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Low-Rank Parity-Check Codes Over Finite Commutative Rings And

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Low-Rank Parity-Check Codes Over Finite Commutative Rings And 1 Low-Rank Parity-Check Codes Over Finite Commutative Rings and Application to Cryptography Hermann Tchatchiem Kamche, Herve´ Tale´ Kalachi, Franck Rivel Kamwa Djomou, Emmanuel Fouotsa Abstract—Low-Rank Parity-Check (LRPC) codes are a class third round of this competition and the recent generic attacks of rank metric codes that have many applications specifically [6], [7] which considerably improved the cost of all known in cryptography. Recently, LRPC codes have been extended to decoding attacks. And this, even if we are only talking about Galois rings which are a specific case of finite rings. In this paper, we first define LRPC codes over finite commutative local rings, a reduction of security levels of rank based candidates. which are bricks of finite rings, with an efficient decoder and Note that the attacks from [6], [7] exploit the fact that the derive an upper bound of the failure probability together with code used is defined on a finite field. A natural question that the complexity of the decoder. We then extend the definition to generally arises is that of knowing what would become these arbitrary finite commutative rings and also provide a decoder attacks if we are dealing with a code defined on another poorer in this case. We end-up by introducing an application of the corresponding LRPC codes to cryptography, together with the algebraic structure such as finite rings for example. In this case new corresponding mathematical problems. of finite rings, it is interesting to precede such a question by defining LRPC codes over finite rings as well as an application Index Terms—Cryptography; Decoding Algorithm; Finite Rings; Galois Extension; LRPC Codes; Local Rings; Rank Metric to cryptography. Codes. It was recently shown in [8] that LRPC codes can be generalized to the ring of integers modulo a prime power. This work was followed by [9] where the authors generalized the I. INTRODUCTION results of [8] to Galois rings. LRPC codes over finite fields were first introduced in [1] si- In this paper, we use the structure theorem for finite commu- multaneously with an application to cryptography. These codes tative rings [10] to generalize the definition of LRPC codes have two peculiarities which make them attractive enough for over finite commutative rings. We also provide a decoding cryptographic applications. First, LRPC codes are rank metric algorithm, a correction capability together with an upper bound codes [2]. In code-based cryptography, this metric is very of the failure probability of our decoding algorithm. We end- famous because it makes generic attacks [3], [4] in the context up by introducing an application of the corresponding LRPC of Hamming metric impracticable. Therefore, decoding attacks codes to cryptography, together with the corresponding new are more expensive in practice for rank metric compared mathematical problems. to Hamming metric. Second, LRPC codes are by definition The rest of the paper is organized as follows: in Section analogues of low-density parity-check codes [5]. Thus, they II, we give some properties and mathematical notions that have a poor algebraic structure, which makes several known will be needed throughout the paper. That is, the resolution structural attacks impracticable. of linear systems, the rank, the free-rank and the product of Rank metric based cryptography and LRPC codes have re- two submodules in the Galois extension of finite commutative ceived a lot of attention these recent years thanks to the NIST 1 local rings. In Section III, we define LRPC codes over finite arXiv:2106.08712v1 [cs.IT] 16 Jun 2021 competition for Post-Quantum Cryptography Standardization, commutative local rings with a decoding algorithm and its where several code based candidates using LRPC codes went complexity. We also give an upper bound for the failure prob- up to the second round of the competition. However, it is ability. In Section IV, we generalize the definition of LRPC today difficult to ignore the existence of a link between the codes to finite commutative rings and propose in Section V a absence of a candidate from rank based cryptography in the cryptosystem for its applications in cryptography and finally, conclusions and perspectives are given in Section VI. Hermann Tchatchiem Kamche is with the Department of Mathemat- ics, Faculty of Science, University of Yaounde I, Cameroon e-mail: II. PRELIMINARIES [email