We present a new distinguisher for alternant and Goppa codes, whose complexity is subexponential in the error-correcting capability, hence better than that of generic decoding algorithms. Moreover it does not suffer from the strong regime limitations of the previous distinguishers or structure recovery algorithms: in particular, it applies to the codes used in the Classic McEliece candidate for postquantum cryptography standardization. The invariants that allow us to distinguish are graded...

This paper presents an optimization of the memory cost of the quantum Information Set Decoding (ISD) algorithm proposed by Bernstein (PQCrypto 2010), obtained by combining Prange's ISD with Grover's quantum search. When the code has constant rate and length $n$, this algorithm essentially performs a quantum search which, at each iterate, solves a linear system of dimension $\mathcal{O}(n)$. The typical code lengths used in post-quantum public-key cryptosystems range from $10^3$ to $10^5$....

We study the hardness of the Syndrome Decoding problem, the base of most code-based cryptographic schemes, such as Classic McEliece, in the presence of side-channel information. We use ChipWhisperer equipment to perform a template attack on Classic McEliece running on an ARM Cortex-M4, and accurately classify the Hamming weights of consecutive 32-bit blocks of the secret error vector. With these weights at hand, we optimize Information Set Decoding algorithms. Technically, we show how to...

Syndrome decoding problem (SDP) is the security assumption of the code-based cryptography. Three out of the four NIST-PQC round 4 candidates are code-based cryptography. Information set decoding (ISD) is known for the fastest existing algorithm to solve SDP instances with relatively high code rate. Security of code-based cryptography is often constructed on the asymptotic complexity of the ISD algorithm. However, the concrete complexity of the ISD algorithm has hardly ever been known....

The first contribution of this work is a generalisation of Stern's information set decoding (ISD) algorithm. Stern's algorithm, a variant of Stern's algorithm due to Dumer, as well as a recent generalisation of Stern's algorithm due to Bernstein and Chou are obtained as special cases of our generalisation. Our second contribution is to introduce the notion of a set of effective time/memory trade-off (TMTO) points for any ISD algorithm for given ranges of values of parameters of the...

Public-key cryptography is widely deployed for encrypting stored files. This paper uses microbenchmarks and purchase costs to predict the performance of various post-quantum KEMs in this application. In particular, this paper concludes that Classic McEliece is (1) the most efficient option and (2) easily affordable. As a quantitative example, the estimated five-year per-user cost of mceliece6960119f is 0.00024 dollars if 100000 files are encrypted and stored for an average user during those...

In recent years, quantum computers and Shor’s quantum algorithm have been able to effectively solve NP (Non-deterministic Polynomial-time) problems such as prime factorization and discrete logarithm problems, posing a threat to current mainstream asymmetric cryptography, including RSA and Elliptic Curve Cryptography (ECC). As a result, the National Institute of Standards and Technology (NIST) in the United States call for Post-Quantum Cryptography (PQC) methods that include lattice-based...

In recent years, the elliptic curve Qu-Vanstone (ECQV) implicit certificate scheme has found application in security credential management systems (SCMS) and secure vehicle-to-everything (V2X) communication to issue pseudonymous certificates. However, the vulnerability of elliptic-curve cryptography (ECC) to polynomial-time attacks posed by quantum computing raises concerns. In order to enhance resistance against quantum computing threats, various post-quantum cryptography methods have been...

The McEliece cryptosystem is a strong contender for post-quantum schemes, including key encapsulation for confidentiality of key exchanges in network protocols. A McEliece secret key is a structured parity check matrix that is transformed via Gaussian elimination into an unstructured public key. We show that this transformation is a highly critical operation with respect to side-channel leakage. We assume leakage of the elementary row operations during Gaussian elimination, motivated by...

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...

We introduce a novel algebraic approach for attacking the McEliece cryptosystem which is currently at the $4$-th round of the NIST competition. The contributions of the article are twofold. (1) We present a new distinguisher on alternant and Goppa codes working in a much broader range of parameters than \cite{FGOPT11}. (2) With this approach we also provide a polynomial--time key recovery attack on alternant codes which are distinguishable with the distinguisher \cite{FGOPT11}. ...

