The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are establi...The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m+l, 6m+5 and 6m+3, m is an odd positive integer and m is an even positive integer 9|6m+3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to thematrices with symmetric partial difference in each direction for orders 6m+1 , 6m+5; to the trisection matrices with symmetric partial difference in each direction for order 6rn+3.展开更多
By means of this approach, a constructive method of pandiagonal magic squares is proposed. Pandiagonalmagic squares of order mn can be generated via two ones which are orders m and n, respectively.
Some patterns of refined epitomes of pansystems methodology were revealed roles and the related of them in problem-solving, modeling, algorithm-generating and theory-constructing were introduced. An important applicat...Some patterns of refined epitomes of pansystems methodology were revealed roles and the related of them in problem-solving, modeling, algorithm-generating and theory-constructing were introduced. An important application of pansystems methodology is to give some methods of constructing the typical pansymmetric-magic squares: 1. a method of recursively constructing magic squares of order n ( n greater than or equal to 5) ; 2. when magic squares of order m( m greater than or equal to 3) and magic squares of order n ( n greater than or equal to 3) are given of formula of obtaining magic squares of order nm; 3. when magic squares of order m ( m greater than or equal to 3) are given, a method of obtaining magic squares of order 2m.展开更多
Nowadays the E-bank systems witnessed huge growth due to the huge developments in the internet and technologies.The transmitted information represents crucial information that is exposed to various kinds of attacks.Th...Nowadays the E-bank systems witnessed huge growth due to the huge developments in the internet and technologies.The transmitted information represents crucial information that is exposed to various kinds of attacks.This paper presents a new block cipher technique to provide security to the transmitted information between the customers and the ebank systems.The proposed algorithm consists of 10 rounds,each round involves 5 operations.The operations involve Add round key,Sub bytes,Zigzag method,convert to vector,and Magic Square of order 11.The purpose of this algorithm is to make use of the complexity of the Magic Square algorithm,the speed of addition operation,the confusion provided by the zigzag,using these operations with Galois field 28 GF(28),and repeating these operations for several rounds to build fast high secure encryption algorithm.This algorithm is designed to provide fast with high complexity and security which is suitable to encrypt the data that is transmitted over the internet.Speed,complexity,and The National Institute of Standards and Technology Framework NIST suite tests were done.The complexity of the proposed algorithm is=((256)32)r+1∗((256)89)r+1+(256)121.The proposed technique gives higher speed and security in the encryption and decryption phases,according to the results of the experiments.The degree of randomness has grown by 31.8 percent.Due to a decrease in the time of encrypting and decrypting,as well as the usage of the central processing unit(CPU),efficiency is improved.The encryption process throughput is enhanced by 13%,while the decryption process throughput is increased by 11.6 percent with the recommended approach.展开更多
An optimal magic cube of order n is a magic cube whose row sums, column sums and oblique sums of 9 n layers are n(n 3+1)/2. The author proved that optimal magic cubes of order n may be constructed as long as n and 2, ...An optimal magic cube of order n is a magic cube whose row sums, column sums and oblique sums of 9 n layers are n(n 3+1)/2. The author proved that optimal magic cubes of order n may be constructed as long as n and 2, 3, 5, 7 are relatively prime, and a formula for making optimal magic cubes by using optimal Latin squares and optimal magic squares was given.展开更多
An n×n matrix A consisting of nonnegative integers is a general magic square of order n if thesum of elements in each row,column,and main diagonal is the same.A general magic square A of order n iscalled a magic ...An n×n matrix A consisting of nonnegative integers is a general magic square of order n if thesum of elements in each row,column,and main diagonal is the same.A general magic square A of order n iscalled a magic square,denoted by MS(n),if the entries of A are distinct.A magic square A of order n is normalif the entries of A are n^2 consecutive integers.Let A^*d denote the matrix obtained by raising each elementof A to the d-th power.The matrix A is a d-multimagic square,dcnoted by MS(n,d),if A^*e is an MS(n)for 1≤e≤d.In this paper we investigate the existence of normal bimagic squares of order 2u and prove that thereexists a normal bimagic square of order 2u,where u and 6 are coprime and u≥5.展开更多
文摘The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m+l, 6m+5 and 6m+3, m is an odd positive integer and m is an even positive integer 9|6m+3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to thematrices with symmetric partial difference in each direction for orders 6m+1 , 6m+5; to the trisection matrices with symmetric partial difference in each direction for order 6rn+3.