protected]. Herve´ Tale´ Kalachi is with the Department of Computer Engineering, Let R be a finite commutative ring. The set of units in R will National Advanced School of Engineering of Yaounde,´ University of Yaounde be denoted by R∗. The set of all m × n matrices with entries I, Cameroon e-mail: (see http://perso.prema-a.org/herve.tale-kalachi/). from R will be denoted by Rm×n. The k × k identity matrix Franck Rivel Kamwa Djomou is with the Department of Mathematics and m×n Computer Science, Faculty of Science, University of Dschang, Cameroon e- is denoted by Ik. Let A 2 R , we denote by row (A) and mail: [email protected]. col (A) the R−submodules generated by the row and column Emmanuel Fouotsa is with the Department of Mathematics, Higher vectors of A, respectively. Teacher Training College, University of Bamenda, Cameroon e-mail: em- n [email protected]. Let fa1;:::; arg be a subset of R . The R−submodule of 1 n National Institute of Standards and Technology R generated by fa1;:::; arg is denoted by ha1;:::; ariR. 2 Recall that fa1;:::; arg is linearly independent over R if The solutions of Equation (3) are (3; 2; 2; 2); (1; 2; 3; 0); whenever α1a1 + ··· + αrar = 0 for some α1; : : : ; αr 2 R, (3; 2; 0; 2); (1; 2; 1; 0). then α1 = 0; : : : ; αr = 0. If fa1;:::; arg is linearly inde- Thus, the solutions of Equation (2) are (3 + 2ξ; 2 + 2ξ); pendent, then we say that it is a basis of the free module (1 + 3ξ; 2); (3; 2 + 2ξ); (1 + ξ; 2). ha1;:::; ariR. If the column vectors of A are linearly independent, then A local ring is a ring with exactly one maximal ideal. As an Equation (1) is easy to solve using the following: k m×n example, the ring Zpk of integers modulo p , p is a prime, is a Proposition 2.3: Let A 2 R . Assume that the column local ring whose the maximal ideal is pZpk . By [10, Theorem vectors of A are linearly independent. Then there is an VI.2] each finite commutative ring can be decomposed as a invertible matrix P 2 Rm×m such that direct sum of a local rings. Using this decomposition, some I PA = n results over local rings can naturally be extended to finite rings. 0 Thus, in this section, we will first study some properties that we will need over finite commutative local rings. Moreover, P can be calculated in time O (mn min fm; ng) = O mn2. Proof. In [12], Fan et al. proved the existence of P and A. Linear Equations Over Finite Commutative Local Rings calculated it using the Gaussian elimination. Thus, as in In the decoding algorithm, we will solve linear equations. the case of fields, the computation can be done in time 2 In [11], Bulyovszky and Horvath´ gave a good method to solve O (mn min fm; ng) = O mn . linear equations over finite local rings. In this subsection, we Corollary 2.4: Assume that the column vectors of the matrix assume that R is a local ring with maximal ideal m and residue A of Equation (1) are linearly independent. Then, Equation (1) can be solved in O (mn min fm; ng) = O mn2 operations. field Fq = R=m. By [10], there is a positive integer υ such that jRj = qυ and jmj = qυ−1 where jRj and jmj are respectively the cardinality of R and m. Set q = pµ where p is a prime B. Rank and Free-Rank of Modules & number. Then the characteristic of R is p where & is a non As in [13], we recall here the notions of Rank and Free-rank negative integer. As in [11], There is a subring R0 of R such for modules. & that R0 is isomorphic to the Galois ring of characteristic p Definition 2.5: (Rank of a module) Let R be a finite µς and cardinality p , that is, a ring of the form Zp& [X] = (hµ) commutative ring. Let M be a finitely generated R−module. & where hµ 2 Zp [X] is a monic polynomial of degree µ, where (i) The rank of M, denoted by rkR (M), or simply by its projection in Zp [X] is irreducible. Considering R as an rk (M), is the smallest number of elements in M which R0−module, there are z1; : : : ; zγ 2 R such that R = R0z1 ⊕ generate M as an R−module. · · · ⊕ R0zγ . (ii) The free rank of M, denoted by frkR (M) or simply by Consider the linear system frk (M), is the maximum of the ranks of free R−submodules of M. Ax = b (1) Proposition 2.6: [14] Let F be a finitely generated with unknown x 2 Rn×1, where A 2 Rm×n and b 2 Rm×1. free R−module and fe1; : : : ; eng be a basis of F . Then In [11], Bulyovszky and Horvath´ transformed Equation (1) frkR (F ) = rkR (F ) = n and any generating set of F to a linear system of γm equations of γn unknowns over consisting of n elements is a basis of F . the Galois ring R0. Then, they used Hermite normal form to Corollary 2.7: Let M be a finitely generated R−module. solve them.
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