Quantitative analyses of the costs of cryptographic attack algorithms play a central role in comparing cryptosystems, guiding the search for improved attacks, and deciding which cryptosystems to standardize. Unfortunately, these analyses often turn out to be wrong. Sometimes errors are not caught until years later. This paper introduces CryptAttackTester (CAT), a software framework for high-assurance quantification of attack effectiveness. CAT enforces complete definitions of attack...

As the technical feasibility of a quantum computer becomes more and more likely, post-quantum cryptography algorithms are receiving particular attention in recent years. Among them, code-based cryptosystems were first considered unsuited for hardware and embedded software implementations because of their very large key sizes. However, recent work has shown that such implementations are practical, which also makes them susceptible to physical attacks. In this article, we propose a horizontal...

The generalized inverses of systematic non-square binary matrices have applications in mathematics, channel coding and decoding, navigation signals, machine learning, data storage, and cryptography, such as the McEliece and Niederreiter public-key cryptosystems. A systematic non-square (n−k)×n matrix H, n > k, has 2 power k×(n−k) different generalized inverse matrices. This paper presents an algorithm for generating these matrices and compares it with two well-known methods, i.e....

The generalized inverses of systematic non-square binary matrices have applications in mathematics, channel coding and decoding, navigation signals, machine learning, data storage and cryptography such as the McEliece and Niederreiter public-key cryptosystems. A systematic non-square $(n-k) \times n$ matrix $H$, $n > k$, has $2^{k\times(n-k)}$ different generalized inverse matrices. This paper presents an algorithm for generating these matrices and compares it with two well-known methods,...

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...

Digital signatures ensure legitimate access through identity authentication. It is also used to build blocks in blockchains and to authenticate transactions. The Courtois-Finiasz-Sendrier (CFS) digital signature is a well-known code-based digital signature scheme based on the Niederreiter cryptosystem. The CFS signature, however, is not widely used due to the long processing time required by its signing algorithm. Most code-based digital signature schemes are based on Niederreiter. The...

The security of cryptographic primitives is an important issue. The Shor algorithm illustrates how quantum attacks threaten the security of these widely used primitives. Code-based cryptography is one of several approaches resistant to quantum attacks. To date, no attack has been able to break a code-based cryptosystem in polynomial time. Despite this level of security, these cryptosystems have not been considered for practical applications such as e-commerce, medical and industrial IoT,...

Among the fourth round finalists of the NIST post-quantum cryptography standardization process for public-key encryption algorithms and key encapsulation mechanisms, three rely on hard problems from coding theory. Key encapsulation mechanisms are frequently used in hybrid cryptographic systems: a public-key algorithm for key exchange and a secret key algorithm for communication. A major point is thus the initial key exchange that is performed thanks to a key encapsulation mechanism. In this...

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...

Developers of computer-aided cryptographic tools are optimistic that formal methods will become a vital part of developing new cryptographic systems. We study the use of such tools to specify and verify the implementation of Classic McEliece, one of the code-based cryptography candidates in the fourth round of the NIST Post-Quantum standardisation Process. From our case study we draw conclusions about the practical applicability of these methods to the development of novel cryptography.

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 present an algebraic attack on a McEliece-like scheme based on BCH codes (BCH-McEliece), where the Goppa code is replaced by a suitably permuted BCH code. Our attack continues the line of work devising attacks against McEliece-like schemes with Goppa-like codes, with the goal of getting a better understanding of why Goppa codes are so intractable. Our starting point is the work of Faugère, Perret and Portzamparc (Asiacrypt 2014). We take their algebraic model and adapt and improve their...

TLS is ubiquitous in modern computer networks. It secures transport for high-end desktops and low-end embedded devices alike. However, the public key cryptosystems currently used within TLS may soon be obsolete as large-scale quantum computers, once realized, would be able to break them. This threat has led to the development of post-quantum cryptography (PQC). The U.S. standardization body NIST is currently in the process of concluding a multi-year search for promising post-quantum...

Recently, the results of the NIST PQC contest were announced. Classic McEliece, one of the 3rd round candidates, was selected as the fourth round candidate. Classic McEliece is the only code-based cipher in the NIST PQC finalists in third round and the algorithm is regarded as secure. However, it has low efficiency. In this paper, we propose an efficient software implementation of Classic McEliece, a code-based cipher, on 64-bit ARMv8 processors. Classic McEliece can be divided into Key...

