Public Key Cryptosystem Based on Low Density Lattice Codes

Abstract

McEliece and Goldreich---Goldwasser---Halevi (GGH) cryptosystems are two instances of code and lattice-based cryptosystems whose security are based on the hardness of coding theoretic and lattice problems, respectively. However, such cryptosystems have a number of drawbacks which make them inefficient in practice. On the other hand, low density lattice codes (LDLCs) are practical lattice codes which can achieve capacity over additive white Gaussian noise channel and also can be encoded and decoded efficiently. This paper introduces a public key cryptosystem based on Latin square LDLCs, by which a relationship can be attained between code and lattice-based cryptography. In this way, we can exploit the efficient properties of codes and lattices, simultaneously to improve the security and efficiency of the proposed scheme. For instance, the security of this scheme is based on the hard problems related to lattices, i.e., closest vector problem and shortest basis problem, which in turn lead to increase the security level. On the other hand, we exploit the low complexity decoding algorithm of LDLCs to reduce the computational complexity. Moreover, this property allows using the larger values of the codeword length. Also, we use the special Gaussian vector, whose variance is upper bounded by Poltyrev bound, as the perturbation (error) vector. These strategies make the proposed scheme to be secure against the conventional cryptanalytic attacks.