Research On McEliece Cryptosystem Based On LDPC Codes

Low-Density Parity Check (LDPC) codes, proposed by Robert Gallager in 1962, have excellent error correcting performance approaching Shannon limit and low decoding complexity. Nowadays, LDPC codes have been widely used in deep space communications, optical communications, satellite digital video and audio broadcasting and other fields and have become a strong competitor of the fourth generation communication system (4G).In 1978, McEliece proposed a public-key cryptosystem based on algebraic coding theory. The security of the cryptosystem relies on the fact that decoding a large linear code with no visible structure is NP-complete. Compared with other competing solutions, such as RSA, McEliece's scheme has the advantage of faster encryption and decryption. But its drawbacks are also clear:large public-key size and low information rate, which prevent it from being widely used. Considering this drawbacks, researchers have made efforts to reduce key size or obtain higher information rate.Security and reliability are two important aspects in the wireless communication system. Combining the two aspects is an attractive idea since it may reduce the processing cost or provide a more efficient implementation. In order to improve the security level of the physical layer, a new McEliece symmetric-key encryption scheme based on irregular low-density parity-check (LDPC) codes is proposed. The degree distribution property of the irregular LDPC codes and a large random number are applied to encrypt message. Analysis shows that the proposed scheme provides an acceptable level of security. The secret key size of this scheme is much smaller than those of the previous symmetric-key McEliece-like schemes. Moreover, there is no trade-off between the error correction performance and the security level in the proposed scheme.Next, a new McEliece-like symmetric-key scheme is proposed in this paper. Modulation is applied as an encryption step to provide two functions:modulation and scrambling. The mapping process of modulation is controlled by error vectors, which is different from we are familiar with. Analysis shows that the new scheme can keep an acceptable security level and achieve a smaller key size in comparison with the previous McEliece-like symmetric-key schemes. Moreover, there is no trade-off between error correcting performance and the security level.