Post-quantum Cryptography (PQC) has reached the verge of standardization competition, with Kyber as a winning candidate. In this work, we demonstrate practical backdoor insertion in Kyber through kleptrography. The backdoor can be inserted using classical techniques like ECDH or post-quantum Classic Mceliece. The inserted backdoor targets the key generation procedure where generated output public keys subliminally leak information about the secret key to the owner of the backdoor. We...

Classic McEliece is a code based encryption scheme and candidate of the NIST post quantum contest. Implementing Classic McEliece on smart card chips is a challenge, because those chips have only a very limited amount of RAM. Decryption is not an issue because the cryptogram size is short and the decryption algorithm can be implemented using very few RAM. However key generation is a concern, because a large binary matrix must be inverted. In this paper, we show how key generation can be done...

QC-MDPC (quasi cyclic moderate density parity check) code-based McEliece cryptosystems are considered to be one of the candidates for post-quantum cryptography. Decreasing DER (decoding error rate) is one of important factor for their security, since recent attacks to these cryptosystems effectively use DER information. In this paper, we pursue the possibility of optimization-base decoding, concretely we examine ADMM (alternating direction method of multipliers), a recent developing...

We present a key-recovery fault injection attack on the Classic McEliece Key Encapsulation Mechanism (KEM). The fault injections target the error-locator polynomial of the Goppa code and the validity checks in the decryption algorithm, making a chosen ciphertext attack possible. Faulty decryption outputs are used to generate a system of polynomial equations in the secret support elements of the Goppa code. After solving the equations, we can determine a suitable Goppa polynomial and form an...

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...

Classic McEliece is a code-based quantum-resistant public-key scheme characterized with relative high encapsulation/decapsulation speed and small cipher- texts, with an in-depth analysis on its security. However, slow key generation with large public key size make it hard for wider applications. Based on this observation, a high-throughput key generator in hardware, is proposed to accelerate the key generation in Classic McEliece based on algorithm-hardware co-design. Meanwhile the storage...

This paper presents a new McEliece-type encryption scheme based on Gabidulin codes, which uses linearized transformations to disguise the private key. When endowing this scheme with the partial cyclic structure, we obtain a public key of the form $GM^{-1}$, where $G$ is a partial circulant generator matrix of Gabidulin code and $M$ as well as $M^{-1}$ is a circulant matrix of large rank weight, even as large as the code length. Another difference from Loidreau's proposal at PQCrypto 2017 is...

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 $\mathcal{C}$,...

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...

We consider the McEliece cryptosystem with a binary Goppa code $C \subset \mathbb{F}_2^n$ specified by an irreducible Goppa polynomial $g(x) \in \mathbb{F}_{2^m}[X]$ and Goppa points $(\alpha_1, \ldots, \alpha_n) \in \mathbb{F}_{2^m}^n$. Since $g(x)$ together with the Goppa points allow for efficient decoding, these parameters form McEliece secret keys. Such a Goppa code $C$ is an $(n-tm)$-dimensional subspace of $\mathbb{F}_2^n$, and therefore $C$ has co-dimension $tm$. For typical...

In this paper, we propose the first key-recovery side-channel attack on Classic McEliece, a KEM finalist in the NIST Post-quantum Cryptography Standardization Project. Our novel idea is to design an attack algorithm where we submit special ciphertexts to the decryption oracle that correspond to cases of single errors. Decoding of such cipher-texts involves only a single entry in a large secret permutation, which is part of the secret key. Through an identified leakage in the additive...

This paper reviews, from bottom to top, a polynomial-time algorithm to correct t errors in classical binary Goppa codes defined by squarefree degree-t polynomials. The proof is factored through a proof of a simple Reed--Solomon decoder, and the algorithm is simpler than Patterson's algorithm. All algorithm layers are expressed as Sage scripts backed by test scripts. All theorems are formally verified. The paper also covers the use of decoding inside the Classic McEliece cryptosystem,...

We present the first specification-compliant constant-time FPGA implementation of the Classic McEliece cryptosystem from the third-round of NIST's Post-Quantum Cryptography standardization process. In particular, we present the first complete implementation including encapsulation and decapsulation modules as well as key generation with seed expansion. All the hardware modules are parametrizable, at compile time, with security level and performance parameters. As the most time consuming...

