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A New Public-Key Cipher System Based Upon the Diophantine Equations
January 1995 (vol. 44 no. 1)
pp. 13-19

Abstract—A new public-key (two-key) cipher scheme is proposed in this paper. In our scheme, keys can be easily generated. In addition, both encryption and decryption procedures are simple. To encrypt a message, the sender needs to conduct a vector product of the message being sent and the enciphering key. On the other hand, the receiver can easily decrypt it by conducting several multiplication operations and modulus operations. For security analysis, we also examine some possible attacks on the presented scheme.

Index Terms—Public keys, private keys, cryptosystems, Diophantine equation problems, integer knapsack problems, one-way functions, trapdoor one-way functions, NP-complete.

[1] E. F. Brickell,“A new knapsack based cryptosystem,”inCrypto '83, rump session, 1983.
[2] C. C. Chang and J. C. Shieh,“Pairwise relatively prime generating polynomials and their applications,”inProc. Int. Workshop on Discrete Algorithms and Complexity,Kyushu, Japan, Nov. 1989, pp. 137–140.
[3] B. Chor, and R. L. Rivest,“Knapsack Type Public Key Cryptosystem Based on Arithmetic in Finite Field,”IEEE Trans. Inform. Theory,vol. 34, No. 5, 1988, pp. 901–909.
[4] S. A. Cook,“The Complexity of Theorem-Proving Procedures,”Proc. 3rd Ann. ACM Symposium on Theory of Computing,New York: Association for Computing Machinery, 1971, pp. 151–155.
[5] D.E.R. Denning, Cryptography and Data Security. Addison-Wesley, 1983.
[6] W. Diffie and M. Hellman,“New directions in cryptography,”IEEE Trans. Inform. Theory,vol. 22, pp. 644–654, 1976.
[7] T. El Gamal,“A public key cryptosystem and signature scheme based on discrete logarithms,”IEEE Trans. Inform. Theory,vol. 31, no. 4, pp. 469–472, 1985.
[8] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[9] S. Goldwasser and S. Micali,“Probabilistic encryption,”J. Comp. Syst. Sci.,vol. 28, no. 2, pp. 270–299, 1984.
[10] E. M. Gurari, and O. H. Ibarra,“An Np-complete number theoretic problem,”inProc. 10th Ann. ACM Symp. Theory Computing.New York: Association for Computing Machinery, 1978, pp. 205–215.
[11] D. Hilbert,“Mathematische Probleme,”Vortrag, gehalten auf dem internationalen Mathematiker Kongrass zu Paris, 1900,Nachr. Akad. Wiss. Gottingen Math.-Phys.,pp. 253–297; Translation:Bull. Am. Math. Soc.,vol. 8, 1901, pp. 437–479.
[12] D.E. Knuth, The Art of Computer Programming, vol. 1,Addison Wesley, second ed. 1973.
[13] ——,The Art of Computer Programming, Vol. 2: Seminumerical Algorithms,2nd ed. Reading, MA: Addison-Wesley, 1981.
[14] K. Manders and L. Adleman,“NP-complete decision problems for binary quadratics,”J. Comput. Syst. Sci.,vol. 16, pp. 168–184, 1978.
[15] Y. Matijasevi$\check{\rm c}$,“Enumerable sets are Diophantine,”Dokl. Akad. Nauk SSSR,vol. 191, 1970, pp. 279–282 (in Russian); English translation inSoviet Math. Dokl.,vol. 11, pp. 354–357.
[16] Y. Matijasevi$\check{\rm c}$and J. Robinson,“Reduction of an arbitrary Diophantine equation to one in 13 unknowns,”Acta Arithmetica,vol. 27, pp. 521–553, 1975.
[17] R. C. Merkle and M. Hellman,“Hiding information and signatures in trap-door knapsacks,”IEEE Trans. Inform. Theory,vol. 24, pp. 525–530, 1978.
[18] L. J. Mordell,Diophantine Equations,vol. 30 inPure and Applied Mathematics,Paul A. Smith and Samuel Eilenberg, Eds. London and New York: Academic Press, 1969.
[19] S. C. Pohlig and M. E. Hellman,“An improved algorithm for computing logarithms over GF(p) and its cryptographic significance,”IEEE Trans. Inform. Theory,vol. 24, no. 1, pp. 106–110.
[20] M. O. Rabin,“Digitalized signatures and public-key functions as intractable as factorization,”Tech. Rep. TR-212, Laboratory for Computer Science, MIT, 1979.
[21] R.L. Rivest,A. Shamir, and L.A. Adleman,"A Method for Obtaining Digital Signatures and Public Key Cryptosystems," Comm. ACM, vol. 21, pp. 120-126, 1978.
[22] A. Shamir,“Embedding cryptographic trapdoors in arbitrary knapsack systems,”Technical memo TM-230, Laboratory for Computer Science, MIT, 1982.
[23] T. Skolem,“Diophatische gleichungen,”Ergebisse d. Math. u. Ihrer Grenzgebiete, Bd. 5,Julius Springer, 1938.
[24] S. P. Tung,“Computational complexities of diophantine equations with parameters,”J. Algorithms,vol. 8, 1987, pp. 324–336.
[25] S. P. Tung,“Complexity of sentences over number rings,”SIAM J. Computing,vol. 20, No. 1, February 1991, pp. 126–143.
[26] H. C. Williams,“A modification of the RSA public-key encryption procedure,”IEEE Trans. Information Theory,vol. 26, 1980, pp. 726–729.

C. H. Lin, C. C. Chang, R. C. T. Lee, "A New Public-Key Cipher System Based Upon the Diophantine Equations," IEEE Transactions on Computers, vol. 44, no. 1, pp. 13-19, Jan. 1995, doi:10.1109/12.368013
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