Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible...Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let<sub></sub><sub></sub> (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .展开更多
Luo et al wrote in a recent paper [A Fast Algorithm for Computing gcd Based on Binary Multi Precision,this journal,2002,Vol.32,No.5,pp.542 545; MR 2003h:11161 ] that “the classical Euclid’s algorithm for computing t...Luo et al wrote in a recent paper [A Fast Algorithm for Computing gcd Based on Binary Multi Precision,this journal,2002,Vol.32,No.5,pp.542 545; MR 2003h:11161 ] that “the classical Euclid’s algorithm for computing the gcd of two integers takes time O(\%ln\% 3N)”, and “present” an improved algorithm (called “binary gcd” for short) based on binary multi precision with time complexity O(\%ln\% 2N). In this paper,we point out two well known facts: firstly,the binary gcd,without usefull implimentation improvements, is identical in mathematical theory to Stein’s Binary GCD algorithm published in 1967; secondly,both Euclid’s algorithm and Binary GCD have the same time complexity O(\%ln\% 2N).展开更多
准循环重复累积(Quasi⁃Cyclic Repeat Accumulate,QC⁃RA)码具有准循环低密度奇偶校验(Low Density Parity Check,LDPC)码的优点,同时能实现差分编码且为系统码,非常适用于编码协作系统,文中研究采用QC⁃RA码的编码协作系统。首先,提出基...准循环重复累积(Quasi⁃Cyclic Repeat Accumulate,QC⁃RA)码具有准循环低密度奇偶校验(Low Density Parity Check,LDPC)码的优点,同时能实现差分编码且为系统码,非常适用于编码协作系统,文中研究采用QC⁃RA码的编码协作系统。首先,提出基于最大公约数(Greatest Common Divisor,GCD)定理的QC⁃RA码构造方法;然后,进一步基于GCD定理联合构造编码协作系统信源节点与中继节点采用的QC⁃RA码,并从理论上证明基于该联合构造方法得到的编码协作系统QC⁃RA码无girth⁃4、girth⁃6环。仿真结果表明,采用QC⁃RA码的编码协作系统相对于点对点系统具有明显的性能增益;同时,与采用大列重构造QC⁃RA的编码协作相比,采用文中基于GCD定理联合构造的QC⁃RA码的编码协作误码率性能更加优异。展开更多
In order to guarantee the user's privacy and the integrity of data when retrieving ciphertext in an untrusted cloud environment, an improved ciphertext retrieval scheme was proposed based on full homomorphic encry...In order to guarantee the user's privacy and the integrity of data when retrieving ciphertext in an untrusted cloud environment, an improved ciphertext retrieval scheme was proposed based on full homomorphic encryption. This scheme can encrypt two bits one time and improve the efficiency of retrieval. Moreover, it has small key space and reduces the storage space. Meanwhile, the homomorphic property of this scheme was proved in detail. The experimental results and comparisons show that the proposed scheme is characterized by increased security, high efficiency and low cost.展开更多
文摘Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let<sub></sub><sub></sub> (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .
文摘Luo et al wrote in a recent paper [A Fast Algorithm for Computing gcd Based on Binary Multi Precision,this journal,2002,Vol.32,No.5,pp.542 545; MR 2003h:11161 ] that “the classical Euclid’s algorithm for computing the gcd of two integers takes time O(\%ln\% 3N)”, and “present” an improved algorithm (called “binary gcd” for short) based on binary multi precision with time complexity O(\%ln\% 2N). In this paper,we point out two well known facts: firstly,the binary gcd,without usefull implimentation improvements, is identical in mathematical theory to Stein’s Binary GCD algorithm published in 1967; secondly,both Euclid’s algorithm and Binary GCD have the same time complexity O(\%ln\% 2N).
文摘准循环重复累积(Quasi⁃Cyclic Repeat Accumulate,QC⁃RA)码具有准循环低密度奇偶校验(Low Density Parity Check,LDPC)码的优点,同时能实现差分编码且为系统码,非常适用于编码协作系统,文中研究采用QC⁃RA码的编码协作系统。首先,提出基于最大公约数(Greatest Common Divisor,GCD)定理的QC⁃RA码构造方法;然后,进一步基于GCD定理联合构造编码协作系统信源节点与中继节点采用的QC⁃RA码,并从理论上证明基于该联合构造方法得到的编码协作系统QC⁃RA码无girth⁃4、girth⁃6环。仿真结果表明,采用QC⁃RA码的编码协作系统相对于点对点系统具有明显的性能增益;同时,与采用大列重构造QC⁃RA的编码协作相比,采用文中基于GCD定理联合构造的QC⁃RA码的编码协作误码率性能更加优异。
基金Supported by the Research Program of Chongqing Education Commission(JK15012027,JK1601225)the Chongqing Research Program of Basic Research and Frontier Technology(cstc2017jcyjBX0008)+1 种基金the Graduate Student Research and Innovation Foundation of Chongqing(CYB17026)the Basic Applied Research Program of Qinghai Province(2019-ZJ-7099)
文摘In order to guarantee the user's privacy and the integrity of data when retrieving ciphertext in an untrusted cloud environment, an improved ciphertext retrieval scheme was proposed based on full homomorphic encryption. This scheme can encrypt two bits one time and improve the efficiency of retrieval. Moreover, it has small key space and reduces the storage space. Meanwhile, the homomorphic property of this scheme was proved in detail. The experimental results and comparisons show that the proposed scheme is characterized by increased security, high efficiency and low cost.