We show how to backdoor the McEliece cryptosystem such that a backdoored public key is indistinguishable from a usual public key, but allows to efficiently retrieve the underlying secret key. For good cryptographic reasons, McEliece uses a small random seed 𝛅 that generates via some pseudo random generator (PRG) the randomness that determines the secret key. Our backdoor mechanism works by encoding an encryption of 𝛅 into the public key. Retrieving 𝛅 then allows to efficiently recover...

The NIST standardization process for post-quantum cryptography has been drawing the attention of researchers to the submitted candidates. One direction of research consists in implementing those candidates on embedded systems and that exposes them to physical attacks in return. The Classic McEliece cryptosystem, which is among the four finalists of round 3 in the Key Encapsulation Mechanism category, builds its security on the hardness of the syndrome decoding problem, which is a classic...

Encryption satisfying CCA2 security is commonly known to be unnecessarily strong for realizing secure channels. Moreover, CCA2 constructions in the standard model are far from being competitive practical alternatives to constructions via random oracle. A promising research area to alleviate this problem are weaker security notions—like IND-RCCA secure encryption or IND-atag-wCCA secure tag-based encryption—which are still able to facilitate secure message transfer (SMT) via authenticated...

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,...

A natural and recurring idea in the knapsack/lattice cryptography literature is to start from a lattice with remarkable decoding capability as your private key, and hide it somehow to make a public key. This is also how the code-based encryption scheme of McEliece (1978) proceeds. This idea has never worked out very well for lattices: ad-hoc approaches have been proposed, but they have been subject to ad-hoc attacks, using tricks beyond lattice reduction algorithms. On the other hand the...

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...

The paper is devoted to the Hadamard square of concatenated linear codes. Such codes consist of codewords that are obtained by concatenation part of the codewords from other codes. It is proved that if the sum of Hadamard squares’ dimensions of the codes used in the concatenation is slightly less than the dimension of the entire space, then the Hadamard square of the concatenated code is equal to the Cartesian product of the Hadamard square of code-components. It means that the cryptanalysis...

This paper presents a new technique for disturbing the algebraic structure of linear codes in code-based cryptography. Specifically, we introduce the so-called semilinear transformations in coding theory and then creatively apply them to the construction of code-based cryptosystems. Note that $\mathbb{F}_{q^m}$ can be viewed as an $\mathbb{F}_q$-linear space of dimension $m$, a semilinear transformation $\varphi$ is therefore defined as an $\mathbb{F}_q$-linear automorphism of...

This paper presents two public-key cryptosystems based on the so-called expanded Gabidulin codes, which are constructed by expanding Gabidulin codes over the base field. Exploiting the fast decoder of Gabidulin codes, we propose an efficient algorithm to decode these new codes when the noise vector satisfies a certain condition. Additionally, these new codes have an excellent error-correcting capability because of the optimality of their parent Gabidulin codes. Based on different masking...

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...

One of the finalists in the NIST post-quantum cryptography competition is the Classic McEliece cryptosystem. Unfortunately, its public key size represents a practical limitation. One option to address this problem is to use different families of error-correcting codes. Most of such attempts failed as those cryptosystems were proved not secure. In this paper, we propose a McEliece type cryptosystem using high minimum distance self-dual codes and punctured codes derived from them. To the best...

While server-only authentication with certificates is the most widely used mode of operation for the Transport Layer Security (TLS) protocol on the world wide web, there are many applications where TLS is used in a different way or with different constraints. For example, embedded Internet-of-Things clients may have a server certificate pre-programmed and be highly constrained in terms of communication bandwidth or computation power. As post-quantum algorithms have a wider range of...

A core goal of the NIST PQC competition is to produce public-key encryption (PKE) schemes which, even if attacked with a large-scale quantum computer, maintain the security guarantees needed by applications. The main security focus in the NIST PQC context has been IND-CCA security, but other applications demand that PKE schemes provide 'anonymity' (Bellare et al., ASIACRYPT 2001), and 'robustness' (Abdalla et al., TCC 2010). Examples of such applications include anonymous communication...

This paper presents a constant-time implementation of Classic McEliece for ARM Cortex-M4. Specifically, our target platform is stm32f4-Discovery, a development board on which the amount of SRAM is not even large enough to hold the public key of the smallest parameter sets of Classic McEliece. Fortunately, the flash memory is large enough, so we use it to store the public key. For the level-1 parameter sets mceliece348864 and mceliece348864f, our implementation takes 582 199 cycles for...