文摘By means of this approach, a constructive method of pandiagonal magic squares is proposed. Pandiagonalmagic squares of order mn can be generated via two ones which are orders m and n, respectively.
文摘Some patterns of refined epitomes of pansystems methodology were revealed roles and the related of them in problem-solving, modeling, algorithm-generating and theory-constructing were introduced. An important application of pansystems methodology is to give some methods of constructing the typical pansymmetric-magic squares: 1. a method of recursively constructing magic squares of order n ( n greater than or equal to 5) ; 2. when magic squares of order m( m greater than or equal to 3) and magic squares of order n ( n greater than or equal to 3) are given of formula of obtaining magic squares of order nm; 3. when magic squares of order m ( m greater than or equal to 3) are given, a method of obtaining magic squares of order 2m.
文摘Nowadays the E-bank systems witnessed huge growth due to the huge developments in the internet and technologies.The transmitted information represents crucial information that is exposed to various kinds of attacks.This paper presents a new block cipher technique to provide security to the transmitted information between the customers and the ebank systems.The proposed algorithm consists of 10 rounds,each round involves 5 operations.The operations involve Add round key,Sub bytes,Zigzag method,convert to vector,and Magic Square of order 11.The purpose of this algorithm is to make use of the complexity of the Magic Square algorithm,the speed of addition operation,the confusion provided by the zigzag,using these operations with Galois field 28 GF(28),and repeating these operations for several rounds to build fast high secure encryption algorithm.This algorithm is designed to provide fast with high complexity and security which is suitable to encrypt the data that is transmitted over the internet.Speed,complexity,and The National Institute of Standards and Technology Framework NIST suite tests were done.The complexity of the proposed algorithm is=((256)32)r+1∗((256)89)r+1+(256)121.The proposed technique gives higher speed and security in the encryption and decryption phases,according to the results of the experiments.The degree of randomness has grown by 31.8 percent.Due to a decrease in the time of encrypting and decrypting,as well as the usage of the central processing unit(CPU),efficiency is improved.The encryption process throughput is enhanced by 13%,while the decryption process throughput is increased by 11.6 percent with the recommended approach.
文摘An optimal magic cube of order n is a magic cube whose row sums, column sums and oblique sums of 9 n layers are n(n 3+1)/2. The author proved that optimal magic cubes of order n may be constructed as long as n and 2, 3, 5, 7 are relatively prime, and a formula for making optimal magic cubes by using optimal Latin squares and optimal magic squares was given.
基金This paper is supported by the National Natural Science Foundation of China(Nos.11871417,11501181)Science Foundation for Youths(Grant No.2014QK05)Ph.D.(Grant No.qd14140)of Henan Normal University.
文摘An n×n matrix A consisting of nonnegative integers is a general magic square of order n if thesum of elements in each row,column,and main diagonal is the same.A general magic square A of order n iscalled a magic square,denoted by MS(n),if the entries of A are distinct.A magic square A of order n is normalif the entries of A are n^2 consecutive integers.Let A^*d denote the matrix obtained by raising each elementof A to the d-th power.The matrix A is a d-multimagic square,dcnoted by MS(n,d),if A^*e is an MS(n)for 1≤e≤d.In this paper we investigate the existence of normal bimagic squares of order 2u and prove that thereexists a normal bimagic square of order 2u,where u and 6 are coprime and u≥5.
