This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is dif...This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.展开更多
In this paper, we establish a general theta function identity. It is a common origin of many theta function identities. From which many classical and new modular equations are derived. All the proofs are elementary.
Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pyth...Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .展开更多
文摘This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.
基金Supported by the National Natural Science Foundation of China（11071107, 11371184）
文摘In this paper, we establish a general theta function identity. It is a common origin of many theta function identities. From which many classical and new modular equations are derived. All the proofs are elementary.
文摘Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .
