A Smart Approach for GPT Cryptosystem Based on Rank Codes

The concept of Public- key cryptosystem was innovated by McEliece's cryptosystem. The public key cryptosystem based on rank codes was presented in 1991 by Gabidulin -Paramonov-Trejtakov(GPT). The use of rank codes in cryptographic applications is advantageous since it is practically impossible to utilize combinatoric decoding. This has enabled using public keys of a smaller size. Respective structural attacks against this system were proposed by Gibson and recently by Overbeck. Overbeck's attacks break many versions of the GPT cryptosystem and are turned out to be either polynomial or exponential depending on parameters of the cryptosystem. In this paper, we introduce a new approach, called the Smart approach, which is based on a proper choice of the distortion matrix X. The Smart approach allows for withstanding all known attacks even if the column scrambler matrix P over the base field Fq.Comment: 5 pages. to appear in Proceedings of IEEE ISIT201

Polynomial-Time Key Recovery Attack on the Faure-Loidreau Scheme based on Gabidulin Codes

Encryption schemes based on the rank metric lead to small public key sizes of order of few thousands bytes which represents a very attractive feature compared to Hamming metric-based encryption schemes where public key sizes are of order of hundreds of thousands bytes even with additional structures like the cyclicity. The main tool for building public key encryption schemes in rank metric is the McEliece encryption setting used with the family of Gabidulin codes. Since the original scheme proposed in 1991 by Gabidulin, Paramonov and Tretjakov, many systems have been proposed based on different masking techniques for Gabidulin codes. Nevertheless, over the years all these systems were attacked essentially by the use of an attack proposed by Overbeck. In 2005 Faure and Loidreau designed a rank-metric encryption scheme which was not in the McEliece setting. The scheme is very efficient, with small public keys of size a few kiloBytes and with security closely related to the linearized polynomial reconstruction problem which corresponds to the decoding problem of Gabidulin codes. The structure of the scheme differs considerably from the classical McEliece setting and until our work, the scheme had never been attacked. We show in this article that this scheme like other schemes based on Gabidulin codes, is also vulnerable to a polynomial-time attack that recovers the private key by applying Overbeck's attack on an appropriate public code. As an example we break concrete proposed $80$ bits security parameters in a few seconds.Comment: To appear in Designs, Codes and Cryptography Journa

Injective Rank Metric Trapdoor Functions with Homogeneous Errors

In rank-metric cryptography, a vector from a finite dimensional linear space over a finite field is viewed as the linear space spanned by its entries. The rank decoding problem which is the analogue of the problem of decoding a random linear code consists in recovering a basis of a random noise vector that was used to perturb a set of random linear equations sharing a secret solution. Assuming the intractability of this problem, we introduce a new construction of injective one-way trapdoor functions. Our solution departs from the frequent way of building public key primitives from error-correcting codes where, to establish the security, ad hoc assumptions about a hidden structure are made. Our method produces a hard-to-distinguish linear code together with low weight vectors which constitute the secret that helps recover the inputs.The key idea is to focus on trapdoor functions that take sufficiently enough input vectors sharing the same support. Applying then the error correcting algorithm designed for Low Rank Parity Check (LRPC) codes, we obtain an inverting algorithm that recovers the inputs with overwhelming probability

Coding Theory

This book explores the latest developments, methods, approaches, and applications of coding theory in a wide variety of fields and endeavors. It consists of seven chapters that address such topics as applications of coding theory in networking and cryptography, wireless sensor nodes in wireless body area networks, the construction of linear codes, and more

The Hardness of LPN over Any Integer Ring and Field for PCG Applications

Learning parity with noise (LPN) has been widely studied and used in cryptography. It was recently brought to new prosperity since Boyle et al. (CCS\u2718), putting LPN to a central role in designing secure multi-party computation, zero-knowledge proofs, private set intersection, and many other protocols. In this paper, we thoroughly studied the security of LPN problems in this particular context. We found that some important aspects have long been ignored and many conclusions from classical LPN cryptanalysis do not apply to this new setting, due to the low noise rates, extremely high dimensions, various types (in addition to $\mathbb{F}_2$) and noise distributions. 1. For LPN over a field, we give a parameterized reduction from exact-noise LPN to regular-noise LPN. Compared to the recent result by Feneuil, Joux and Rivain (Crypto\u2722), we significantly reduce the security loss by paying only a small additive price in dimension and number of samples. 2. We analyze the security of LPN over a ring $\mathbb{Z}_{2^\lambda}$. Existing protocols based on LPN over integer rings use parameters as if they are over fields, but we found an attack that effectively reduces the weight of a noise by half compared to LPN over fields. Consequently, prior works that use LPN over $\mathbb{Z}_{2^\lambda}$ overestimate up to 40 bits of security. 3. We provide a complete picture of the hardness of LPN over integer rings by showing: 1) the equivalence between its search and decisional versions; 2) an efficient reduction from LPN over $\mathbb{F}_2$ to LPN over $\mathbb{Z}_{2^\lambda}$; and 3) generalization of our results to any integer ring. Finally, we provide an all-in-one estimator tool for the bit security of LPN parameters in the context of PCG, incorporating the recent advanced attacks

Actas de las VI Jornadas Nacionales (JNIC2021 LIVE)

Estas jornadas se han convertido en un foro de encuentro de los actores más relevantes en el ámbito de la ciberseguridad en España. En ellas, no sólo se presentan algunos de los trabajos científicos punteros en las diversas áreas de ciberseguridad, sino que se presta especial atención a la formación e innovación educativa en materia de ciberseguridad, y también a la conexión con la industria, a través de propuestas de transferencia de tecnología. Tanto es así que, este año se presentan en el Programa de Transferencia algunas modificaciones sobre su funcionamiento y desarrollo que han sido diseñadas con la intención de mejorarlo y hacerlo más valioso para toda la comunidad investigadora en ciberseguridad