This paper presents two modifications for Loidreau’s code-based cryptosystem. Loidreau’s cryptosystem is a rank metric code-based cryptosystem constructed by using Gabidulin codes in the McEliece setting. Recently a polynomial-time key recovery attack was proposed to break Loidreau’s cryptosystem in some cases. To prevent this attack, we propose the use of subcodes to disguise the secret codes in Modification I. In Modification II, we choose a random matrix of low column rank over F q to mix...

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...

The Classic McEliece cryptosystem is one of the most trusted quantum-resistant cryptographic schemes. Deploying it in practical applications, however, is challenging due to the size of its public key. In this work, we bridge this gap. We present an implementation of Classic McEliece on an ARM Cortex-M4 processor, optimized to overcome memory constraints. To this end, we present an algorithm to retrieve the public key ad-hoc. This reduces memory and storage requirements and enables the...

Code-based public-key cryptosystems are promising candidates for standardization as quantum-resistant public-key cryptographic algorithms. Their security is based on the hardness of the syndrome decoding problem. Computing the syndrome in a finite field, usually $\mathbb{F}_2$, guarantees the security of the constructions. We show in this article that the problem becomes considerably easier to solve if the syndrome is computed in $\mathbb{N}$ instead. By means of laser fault injection, we...

The existence of some structure in a code can lead to the decrease of security of the whole system built on it. Often subcodes are used to ``disguise'' the code as a ``general-looking'' one. However, the security of subcodes, whose Hadamard square is equal to the square of the base code, is reduced to the security of this code, i.e. this condition is undesirable. The paper finds the limiting conditions on the number of vectors of degree $ r $ removing of which retains this weakness for...

We report on the concrete cryptanalysis of LEDAcrypt, a 2nd Round candidate in NIST's Post-Quantum Cryptography standardization process and one of 17 encryption schemes that remain as candidates for near-term standardization. LEDAcrypt consists of a public-key encryption scheme built from the McEliece paradigm and a key-encapsulation mechanism (KEM) built from the Niederreiter paradigm, both using a quasi-cyclic low-density parity-check (QC-LDPC) code. In this work, we identify a large...

In this paper we present PQ-WireGuard, a post-quantum variant of the handshake in the WireGuard VPN protocol (NDSS 2017). Unlike most previous work on post-quantum security for real-world protocols, this variant does not only consider post-quantum confidentiality (or forward secrecy) but also post-quantum authentication. To achieve this, we replace the Diffie-Hellman-based handshake by a more generic approach only using key-encapsulation mechanisms (KEMs). We establish security of...

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...

NTS-KEM is one of the 17 post-quantum public-key encryption (PKE) and key establishment schemes remaining in contention for standardization by NIST. It is a code-based cryptosystem that starts with a combination of the (weakly secure) McEliece and Niederreiter PKE schemes and applies a variant of the Fujisaki-Okamoto (Journal of Cryptology 2013) or Dent (IMACC 2003) transforms to build an IND-CCA secure key encapsulation mechanism (KEM) in the classical random oracle model (ROM). Such...

This paper presents an attack based on side-channel information and Information Set Decoding (ISD) on the Niederreiter cryptosystem and an evaluation of the practicality of the attack using an electromagnetic side channel. First, we describe a basic plaintext-recovery attack on the decryption algorithm of the Niederreiter cryptosystem. In case the cryptosystem is used as Key-Encapsulation Mechanism (KEM) in a key exchange, the plaintext corresponds to a session key. Our attack is an...

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,...

Verifiable random functions (VRFs) are essentially digital signatures with additional properties, namely verifiable uniqueness and pseudorandomness, which make VRFs a useful tool, e.g., to prevent enumeration in DNSSEC Authenticated Denial of Existence and the CONIKS key management system, or in the random committee selection of the Algorand blockchain. Most standard-model VRFs rely on admissible hash functions (AHFs) to achieve security against adaptive attacks in the standard model. Known...

