PKC-PC: A Variant of the McEliece Public Key Cryptosystem based on Polar Codes

Index Terms— Channel Coding, McEliece Cryptosystem, Polar Codes, Public Key Cryptography I. INTRODUCTION AND MOTIVATION T has been revealed that the conventional used public key cryptosystems, whose security are based on the difficulty of discrete logarithm or factoring problems, are broken by the quantum computers in polynomial-time [1]. One of the important categories of cryptosystems which can resist quantum computer-based attacks is code-based cryptosystems. These kinds of cryptosystems can be considered as alternatives to the conventional public key cryptosystems, such as RSA and ElGamal [2]. The security of most of these cryptosystems relies on the hardness of some conventional problems in coding theory [3]. For example, it was previously shown that the decoding of a linear code with no clear structure is NP-complete problem [4]. The first public key

The material in this paper was presented in part at the Eleventh International ISC Conference on Information Security and Cryptology (ISCISC 2014), Tehran, Iran, September 2014. This work was supported in part by Iranian National Science Foundation (INSF) under Grant 92.32575. Reza Hooshmand is with the Department of Electrical Engineering, Shahid Sattari Aeronautical University of Science and Technology, Tehran 1384663113, Iran (e-mail: rhooshmand@ssau.ac.ir).
Masoumeh Koochak Shooshtari is with the Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran 16315-1355, Iran (e-mail: m-koochak@ee.kntu.ac.ir). Mohammad Reza Aref is with the Department of Electrical Engineering, Sharif University of Technology, Tehran 11365/8639, Iran, (e-mail: aref@sharif.edu)

code-based cryptosystem originally proposed based on the binary Goppa codes is McEliece cryptosystem [5]. This cryptosystem applies a binary Goppa codes’ generator matrix, a scrambling matrix, and a permuting matrix as the private key. The scrambling and permuting matrices are employed to convert the private key into the public matrix. The McEliece cryptosystem applies the generator matrix and encodes the information vector into the public code’s codewords. Compared to the conventional public key cryptosystems, McEliece cryptosystem has low complexity encryption/ decryption algorithms. Nevertheless, due to the use of binary Goppa codes, this cryptosystem has two major weaknesses [6]: (i) low transmission rate, and (ii) huge key size.
One of efficient approaches to resolve the weaknesses of McEliece cryptosystem is exchanging binary Goppa codes with the other linear block codes. However, such replacement can yield serious flaws in its security level. Thus far, several schemes have been proposed to dominate the weaknesses of McEliece scheme by exchanging the Goppa codes with the different linear codes such as generalized Reed–Solomon (GRS) codes [7], Reed–Muller codes [8], quasi cyclic low density parity check (QC-LDPC) codes [9-11], wild Goppa codes [12, 13], p-adic Goppa codes [11, 14], moderate density parity check (MDPC) codes [15, 16], convolutional codes [17] and more recently low density lattice codes (LDLCs) [18]. Some of these suggested yields decrease the public key length while keeping the same security level against the conventional attacks. However, most of them exposed the McEliece cryptosystem to security threats and yield serious flaws in its security level. For example, public key schemes based on GRS and Reed-Muller codes were broken in [19] and [20], respectively. A number of versions of LDPC code-based schemes [9, 10] have been successfully cryptanalyzed with efficient attacks in [21, 22]. Some of the parameters that can be found in public key schemes based on wild Goppa codes [12, 13]

…(Full text truncated)…