In this paper, we revisit the Rank Quasi-Cyclic (RQC) (Melchor et al., IEEE IT, 2018) encryption scheme by proposing three possible variations for its design. Our first improvement relies on the introduction of Augmented Gabidulin codes, a new family of decodable codes exploiting the concept of support erasure for the rank metric. Following the work of Melchor et al. (PQCrypto, 2022), our second improvement uses multiple syndromes to increase the weight of the error to be decoded. As pioneered in Melchor et al. (NIST PQC, 2020), our third variation considers non-homogeneous error weights in order to decrease the parameters. These improvements can be combined together to design schemes offering various trade-offs in term of security and size. Our Multi-UR-AG (multiple syndromes, unstructured, augmented Gabidulin) scheme achieves a size of 11kB (public key + ciphertext) for 128 bits of security while featuring a conservative design as it relies on pure random instances without any ideal structure. Besides, our NH- Multi-RQC-AG (non-homogeneous error, multiple syndromes, ideal structure, augmented Gabidulin) achieves a size of 2.7 kB for 128 bits of security, namely a 50 % improvement with respect to classical RQC. Our second and third variations respectively rely on the security of the$\textsf {RSL} $and$\textsf {NHRSD} $problems (or$\textsf {NHRSL} $when considered together). In this paper, we also provide new security analysis and attacks for these problems. While these results are important for our new schemes, they are of independent interest as well. Our security analysis for the$\textsf {RSL} $problem provides an improvement on the recent algebraic attacks for some instances. In addition, we show that the$\textsf {RSL} $problem can be solved in polynomial time when$N \geq (k+1) r\frac {m}{m-r}$, this improves the best known combinatorial attack (Gaborit et al., Crypto, 2017). We also propose the first combinatorial attack against the$\textsf {NHRSD} $problem along with a precise complexity analysis of the algebraic attack described Melchor et al. (NIST PQC, 2020). At last, we combine these analysis to provide an attack against the$\textsf {NHRSL} $problem.

The ongoing NIST standardization process has shown that Proof of Knowledge (PoK) based signatures have become an important type of possible post-quantum signatures. Regarding code-based cryptography, the main original approach for PoK based signatures is the Stern protocol which allows to prove the knowledge of a small weight vector solving a given instance of the Syndrome Decoding (SD) problem over ${\mathbb{F}_2}$. It features a soundness error equal to 2/3. This protocol was improved a few years later by Véron who proposed a variation of the scheme based on the General Syndrome Decoding (GSD) problem which leads to better results in terms of communication. A few years later, the AGS protocol introduced a variation of the Véron protocol based on Quasi-Cyclic (QC) matrices. The AGS protocol permits to obtain an asymptotic soundness error of 1/2 and an improvement in terms of communications.In the present paper, we introduce the Quasi-Cyclic Stern PoK which constitutes an adaptation of the AGS scheme in a SD context, as well as several new optimizations for code-based PoK. Our main optimization on the size of the signature cannot be applied to GSD based protocols such as AGS and therefore motivated the design of our new protocol. In addition, we also provide a special soundness proof that is compatible with the use of the Fiat-Shamir transform for 5-round protocols. This approach is valid for our protocol but also for the AGS protocol which was lacking such a proof. We compare our results with existing signatures including the recent code-based signatures based on PoK leveraging the MPC in the head paradigm. In practice, our new protocol is as fast as AGS while reducing its associated signature length by 20%. As a consequence, it constitutes an interesting trade-off between signature length and execution time for the design of a code-based signature relying only on the difficulty of the SD problem.

Decision forests are classical models to efficiently make decision on complex inputs with multiple features. While the global structure of the trees or forests is public, sensitive information have to be protected during the evaluation of some client inputs with respect to some server model. Indeed, the comparison thresholds on the server side may have economical value while the client inputs might be critical personal data. In addition, soundness is also important for the receiver. In our case, we will consider the server to be interested in the outcome of the model evaluation so that the client should not be able to bias it. In this paper, we propose a new offline/online protocol between a client and a server with a constant number of rounds in the online phase, with both privacy and soundness against malicious clients.

Designing code-based signatures is both an important and challenging problem. A standard way to tackle it consists to use the Fiat-Shamir heuristic along with an identification scheme that is required to be zero-knowledge. The authors of [1] have highlighted an issue within the zero-knowledge proof of the Veron identification scheme [2]. It turns out that the zero-knowledge proof of the AGS protocol [3] is impacted in a similar way. In this paper, we present a masking technique that solves the aforementioned issue without inducing any performance penalty. We introduce the Masked Veron and Masked AGS protocols that both leverage this masking technique and provide their zero-knowledge proofs. In addition, we present a new technique improving the performances of signatures built from code-based identification schemes subject to the attack described in [4]. The Masked Veron and Masked AGS protocols feature all the existing performance improvements from the literature.

In this paper, we introduce a new hard problem opening up the construction for new quantum resistant cryptographic schemes. The latter is called Learning Rank with Errors (LRE) and can be seen as an adaptation of the LPN problem to the rank metric setting. In addition, we describe HBLRE, an HB-like authentication protocol that constitutes an application of the aforementioned problem. We also prove that HTLRE is secure against passive attacks and compare its parameters to those of the initial HB scheme.