The use of rank instead of Hamming metric has been proposed to address the main drawback of code-based cryptography: large key sizes. There exist several Key Encapsulation Mechanisms (KEM) and Public Key Encryption (PKE) schemes using rank metric including some submissions to the NIST call for standardization of Post-Quantum Cryptography. In this work, we present an IND-CCA PKE scheme based on the McEliece adaptation to rank metric proposed by Loidreau at PQC 2017. This IND-CCA PKE scheme...

Since its invention by McEliece in 1978, cryptography based on Error Correcting Codes (ECC) has suffered from the reputation of not being suitable for constrained devices. Indeed, McEliece's scheme and its variants have large public keys and relatively long ciphertexts. Recent works on these downsides explored the possible use of ECC based on rank metric instead of Hamming metric. These codes were introduced in the late 80's to eliminate errors with repeating patterns, regardless of their...

McEliece and Niederreiter cryptosystems are robust and versatile cryptosystems. These cryptosystems work with any linear error-correcting codes. They are popular these days because they can be quantum-secure. In this paper, we study the Niederreiter cryptosystem using quasi-cyclic codes. We prove, if these quasi-cyclic codes satisfy certain conditions, the corresponding Niederreiter cryptosystem is resistant to the hidden subgroup problem using quantum Fourier sampling. Our proof requires...

Twisted Reed-Solomon (TRS) codes are a family of codes that contains a large number of maximum distance separable codes that are non-equivalent to Reed--Solomon codes. TRS codes were recently proposed as an alternative to Goppa codes for the McEliece code-based cryptosystem, resulting in a potential reduction of key sizes. The use of TRS codes in the McEliece cryptosystem has been motivated by the fact that a large subfamily of TRS codes is resilient to a direct use of known algebraic...

In this paper, we present a high-speed constant time hardware implementation of NTRUEncrypt Short Vector Encryption Scheme (SVES), fully compliant with the IEEE 1363.1 Standard Specification for Public Key Cryptographic Techniques Based on Hard Problems over Lattices. Our implementation follows an earlier proposed Post-Quantum Cryptography (PQC) Hardware Application Programming Interface (API), which facilitates its fair comparison with implementations of other PQC schemes. The paper...

This paper introduces a constant-time implementation for a quasi-cyclic moderate-density-parity-check (QC-MDPC) code based encryption scheme. At a $2^{80}$ security level, the software takes 14679937 Cortex-M4 and 1560072 Haswell cycles to decrypt a short message, while the previous records were 18416012 and 3104624 (non-constant-time) cycles. Such speed is achieved by combining two techniques: 1) performing each polynomial multiplication in $\mathbb{F}_2[x]/(x^r-1)$ and...

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...

In this paper, we propose a new general construction to reduce the public key size of McEliece cryptosystems constructed from automorphism-induced Goppa codes. In particular, we generalize the ideas of automorphism-induced Goppa codes by considering nontrivial subsets of automorphism groups to construct Goppa codes with a nice block structure. By considering additive and multiplicative automorphism subgroups, we provide explicit constructions to demonstrate our technique. We show that our...

We present and analyze the performance of DRANKULA, a McEliece-like cryptosystem implementation using \textit{rank metric} instead of Hamming distance. Namely, we use the scheme proposed by Loidreau in PQCrypto 2017 using Gabidulin codes. We propose a set of carefully selected parameters and we address several non-trivial issues when porting this scheme into real-world systems as, for example, the generation of errors of a given rank. We provide the pseudo-code of the core algorithms of the...

We present a key recovery attack against Y. Wang's Random Linear Code Encryption (RLCE) scheme recently submitted to the NIST call for post-quantum cryptography. This attack recovers the secret key for all the short key parameters proposed by the author.

We present an efficient key recovery attack on code based encryption schemes using some quasi–dyadic alternant codes with extension degree 2. This attack permits to break the proposal DAGS recently submitted to NIST.

In 2013, Misoczki, Tillich, Sendrier and Barreto proposed a variant of the McEliece cryptosystem based on quasi-cyclic moderate-density parity-check (QC-MDPC) codes. This proposal uses an iterative bit-flipping algorithm in its decryption procedure. Such algorithms fail with a small probability. At Asiacrypt 2016, Guo, Johansson and Stankovski (GJS) exploited these failures to perform a key recovery attack. They introduced the notion of the distance spectrum of a sparse vector and showed...

