The Application Of Rank Distance Codes To Cryptography

A vast amount of data to be collected and stored by data-base or being transmitted over channel are vulnerable to attacks because public channel for transmitting message and computer system for storing data are very much fragile. In order to withstand attacks that they may be subjected to, proper safeguarding measures are in demand besides making laws. Cryptotechnique are exactly one kind of effective measure. Under insecure circumstances, it can ensure security of communication. Therefore, the design of cryptosystems of secure and effective plays a key role for security of communication systems. This paper centers round the design of cryptosystems that are secure and effecti~ Based on the decoding problem of syndrome regarding rank distances codes, we have designed some new public-key and secret-key cryptosystems. Based on the problem of modular knapsack regarding error-correcting codes, we have designed two kinds of identification schemes. Except of these, we have made some theoretical research regarding rank distance codes and finite field. The main research work and results are as follows: I .The new concept of rank distance BCH codes are put forward. The parity check matrix and minimum rank distance of them are given. The generator matrix of maximum rank distance Reed-Solomon code given by E.M.Gabidulin is corrected. The existence of roots of a linearized polynomial over GF(2~? is researched and the necessary and sufficient condition is given that some element over GF(2~) is a root of a given linearized polynomial. The method of finding out the solution of linear equations and inverse of some common matrices over GF(p) is given. 2.By improving the identification scheme given by J.Stern. the limitation on the weight of mysterious datum to be changed into the limitation on the distribution of code elements of it, we have constructed a new identification scheme based on a parity check matrix of error-correcting codes over GF(q). By improving the identification scheme given by Pascal Vé–žon, we have constructed a new identification scheme based on a generator matrix of error-correcting codes over GF(q). We have proved that the two given protocol are zero-knowledge interactive proof in the random oracle model. We have discussed security, cornplicacy of space and communication of the two schemes. A series of identification schemes can be constructed on the basis of the two schemes. 3 .By assuming that the decoding problem of syndrome regarding rank distance codes is NP complete, a new identification scheme has been constructed based on a parity check matrix of rank distance codes over GF(q j. In the new identification scheme, the limitation on the weight of mysterious datum in the scheme given by J.Stem has been changed into the limitation on the rank of it. A new identification scheme has also been constructed based on a generator matrix of rank distance codes over GF(q N). We have proved that the two given protocol are zero-knowledge interactive proof in the random oracle model. We have discussed security, complicacy of space and cormnunication of the two schemes. 4.Based on maximum rank distance codes, we have constructed two new kinds of McEliece public-key cryptosystem, two new kinds of Niederreiter public-key crvptosystem, one new kind of Stem scheme, four new kinds of secret-code encryption schemes. two new kinds of Rao-Nam schemes. We have analyzed the feasibility and security of these cryptosystems.