arXiv subjects
Publications and source records attributed to Pierre Loidreau.
In this paper we show that in the case where the public-key can be distinguished from a random code in Loidreau's encryption scheme, then Coggia-Couvreur attack can be extended to recover an equivalent secret key. This attack can be conducted in polynomial-time if the masking vector space has dimension 3, thus recovering the results of Ghatak.
We present a rank metric code-based encryption scheme with key and ciphertext sizes comparable to that of isogeny-based cryptography for an equivalent security level. The system also benefits from efficient encryption and decryption algorithms, which rely on linear algebra operations over finite fields of moderate sizes. The security only relies on rank metric decoding problems, and does not require to hide the structure of a code. Based on the current knowledge, those problems cannot be efficiently solved by a quantum computer. Finally, the proposed scheme admits a failure probability that can be precisely controlled and made as low as possible.
We address the problem of decoding Gabidulin codes beyond their unique error-correction radius. The complexity of this problem is of importance to assess the security of some rank-metric code-based cryptosystems. We propose an approach that introduces row or column erasures to decrease the rank of the error in order to use any proper polynomial-time Gabidulin code error-erasure decoding algorithm. This approach improves on generic rank-metric decoders by an exponential factor.
We generalise Gabidulin codes to the case of infinite fields, eventually with characteristic zero. For this purpose, we consider an abstract field extension and any automorphism in the Galois group. We derive some conditions on the automorphism to be able to have a proper notion of rank metric which is in coherence with linearized polynomials. Under these conditions, we generalize Gabidulin codes and provide a decoding algorithm which decode both errors and erasures. Then, we focus on codes over integer rings and how to decode them. We are then faced with the problem of the exponential growth of intermediate values, and to circumvent the problem, it is natural to propose to do computations modulo a prime ideal. For this, we study the reduction of generalized Gabidulin codes over number ideals codes modulo a prime ideal, and show they are classical Gabidulin codes. As a consequence, knowing side information on the size of the errors or the message, we can reduce the decoding problem over the integer ring to a decoding problem over a finite field. We also give examples and timings.
We transpose the theory of rank metric and Gabidulin codes to the case of fields of characteristic zero. The Frobenius automorphism is then replaced by any element of the Galois group. We derive some conditions on the automorphism to be able to easily transpose the results obtained by Gabidulin as well and a classical polynomial-time decoding algorithm. We also provide various definitions for the rank-metric.