS) in the near future. Herein we consider a catch-22 situation, wherein the cache server checks whether content has been cached or not, i.e., the cache server needs to observe it, thereby...

With the recent shift to post-quantum algorithms it becomes increasingly important to provide precise bit-security estimates for code-based cryptography such as McEliece and quasi-cyclic schemes like BIKE and HQC. While there has been significant progress on information set decoding (ISD) algorithms within the last decade, it is still unclear to which extent this affects current cryptographic security estimates. We provide the first concrete implementations for representation-based ISD,...

The pseudorandom sampling of constant weight words, as it is currently implemented in cryptographic schemes like BIKE or HQC, is prone to the leakage of information on the seed being used for the pseudorandom number generation. This creates a vulnerability when the semantic security conversion requires a deterministic re-encryption. This observation was first made in [HLS21] about HQC and a timing attack was presented to recover the secret key. As suggested in [HLS21] a similar attack...

Well before large-scale quantum computers will be available, traditional cryptosystems must be transitioned to post-quantum (PQ) secure schemes. The NIST PQC competition aims to standardize suitable cryptographic schemes. Candidates are evaluated not only on their formal security strengths, but are also judged based on the security with regard to resistance against side-channel attacks. Although round 3 candidates have already been intensively vetted with regard to such attacks, one...

The aim of this document is to clarify the DFR (Decoding Failure Rate) claims made for BIKE, a third round alternate candidate KEM (Key Encapsulation Mechanism) to the NIST call for post-quantum cryptography standardization. For the most part, the material presented here is not new, it is extracted from the relevant scientific literature, in particular [V21]. Even though a negligible DFR is not needed for a KEM using ephemeral keys (e.g. TLS) which only requires IND-CPA security, it seems...

BIKE is a Key Encapsulation Mechanism selected as an alternate candidate in NIST’s PQC standardization process, in which performance plays a significant role in the third round. This paper presents FPGA implementations of BIKE with the best area-time performance reported in literature. We optimize two key arithmetic operations, which are the sparse polynomial multiplication and the polynomial inversion. Our sparse multiplier achieves time-constancy for sparse polynomials of indefinite Hamming...

This paper investigates __anonymity__ of all NIST PQC Round 3 KEMs: Classic McEliece, Kyber, NTRU, Saber, BIKE, FrodoKEM, HQC, NTRU Prime (Streamlined NTRU Prime and NTRU LPRime), and SIKE. We show the following results: * NTRU is anonymous in the quantum random oracle model (QROM) if the underlying deterministic PKE is strongly disjoint-simulatable. NTRU is collision-free in the QROM. A hybrid PKE scheme constructed from NTRU as KEM and appropriate DEM is anonymous and robust. (Similar...

This paper presents a side-channel analysis (SCA) on key encapsulation mechanism (KEM) based on the Fujisaki–Okamoto (FO) transformation and its variants. The FO transformation has been widely used in actively securing KEMs from passively secure public key encryption (PKE), as it is employed in most of NIST post-quantum cryptography (PQC) candidates for KEM. The proposed attack exploits side-channel leakage during execution of a psuedorandom function (PRF) in the re-encryption of KEM...

We investigate __all__ NIST PQC Round 3 KEM candidates from the viewpoint of fault-injection attacks: Classic McEliece, Kyber, NTRU, Saber, BIKE, FrodoKEM, HQC, NTRU Prime, and SIKE. All KEM schemes use variants of the Fujisaki-Okamoto transformation, so the equality test with re-encryption in decapsulation is critical. We survey effective key-recovery attacks when we can skip the equality test. We found the existing key-recovery attacks against Kyber, NTRU, Saber, FrodoKEM, HQC, one of two...

A new ``Weighted Bit-flipping'' (WBF) iterative decoder is presented and analyzed with respect to its Decoding Failure Rate (DFR). We show that the DFR is indeed lower than that of the BGF decoder as suggested by the BIKE third round submission to the NIST PQC standardization process. The WBF decoder requires more iterations to complete than BGF, but by creating a hybrid decoder we show that a lower DFR compared to that of the BGF decoder can still be achieved while keeping the computational...

BIKE is a key encapsulation mechanism that entered the third round of the NIST post-quantum cryptography standardization process. This paper presents two constant-time implementations for BIKE, one tailored for the Intel Haswell and one tailored for the ARM Cortex-M4. Our Haswell implementation is much faster than the avx2 implementation written by the BIKE team: for bikel1, the level-1 parameter set, we achieve a 1.39x speedup for decapsulation (which is the slowest operation) and a 1.33x...

The KEM BIKE is a Round-3 alternative finalist in the NIST Post-Quantum Cryptography project. It uses the FO$^{\not \bot}$ transformation so that an instantiation with a decoder that has a DFR of $2^{-128}$ will make it IND-CCA secure. The current BIKE design does not bind the randomness of the ciphertexts (i.e., the error vectors) to a specific public key. We propose to change this design, although currently, there is no attack that leverages this property. This modification can be...

