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On the Equivalence of Mceliece's and Niederreiter's Public-Key Cryptosystems Y Singapore Management University Institutional Knowledge at Singapore Management University Research Collection School Of Information Systems School of Information Systems 1-1994 On the equivalence of McEliece's and Niederreiter's public-key cryptosystems Y. X. LI Robert H. DENG Singapore Management University, [email protected] X. M. WANG DOI: https://doi.org/10.1109/18.272496 Follow this and additional works at: https://ink.library.smu.edu.sg/sis_research Part of the Information Security Commons Citation LI, Y. X.; DENG, Robert H.; and WANG, X. M.. On the equivalence of McEliece's and Niederreiter's public-key cryptosystems. (1994). IEEE Transactions on Information Theory. 40, (1), 271-273. Research Collection School Of Information Systems. Available at: https://ink.library.smu.edu.sg/sis_research/94 This Journal Article is brought to you for free and open access by the School of Information Systems at Institutional Knowledge at Singapore Management University. It has been accepted for inclusion in Research Collection School Of Information Systems by an authorized administrator of Institutional Knowledge at Singapore Management University. For more information, please email [email protected]. IKANJALIIUNJ UN INrUKb’lAIlUN IHCUKX, VUL. W,IYU. 1, JAIYUAKX IYY4 LI1 REERENCES of Niederreiter’s cryptosystem, by Niederreiter [3] and by Brickell and Odlyzko [4]. Furthermore, we employ the best known attack, the J. Ziv and A. Lempel, “Compression of individual sequences via Lee-Brickell attack [5], to cryptanalyze the two systems. Some new variable-rate coding,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 530-536, Sept. 1978. optimal parameter values and work factors are obtained. I. Rissanen, “A universal data compression system,” IEEE Trans. We briefly review McEliece’s and Niederreiter’s cryptosystems in Inform. Theory, vol. IT-29, pp. 656-664. Sept. 1983. Section 11. The equivalence of the two systems is derived in Section R. G. Gallager, Information Theory and Reliable Communication. New III. Finally, we cryptanalyze the two systems and comment on the York Wiley, 1968. selection of optimal system.parameters in Section P. Elias, “Universal codeword sets and representations of the integers,” IV. IEEE Trans. Inform. Theory, vol. IT-21, pp. 194-203, Mar. 1975. T. M. Cover and I. A. Thomas, Elements of Information Theory. New MCELIECE’S AND NIEDERREITER’SCRWroSYSTEMS York Wiley, 1991. II. W. F. Stout, Almost Sure Convergence. New York Academic, 1974. In this section, brief descriptions of McEliece’s and Niederreiter’s L. D. Davisson, “Universal noiseless coding,” IEEE Trans. Inform cryptosystems are presented in order to facilitate discussions in later Theory, vol. IT-19, pp. 783-795, Nov. 1973. sections. Both ctyptosystems are algebraic-coded two-key systems. I. C. Kieffer, “A unified approach to weak universal source coding,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 674-682, Nov. 1978. The basic idea behind them was to construct a linear error-correcting A. D. Wyner, “An upper bound on the entropy series.’’ Inform. Contr., code for which a fast decoding algorithm is known, and then to vol. 20, pp. 176181. 1972. disguise it as a general linear code whose decoding problem is L. Gyorfi, I. Pfili, and E. C. van der Meulen, “On universal noise- NP-complete. less source coding for infinite source alphabets,” in European Trans. Telecommun. Related Technol., vol. 4, pp. 125-132, Mar.-Apr. 1993. A. McEliece ’s Cryptosystem [2] On the Equivalence of McEliece’s and This system uses a binary (n, k, 2t+ 1) Goppa code C where n is Niederreiter’s Public-Key Cryptosystems the code length, k is the code dimension, and t is the error-correcting capability of C. C is constructed by randomly selecting an irreducible Yuan Xing Li, Member, IEEE, Robert H. Deng, polynomial of degree t over GF(2’) as the Goppa polynomial (note and Xin Mei Wang, Member, IEEE that n = 2’). Let G be a k x n generator matrix of C [6], S any k x k nonsingular matrix, and P any n x n permutation matrix. Private Key: G, S, P. Abstract-It is shown that McEliece’s and Niederreiter’s public-key Public Key: G‘ = SGP and t. cryptosystems are equivalent when set up for corresponding choices Messages: k bit vectors m over GF(2). of parameters. A security analysis for the two systems, based on this equivalence observation, is presented. Encryption: c = mG‘ + e. e, an n bit error vector with (Hamming) weight t. c, the n bit ciphertext. Index Tenns-Cryptosystems, McEliece’s cryptosystem, Niederreiter’s cryptosystem, security, algebraic codes. Decryption: Since c = mSGP + e, cP-’ = (mS)G+ eP-’ . Use a fast decoding algorithm for C to correct the error “eP-’, find mS