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不定方程是数论的一个分支，它有着悠久的历史与丰富的内容。On the diophantine equation is the number one branch, it has a long history and rich content.

因此，国内外有诸多学者对不定方程有着广泛而深入的研究。So many researchers study the Diophantine Equation extensively and highly in the domestic and abroad.

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这种方法无需求解丢番图方程，减少了控制算法的计算量。The method does not need solve the Diophantine equation, but reduces the computation of the algorithm.

通过有步骤地解两个刁番都方程来直接确定控制器的各个参数。The parameters of the controllers are directly determined by successively solving two Diophantine equations.

本文介绍了广义预测控制的基本算法，在此基础上给出了基于Diophantine方程的递推算法。Here introduce the basic algorithm of GPC, next give the introduction of recursive algorithm of Diophantine equations.

三世纪的丢番图的杰出贡献之一，就是把希腊代数学简化，开创了简化代数。Three century Diophantine one of the outstanding contribution that Greece algebra simplify, simplify algebraic created.

第五章给出了求解超定线性丢番图方程组和不等式组的修正ABS算法。Chapter five presents the modified ABS algorithms for solving linear overdetermined linear Diophantine equations and inequations.

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第四章给出了求解线性丢番图不等式组的ABS算法及其在整线性规划中的应用。Chapter four gives ABS algorithms for solving linear Diophantine inequations and their application in integer linear programming.

最近十余年，不定方程不仅自身的发展异常活跃，而且全面应用于其他各个领域。In the recent ten years, Diophantine equation not only develops itself very actively, but also promotes the development of other fields.

最近十余年，不定方程不仅自身的发展异常活跃，而且全面应用于离散数学的其他各个领域。In the recent ten years, not only Diophantine equation itself develops very actively, but also promotes the development of other fields.

传统的预测控制算法虽然有着良好的性能，但是需要求解复杂的丢番图方程。Conventional predictive control algorithm possesses good overall performance, but it needs to solve the complicated Diophantine equation.

运用矩阵的初等变换法和不定方程求解法，给出了求解同余式组的两种简便方法。In this paper, by the elementary transformation of matrix and diophantine equation, two simple methods of solving linear congruence equation system are presented.

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本文研究了和素数有关的方程组和不等式组。通过更细致的指数和估计，我们改进了以前的结果。In this paper we study systems of Diophantine inequalities and equations with prime numbers. We improve previous results by more delicate estimates of exponential sums.

这些体制的安全性依赖于求解丢番图方程中的一些NP完全问题和大数的素因子分解的困难性。The security of these new cryptosystems depends on some NP-complete problems in the theory of Diophantine equations and on the difficulty of large number decomposition.

通过这些方程的求解，对数论中解决其他丢番图方程有一定的帮助和借鉴作用，对丢番图方程研究有一定的贡献。It will help the study of other Diophantine equations, provide some references and make some contributions to the research of Diophantine equation by the solutions of all these problems.

利用矩阵的初等列变换，给出了求解多元线性不定方程的一种方法，该方法改进了传统方法计算量大、步骤多的缺点。By elementary rank transformations of matrix, a method has been obtained to solve linear diophantine equation with some variables. The method overcomes the deficiency of traditional method.

利用不定方程理论及中国剩余定理，我们设计了一类陷门背包公开钥密码系统，它们具有更好的安全性。Using the theorem of linear Diophantine equations and the Chinese remainder theorem, we have constructed a class of trap-door knapsack public-key cryptosystems, which have a better security.

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第三章在ABS算法的基础上给出了求解线性丢番图方程组的整隐式LU算法和整隐式LX算法，讨论了相应的ABS性质，并讨论了复杂性分析及其应用。In chapter three, the DLU and IILX algorithms for linear Diophantine equations are given based on the ABS algorithms. Corresponding properties, complexity and their application are discussed.

然而，这些背包算法易于遭受低密度子集和攻击、GCD攻击、联立丢番图逼近攻击以及正交格攻击等。However, these cryptosystems are vulnerable against low density subset-sum attack, subset of and attacked, GCD attack, simultaneous Diophantine approximation attack and the orthogonal grid attack.