An algebraic approach to the Rank Support Learning problem

Rank-metric code-based cryptography relies on the hardness of decoding a random linear code in the rank metric. The Rank Support Learning problem (RSL) is a variant where an attacker has access to N decoding instances whose errors have the same support and wants to solve one of them. This problem is for instance used in the Durandal signature scheme. In this paper, we propose an algebraic attack on RSL which clearly outperforms the previous attacks to solve this problem. We build upon Bardet et al., Asiacrypt 2020, where similar techniques are used to solve MinRank and RD. However, our analysis is simpler and overall our attack relies on very elementary assumptions compared to standard Gr{\"o}bner bases attacks. In particular, our results show that key recovery attacks on Durandal are more efficient than was previously thought

Envisioning the Future of Cyber Security in Post-Quantum Era: A Survey on PQ Standardization, Applications, Challenges and Opportunities

The rise of quantum computers exposes vulnerabilities in current public key cryptographic protocols, necessitating the development of secure post-quantum (PQ) schemes. Hence, we conduct a comprehensive study on various PQ approaches, covering the constructional design, structural vulnerabilities, and offer security assessments, implementation evaluations, and a particular focus on side-channel attacks. We analyze global standardization processes, evaluate their metrics in relation to real-world applications, and primarily focus on standardized PQ schemes, selected additional signature competition candidates, and PQ-secure cutting-edge schemes beyond standardization. Finally, we present visions and potential future directions for a seamless transition to the PQ era

Some Notes on Code-Based Cryptography

This thesis presents new cryptanalytic results in several areas of coding-based cryptography. In addition, we also investigate the possibility of using convolutional codes in code-based public-key cryptography. The first algorithm that we present is an information-set decoding algorithm, aiming towards the problem of decoding random linear codes. We apply the generalized birthday technique to information-set decoding, improving the computational complexity over previous approaches. Next, we present a new version of the McEliece public-key cryptosystem based on convolutional codes. The original construction uses Goppa codes, which is an algebraic code family admitting a well-defined code structure. In the two constructions proposed, large parts of randomly generated parity checks are used. By increasing the entropy of the generator matrix, this presumably makes structured attacks more difficult. Following this, we analyze a McEliece variant based on quasi-cylic MDPC codes. We show that when the underlying code construction has an even dimension, the system is susceptible to, what we call, a squaring attack. Our results show that the new squaring attack allows for great complexity improvements over previous attacks on this particular McEliece construction. Then, we introduce two new techniques for finding low-weight polynomial multiples. Firstly, we propose a general technique based on a reduction to the minimum-distance problem in coding, which increases the multiplicity of the low-weight codeword by extending the code. We use this algorithm to break some of the instances used by the TCHo cryptosystem. Secondly, we propose an algorithm for finding weight-4 polynomials. By using the generalized birthday technique in conjunction with increasing the multiplicity of the low-weight polynomial multiple, we obtain a much better complexity than previously known algorithms. Lastly, two new algorithms for the learning parities with noise (LPN) problem are proposed. The first one is a general algorithm, applicable to any instance of LPN. The algorithm performs favorably compared to previously known algorithms, breaking the 80-bit security of the widely used (512,1/8) instance. The second one focuses on LPN instances over a polynomial ring, when the generator polynomial is reducible. Using the algorithm, we break an 80-bit security instance of the Lapin cryptosystem

Revisiting Algebraic Attacks on MinRank and on the Rank Decoding Problem

The Rank Decoding problem (RD) is at the core of rank-based cryptography. Cryptosystems such as ROLLO and RQC, which made it to the second round of the NIST Post-Quantum Standardization Process, as well as the Durandal signature scheme, rely on it or its variants. This problem can also be seen as a structured version of MinRank, which is ubiquitous in multivariate cryptography. Recently, [1,2] proposed attacks based on two new algebraic modelings, namely the MaxMinors modeling which is specific to RD and the Support-Minors modeling which applies to MinRank in general. Both improved significantly the complexity of algebraic attacks on these two problems. In the case of RD and contrarily to what was believed up to now, these new attacks were shown to be able to outperform combinatorial attacks and this even for very small field sizes. However, we prove here that the analysis performed in [2] for one of these attacks which consists in mixing the MaxMinors modeling with the Support-Minors modeling to solve RD is too optimistic and leads to underestimate the overall complexity. This is done by exhibiting linear dependencies between these equations and by considering an Fqm version of these modelings which turns out to be instrumental for getting a better understanding of both systems. Moreover, by working over Fqm rather than over Fq, we are able to drastically reduce the number of variables in the system and we (i) still keep enough algebraic equations to be able to solve the system, (ii) are able to analyze rigorously the complexity of our approach. This new approach may improve the older MaxMinors approach on RD from [1,2] for certain parameters. We also introduce a new hybrid approach on the Support-Minors system whose impact is much more general since it applies to any MinRank problem. This technique improves significantly the complexity of the Support-Minors approach for small to moderate field sizes. References: [1] An Algebraic Attack on Rank Metric Code-Based Cryptosystems, Bardet, Briaud, Bros, Gaborit, Neiger, Ruatta, Tillich, EUROCRYPT 2020. [2] Improvements of Algebraic Attacks for solving the Rank Decoding and MinRank problems, Bardet, Bros, Cabarcas, Gaborit, Perlner, Smith-Tone, Tillich, Verbel, ASIACRYPT 2020