This paper studies how to incorporate small information leakages (called “hints”) into information-set decoding (ISD) algorithms. In particular, the influence of these hints on solving the (n, k, t)-syndrome-decoding problem (SDP), i.e., generic syndrome decoding of a code of length n, dimension k, and an error of weight t, is analyzed. We motivate all hints by leakages obtainable through realistic side-channel attacks on code-based post-quantum cryptosystems. One class of studied hints...

We study in this work a particular class of QC-MDPC codes for which the decoding failure rate is significantly larger than for typical QC-MDPC codes of same parameters. Our purpose is to figure out whether the existence of such weak codes impacts the security of cryptographic schemes using QC-MDPC codes as secret keys. A class of weak keys was exhibited in [DGK19]. We generalize it and show that, though their Decoding Failure Rate (DFR) is higher than normal, the set is not large enough to...

Contemporary digital infrastructures and systems use and trust PKC to exchange keys over insecure communication channels. With the development and progress in the research field of quantum computers, well established schemes like RSA and ECC are more and more threatened. The urgent demand to find and standardize new schemes - which are secure in a post-quantum world - was also realized by the NIST which announced a PQC Standardization Project in 2017. Recently, the round three candidates...

The QC-MDPC code-based KEM BIKE is one of the Round-2 candidates of the NIST PQC standardization project. Its specification document describes a version that is claimed to have IND-CCA security. The security proof uses the Fujisaki-Okamoto transformation and a de-coder that targeted a Decoding Failure Rate (DFR) of 2^{-128} (for Level-1 security). However, there are several aspects that need to be amended in order for the IND-CCA proof to hold. The main issue is that using a decoder with DFR...

The NIST PQC standardization project evaluates multiple new designs for post-quantum Key Encapsulation Mechanisms (KEMs). Some of them present challenging tradeoffs between communication bandwidth and computational overheads. An interesting case is the set of QC-MDPC based KEMs. Here, schemes that use the Niederreiter framework require only half the communication bandwidth compared to schemes that use the McEliece framework. However, this requires costly polynomial inversion during the key...

BIKE (Bit-flipping Key Encapsulation) is a promising candidate running in the NIST Post-Quantum Cryptography Standardization process. It is a code-based cryptosystem that enjoys a simple definition, well-understood underlying security, and interesting performance. The most critical step in this cryptosystem consists of correcting errors in a QC-MDPC linear code. The BIKE team proposed variants of the Bit-Flipping Decoder for this step for Round 1 and 2 of the standardization process. In this...

McEliece-like code-based key exchange mechanisms using QC-MDPC codes can reach IND-CPA security under hardness assumptions from coding theory, namely quasi-cyclic syndrome decoding and quasi-cyclic codeword finding. To reach higher security requirements, like IND-CCA security, it is necessary in addition to prove that the decoding failure rate (DFR) is negligible, for some decoding algorithm and a proper choice of parameters. Getting a formal proof of a low DFR is a difficult task. Instead,...

QC-MDPC code-based KEMs rely on decoders that have a small or even negligible Decoding Failure Rate (DFR). These decoders should be efficient and implementable in constant-time. One example for a QC-MDPC KEM is the Round-2 candidate of the NIST PQC standardization project, "BIKE". We have recently shown that the Black-Gray decoder achieves the required properties. In this paper, we deffine several new variants of the Black-Gray decoder. One of them, called Black-Gray-Flip, needs only 7 steps...

The QC-MDPC code-based KEM Bit Flipping Key Encapsulation (BIKE) is one of the Round-2 candidates of the NIST PQC standardization project. It has a variant that is proved to be IND-CCA secure. The proof models the KEM with some black-box ("ideal") primitives. Specifically, the decapsulation invokes an ideal primitive called "decoder", required to deliver its output with a negligible Decoding Failure Rate (DFR). The concrete instantiation of BIKE substitutes this ideal primitive with a new...

Public key algorithms based on QC-MPDC and QC-LDPC codes for key encapsulation/encryption submitted to NIST post-quantum competition (BIKE, QC-MDPC KEM, LEDA) are vulnerable against reaction attacks based on decoding failures. To protect algorithms, authors propose to limit the key usage, in the extreme (BIKE) to only use ephemeral public keys. In some authenticated protocols, we need to combine each key with a signature, which can lead to increased traffic overhead, especially given large...

Continuous Key Agreement (CKA) is a two-party procedure used by Double Ratchet protocols (e. g., Signal). This is a continuous and synchronous protocol that generates a fresh key for every sent/received message. It guarantees forward secrecy and Post-Compromise Security (PCS). PCS allows for reestablishing the security within a few rounds after the state of one of the parties has been compromised. Alwen et al. have recently proposed a new KEM-based CKA construction where every message...

Quasi-cyclic moderate density parity check codes allow the design of McEliece-like public-key encryption schemes with compact keys and a security that provably reduces to hard decoding problems for quasi-cyclic codes. In particular, QC-MDPC are among the most promising code-based key encapsulation mechanisms (KEM) that are proposed to the NIST call for standardization of quantum safe cryptography (two proposals, BIKE and QC-MDPC KEM). The first generation of decoding algorithms suffers...