We propose a new reaction attack on the public-key cryptosystem LEDApkc. The adversary uses the decoding failure rate (DFR) analysis to learn information about the secret masking matrix $Q$. Provided the adversary learns information about $Q$ within $10^4\times \text{DFR}^{-1}$ decryptions (as prescribed by LEDApkc design to thwart previously known attacks), the adversary builds a small set of candidates for $Q$. Using these candidates, the adversary obtains candidates for a generator matrix...

The anticipated emergence of quantum computers in the foreseeable future drives the cryptographic community to start considering cryptosystems, which are based on problems that remain intractable even with strong quantum computers. One example is the family of code-based cryptosystems that relies on the Syndrome Decoding Problem (SDP). Recent work by Misoczki et al. [34] showed a variant of McEliece encryption which is based on Quasi Cyclic - Moderate Density Parity Check (MDPC) codes, and...

Oblivious Transfer (OT) is a fundamental cryptographic protocol that finds a number of applications, in particular, as an essential building block for two-party and multi-party computation. We construct a universally composable (UC) protocol for oblivious transfer secure against active adaptive adversaries from any OW-CPA secure public-key encryption scheme with certain properties in the random oracle model (ROM). In terms of computation, our protocol only requires the generation of a...

This paper presents a constant-time fast implementation for a high-security code-based encryption system. The implementation is based on the “McBits” paper by Bernstein, Chou, and Schwabe in 2013: we use the same FFT algorithms for root finding and syndrome computation, similar algorithms for secret permutation, and bitslicing for low-level operations. As opposed to McBits, where a high decryption throughput is achieved by running many decryption operations in parallel, we take a different...

Current widely-used key exchange (KE) mechanisms will be vulnerable to quantum attacks when sufficiently strong quantum computers become available. Therefore, devising quantum-resistant replacements that combine efficiency with solid security guarantees is an important and challenging task. This paper proposes several contributions towards this goal. First, we introduce CAKE, a key encapsulation algorithm based on the QC-MDPC McEliece encryption scheme, with two major improvements: a) the...

QcBits is a code-based public key algorithm based on a problem thought to be resistant to quantum computer attacks. It is a constant time implementation for a quasi-cyclic moderate density parity check (QC-MDPC) Niederreiter encryption scheme, and has excellent performance and small key sizes. In this paper, we present a key recovery attack against QcBits. We first used differential power analysis (DPA) against the syndrome computation of the decoding algorithm to recover partial information...

Guo et al. recently presented a reaction attack against the QC-MDPC McEliece cryptosystem. Their attack is based on the observation that when a bit-flipping decoding algorithm is used in the QC-MDPC McEliece, then there exists a dependence between the secret matrix $H$ and the failure probability of the bit-flipping algorithm. This dependence can be exploited to reveal the matrix $H$ which constitutes the private key in the cryptosystem. It was conjectured that such dependence is present...

We design a new McEliece-like rank metric based encryption scheme from Gabidulin codes. We explain why it is not affected by the invariant subspace attacks also known as Overbeck's attacks. The idea of the design mixes two existing approaches designing rank metric based encryption schemes. For a given security our public-keys are more compact than for the same security in the Hamming metric based settings.

The security of code-based cryptosystems such as the McEliece cryptosystem relies primarily on the difficulty of decoding random linear codes. The best decoding algorithms are all improvements of an old algorithm due to Prange: they are known under the name of information set decoding techniques. It is also important to assess the security of such cryptosystems against a quantum computer. This research thread started in Overbeck and Sendrier's 2009 survey on code-based cryptography, and the...

Recently, Wang (2016) introduced a random linear code based quantum resistant public key encryp- tion scheme RLCE which is a variant of McEliece encryption scheme. In this paper, we introduce a revised version of the RLCE encryption scheme. The revised RLCE schemes are more efficient than the original RLCE scheme. Specifically, it is shown that RLCE schemes have smaller public key sizes com- pared to binary Goppa code based McEliece encryption schemes for corresponding security levels. The...

The BBCRS scheme is a variant of the McEliece public-key encryption scheme where the hiding phase is performed by taking the inverse of a matrix which is of the form T R where T is a sparse matrix with average row/column weight equal to a very small quantity m, usually m<2, and R is a matrix of small rank z&#10878;1. The rationale of this new transformation is the reintroduction of families of codes, like generalized Reed-Solomon codes, that are famously known for representing insecure...