and thus m. I. INTRODUCTION McEliece investigated several attacks against his system. One of Since the concept of public-key cryptosystems appeared in the those was to factor the public key to obtain the private key, but fundamental paper of Diffie and Hellman [l] in 1977, the field of this approach was thought to be hopeless. Another attack, considered cryptology has undergone a dramatic development. The last decade as the most promising, was to pick k “error-free” elements of the has seen explosive growth in unclassified research in all aspects of ciphertext c, and then to solve a set of A linear equations to recover the cryptology. Public-key cryptosystem and cryptanalysis have been message m. Using this attack, McEliece suggested using n = 1024 two of the most active areas. So far many kinds of public-key and t = 50, i.e., (1024, 524, 101) Goppa code in his system. The cryptosystems have been proposed, and many of them that had been corresponding work factor of the system is approximately 2’O.’ [7]. thought to be secure have been broken. A special class of public-key cryptosystems were constructed B. Nederreiter’s Cryptosystem [3] based on algebraic error-correcting codes. In the present paper, This is a knapsack-type cryptosystem which employs we focus on two such systems, McEliece’s [2] and Niederreiter’s an (n, k, 2t 1) linear code C over GF(q). Let E be an (n - k) x n [3] cryptosystems, examine the relationship between the two, and + parity check matrix of any (n - k) x (n - k) nonsingular derive the interesting result that the two systems are equivalent and C,M matrix, and any n x n permutation matrix, all over GF(q). have the same security when set up for corresponding choices of P parameters. This result allows us to clarify the security evaluations Private Key: E, M, and P. Public Key: H‘ = MHP and t. Manuscript received May 4, 1993; revised November 17, 1993. This work Messages: n dimensional vectors J over GF(q) with weight t. was supported in part by the National Natural Science Foundation of China Encryption: z = #HIT. z, the ciphertext of dimension n - k. under Grant 69072910. z Y. X. Li is with the Magnetic Technology Centre, National University of Decryption: Since = J(MHP)~,z(MT)-’ = (@“)BT. Singapore, Kent Ridge, Singapore 05 1 1, Republic of Singapore. Use a fast decoding algorithm for C to find g$ and thus g. R. H. Deng is with the Department of Electrical Engineering, National Niederreiter [3] cryptanalyzed his system and mentioned two University of Singapore, Kent Ridge, Singapore 05 11, Republic of Singapore. X. M. Wang is with the Department of Information Engineering, Xidian example systems, one using a binary concatenated (104,24, 31) code University, People’s Republic of China. and the other using a (30, 12, 19) ReedSolomon code over GF(31). IEEE Log Number 9215123. The examples were later verified as insecure by Brickell and Odlyzko 0018-9448/94$04.00 Q 1994 IEEE 212 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 1, JANUARY 1994 [4]. The attack used in Niederreiter’s cryptanalysis was to factorize provided that the two cryptosystems are based on the same error- H’ in the hope of getting M,H, and P. However, this cryptanalysis correcting code. Without loss of generality, we assume, throughout was not sufficient to ensure the security of his cryptosystem. Actually, the rest of the paper, that the same binary error-correcting codes are McEliece has shown that this attack was not the most threatening. used in the two systems. In the following, we derive the equivalence relationship between Attack 1 (Lee-Brick11 Attack): Picke a k x k submatrix G6 of the two cryptosystems, and re-evaluate the securities of the two G’ consisting of the jlth, j~th, . , and the jkth columns of G’. systems using the best known attack, the LeeBrickell attack. Let ek and ek be the k dimensional vectors consisting of the jl th, jzth,. , and the jhth bits of e and e, respectively. It then follows that ek +ek = mG6 and that (ck +ek)(GL)-lG = mGL(GL)-lG = EQUIVALENCEOF THE MCELIECE’S m. mG. Choose a k-bit vector ei with t) or fewer ones. If AND NIEDERREITERSCRYFTOSYSTEMS j(< e+ (ek +ei)(GL-l)Ghas weight t, then ei= ek and the message A fair comparison of the two systems dictates that they both be m = (ck + ek)(GL)-’ is recovered; otherwise, choose another e6 based on the same error-correcting (n, k, 2t + 1) linear code C. Let and repeat the above process. If all the bit error pattems eL of H and G denote the parity check matrix and the generator matrix, k weight 5 j have been exhausted and the message still can not be respectively, of C. Then finding H from G (or equivalently finding recovered, then pick another submatrix of G’ and the above process G from H) can be achieved by linear algebra [6].
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