An algebraic approach to the Rank Support Learning problem

Rank-metric code-based cryptography relies on the hardness of decoding a random linear code in the rank metric. The Rank Support Learning problem (RSL) is a variant where an attacker has access to N decoding instances whose errors have the same support and wants to solve one of them. This problem is for instance used in the Durandal signature scheme. In this paper, we propose an algebraic attack on RSL which clearly outperforms the previous attacks to solve this problem. We build upon Bardet et al., Asiacrypt 2020, where similar techniques are used to solve MinRank and RD. However, our analysis is simpler and overall our attack relies on very elementary assumptions compared to standard Gröbner bases attacks. In particular, our results show that key recovery attacks on Durandal are more efficient than was previously thought

Studies on the Security of Selected Advanced Asymmetric Cryptographic Primitives

The main goal of asymmetric cryptography is to provide confidential communication, which allows two parties to communicate securely even in the presence of adversaries. Ever since its invention in the seventies, asymmetric cryptography has been improved and developed further, and a formal security framework has been established around it. This framework includes different security goals, attack models, and security notions. As progress was made in the field, more advanced asymmetric cryptographic primitives were proposed, with other properties in addition to confidentiality. These new primitives also have their own definitions and notions of security. This thesis consists of two parts, where the first relates to the security of fully homomorphic encryption and related primitives. The second part presents a novel cryptographic primitive, and defines what security goals the primitive should achieve. The first part of the thesis consists of Article I, II, and III, which all pertain to the security of homomorphic encryption schemes in one respect or another. Article I demonstrates that a particular fully homomorphic encryption scheme is insecure in the sense that an adversary with access only to the public material can recover the secret key. It is also shown that this insecurity mainly stems from the operations necessary to make the scheme fully homomorphic. Article II presents an adaptive key recovery attack on a leveled homomorphic encryption scheme. The scheme in question claimed to withstand precisely such attacks, and was the only scheme of its kind to do so at the time. This part of the thesis culminates with Article III, which is an overview article on the IND-CCA1 security of all acknowledged homomorphic encryption schemes. The second part of the thesis consists of Article IV, which presents Vetted Encryption (VE), a novel asymmetric cryptographic primitive. The primitive is designed to allow a recipient to vet who may send them messages, by setting up a public filter with a public verification key, and providing each vetted sender with their own encryption key. There are three different variants of VE, based on whether the sender is identifiable to the filter and/or the recipient. Security definitions, general constructions and comparisons to already existing cryptographic primitives are provided for all three variants.Doktorgradsavhandlin

Design and Cryptanalysis of Symmetric-Key Algorithms in Black and White-box Models

Cryptography studies secure communications. In symmetric-key cryptography, the communicating parties have a shared secret key which allows both to encrypt and decrypt messages. The encryption schemes used are very efficient but have no rigorous security proof. In order to design a symmetric-key primitive, one has to ensure that the primitive is secure at least against known attacks. During 4 years of my doctoral studies at the University of Luxembourg under the supervision of Prof. Alex Biryukov, I studied symmetric-key cryptography and contributed to several of its topics. Part I is about the structural and decomposition cryptanalysis. This type of cryptanalysis aims to exploit properties of the algorithmic structure of a cryptographic function. The first goal is to distinguish a function with a particular structure from random, structure-less functions. The second goal is to recover components of the structure in order to obtain a decomposition of the function. Decomposition attacks are also used to uncover secret structures of S-Boxes, cryptographic functions over small domains. In this part, I describe structural and decomposition cryptanalysis of the Feistel Network structure, decompositions of the S-Box used in the recent Russian cryptographic standard, and a decomposition of the only known APN permutation in even dimension. Part II is about the invariant-based cryptanalysis. This method became recently an active research topic. It happened mainly due to recent extreme cryptographic designs, which turned out to be vulnerable to this cryptanalysis method. In this part, I describe an invariant-based analysis of NORX, an authenticated cipher. Further, I show a theoretical study of linear layers that preserve low-degree invariants of a particular form used in the recent attacks on block ciphers. Part III is about the white-box cryptography. In the white-box model, an adversary has full access to the cryptographic implementation, which in particular may contain a secret key. The possibility of creating implementations of symmetric-key primitives secure in this model is a long-standing open question. Such implementations have many applications in industry; in particular, in mobile payment systems. In this part, I study the possibility of applying masking, a side-channel countermeasure, to protect white-box implementations. I describe several attacks on direct application of masking and provide a provably-secure countermeasure against a strong class of the attacks. Part IV is about the design of symmetric-key primitives. I contributed to design of the block cipher family SPARX and to the design of a suite of cryptographic algorithms, which includes the cryptographic permutation family SPARKLE, the cryptographic hash function family ESCH, and the authenticated encryption family SCHWAEMM. In this part, I describe the security analysis that I made for these designs

Multilinear Maps in Cryptography

Multilineare Abbildungen spielen in der modernen Kryptographie eine immer bedeutendere Rolle. In dieser Arbeit wird auf die Konstruktion, Anwendung und Verbesserung von multilinearen Abbildungen eingegangen