Recently we have defined Staircase-Generator codes (St-Gen codes) and their variant with a random split of the generator matrix of the codes. One unique property of these codes is that they work with arbitrary error sets. In this paper we give a brief overview of St-Gen codes and the list decoding algorithm for their decoding. We also analyze the semantic security against chosen plaintext attack (IND-CPA) and key-privacy i.e. indistinguishability of public keys under chosen plaintext attack...

The McEliece cryptosystem is an asymmetric type of cryptography based on error correction code. The classical McEliece used irreducible binary Goppa code which considered unbreakable until now especially with parameter [1024, 524, and 101], but it is suffering from large public key matrix which leads to be difficult to be used practically. In this work Irreducible and Separable Goppa codes have been introduced. The Irreducible and Separable Goppa codes used are with flexible parameters and...

Recently, Gligoroski et al. proposed code-based encryption and signature schemes using list decoding, blockwise triangular private keys, and a nonuniform error pattern based on ``generalized error sets." The general approach was referred to as \emph{McEliece in the World of Escher.} This paper demonstrates attacks which are significantly cheaper than the claimed security level of the parameters given by Gligoroski et al. We implemented an attack on the proposed 80-bit parameters which was...

Instantiations of the McEliece cryptosystem which are considered computationally secure even in a post-quantum era still require hardening against side channel attacks for practical applications. Recently, the first differential power analysis attack on a McEliece cryptosystem successfully recovered the full secret key of a state-of-the-art FPGA implementation of QC-MDPC McEliece. In this work we show how to apply masking countermeasures to the scheme and present the first masked FPGA...

In this paper, we first present a new class of code based public key cryptosystem(PKC) based on Reed-Solomon code over extension field of less than m=9, referred to as K(XVI)SE(1)PKC. We then present a new class of quadratic multivariate PKC, K(XVI)SE(2)PKC, based on binary cyclic code. We show that both K(XVI)SE(1)PKC and K(XVI)SE(2)PKC can be secure against the various linear transformation attacks such as Grobner bases attack due to a non-linear structure introduced when constructing the...

This paper presents extremely fast algorithms for code-based public-key cryptography, including full protection against timing attacks. For example, at a 2^128 security level, this paper achieves a reciprocal decryption throughput of just 60493 cycles (plus cipher cost etc.) on a single Ivy Bridge core. These algorithms rely on an additive FFT for fast root computation, a transposed additive FFT for fast syndrome computation, and a sorting network to avoid cache-timing attacks.

We solve an open question in code-based cryptography by introducing the first provably secure group signature scheme from code-based assumptions. Specifically, the scheme satisfies the CPA-anonymity and traceability requirements in the random oracle model, assuming the hardness of the McEliece problem, the Learning Parity with Noise problem, and a variant of the Syndrome Decoding problem. Our construction produces smaller key and signature sizes than the existing post-quantum group signature...

In the last years code-based cryptosystems were established as promising alternatives for asymmetric cryptography since they base their security on well-known NP-hard problems and still show decent performance on a wide range of computing platforms. The main drawback of code-based schemes, including the popular proposals by McEliece and Niederreiter, are the large keys whose size is inherently determined by the underlying code. In a very recent approach, Misoczki et al. proposed to use...

We analyse the security regarding timing attacks of implementations of the decryption in McEliece PKC with binary Goppa codes. First, we review and extend the existing attacks, both on the messages and on the keys. We show that, until now, no satisfactory countermeasure could erase all the timing leakages in the Extended Euclidean Algorithm (EEA) step. Then, we describe a version of the EEA never used for McEliece so far. It uses a constant number of operations for given public...

Goldreich-Goldwasser-Halevi (GGH) public key cryptosystem is an instance of lattice-based cryptosystems whose security is based on the hardness of lattice problems. In fact, GGH cryptosystem is the lattice version of the first code-based cryptosystem, proposed by McEliece. However, it has a number of drawbacks such as; large public key length and low security level. On the other hand, Low Density Lattice Codes (LDLCs) are the practical classes of lattice codes which can achieve capacity on...

We show that the plaintext of some of the proposed CCA2 conversions of McEliece cryptosystem with a public key in systematic form can be recovered faster than with a general linear decoding. This is due to the fact that an attacker only needs to recover a part of the cleartext to decrypt the relevant plaintext.