| The diophantine equation is the oldest branch in number theory, whose content is extremely abundant, and it has close connections with the algebraic number theory, the algebraic geometry, the combinatorics and so on. In the recent 30 years, the number theory cryptography has developed too much in the modern cryptography, the information encoding theory and computer science, especially in number theory cryptography. So there are still many people who have great interested in diophantine equation. In 1979, Bender E.A and Herzberg N.P discussed the integer solution of diophantine equation ax~2 + by~2 = cp~n, c= 1,2,4, which inspire us to study such kind of diophantine equation ax~2 + bD~m = cp~n. At the same time, we are also inspired to search a better theory, using the number theory to modern cryptography.In this paper, I discussed the integer solutions of the diophantine equation x~p -1 = Dy~n and ax~2+bD~m = p~n with decomposition factor method and primitive prime divisor theory. At the same time, I raised a new number theory cryptography by using of Euler (?) function and continued fraction in number theory.The whole paper is divided into three parts, and its content is as follows:In the first chapter, the development survey of the diophantine equation and num-ber theory cryptography, principle and difficult of solving diophantine equation are set forth, which has made preparation for the following conclusions.In the second chapter, the solution of diophantine equation x~p - 1 = Dy~n and ax~2 + bD~m = p~n are given, which including three conclusions, a deduction and the strict proofs are given.In the third chapter, Euler (?) binary cryptosystem and continued fraction binary cryptosystem have raised, by using of Euler (?) function and continued fraction. |
