With the increasingly sharp and complex competition and confrontation in cyberspace, security issues have extended to traditional fields with unprecedented depth and breadth. With the rise of new network forms and network services represented by mobile Internet, next generation Internet, Internet of Things, cloud computing, named data network and big data. The mode of security requirements has changed from single-user mode for both parties to multi-user mode for at least one party. If you want to know more about this knowledge, you can look at the following cryptography papers.

Topic: Analysis of the Application of Mathematics in Cryptography

Cryptography, as an interdisciplinary subject, involves a wide range of disciplines, among which applied mathematics accounts for a large proportion and its position is becoming more and more important. This paper briefly introduces the basic theory and application of various algorithms involved in mathematical theory and method calculation in cryptography, divides the development history of cryptography into modern cryptography and traditional cryptography, lists two representative plaintext encryption methods, and summarizes and expounds one of them respectively.

Keywords: Cryptography Applied Mathematics Application

With the rapid development of the information age, information security has become more and more important, from personal information to national security. Information security is mainly about the mathematical calculation and processing of all kinds of information in computer systems and information exchange networks to protect information security, and cryptography is the technical core to accomplish these functions. In the initial study, advanced mathematics, linear algebra, probability theory and so on. They are all basic subjects that must be studied, but the practical operation of cryptography, the mathematical knowledge of number theory and modern algebra will still be involved and applied to varying degrees. On this basis, this paper discusses the application of some basic theories in cryptography.

First, the meaning and characteristics of cryptography

Cryptography is a science developed for the need of secure communication. The acceptance process of secure communication is as follows: the initial sender converts (encrypts) the original information in a certain way and sends it, and the receiver receives the encrypted information and decodes (decrypts) it to complete all the processes of secure transmission. However, because the transmission process is information transmission by wired electricity or radio, it is easy for the thief to steal the encrypted information during the information transmission process and recover the original information without knowing the algorithm.

The quality of decrypting confidential information depends on the skill and experience of the decryptor and the quality of the encryption algorithm. The practical application of secure communication includes two important aspects: one is to encrypt the original information by knowing the plaintext to achieve the effect of secure transmission; The second is to decrypt the intercepted encrypted information and obtain useful information. They are called cryptography and cryptanalysis respectively. They are mutual and reflect each other, but their characteristics are different.

In the history of cryptography, cryptography refers to encryption algorithms and devices that realize transmission. There are five typical cryptosystems, namely: character substitution cryptosystem, mechanical cryptosystem, sequential cryptosystem, block cryptosystem and public key cryptosystem, among which public key cryptosystem, which appeared in the mid-1970s, is the most active one.

Second, the traditional password application password system

Before Shannon's Communication Theory of Security System was published in 1949, password transmission was mainly realized by simple substitution and character substitution. Such a simple encryption form generally belongs to the category of traditional passwords.

Substitution cipher can achieve encryption effect by changing the arrangement order of plaintext, and substitution cipher involves the basic theoretical application of modular operation, modular inverse and Euler function in affine cipher.

Traditional password applications are represented by affine password and Hill password. Due to the limitation of space, this paper takes Hill cipher as an example, which uses the idea of linear algebra to encrypt plaintext and simply describes its encryption idea.

Hill Code, or Hill Code, was written by mathematician Lester Hill in the American Mathematical Monthly of 1929.

It is proposed for the first time in an article published on the Internet that its basic application idea is to use linear replacement to replace n consecutive plaintext letters with the same number of secret letters, and the replacement key is a transformation matrix, which only needs to do the same inverse transformation on the encrypted information once.

Third, the application of modern cryptography.

Shannon divided the development of cryptography into traditional cryptography and modern cryptography in Communication Theory of Security Systems published by 1949. This paper also marks the rise of modern cryptography.

Shannon introduced information theory into the research of cryptography for the first time in this paper, in which the concepts of probability statistics and entropy describe and quantitatively analyze the security of information source, key source, transmission ciphertext and cryptosystem, and then put forward the application model of related cryptosystem.

His achievements laid a theoretical foundation for the development of modern cryptography and cryptanalysis of information decoding. Modern symmetric cryptography and public key cryptography are involved in Shannon theory and number theory to varying degrees.

Representatives of modern cryptographic applications include AES algorithm based on byte processing, RSA public key algorithm based on Euler function, and El Gamal signature system which uses uncertain scheme to select random numbers for digital signature and verify its effectiveness. Taking AES algorithm as an example, this paper briefly describes the basic idea of modern password application.

The processing unit of AES algorithm is computer unit byte. The plaintext is input with 128 bits, and then the plaintext is divided into 16 bytes by inputting the key K. After ten rounds of integral operation, the turn functions from the first round to the ninth round are the same, including byte replacement, line shift, column mixing and round key addition, and column mixing is not performed in the last round of iteration.

Moreover, it is worth mentioning that the S-box permutation used in byte substitution uses the knowledge of modern algebra to complete the encryption calculation.

Four. conclusion

This paper defines the encryption and operation of cryptography in different development stages, and then mainly introduces the mathematical methods and theories in cryptography, including the application theories of number theory and probability theory.

With the positive development of modern cryptography, the mathematical foundation as an information encryption tool is more and more closely related to cryptography, and all steps of cryptography are closely related to mathematical theory. Mathematical cryptography has become the mainstream subject of modern cryptography.

Of course, the application of mathematical theory and cryptography discussed in this paper is only a superficial relationship between them. I hope to see relevant experts discuss this issue more deeply, so as to promote the deeper communication and development between applied mathematical theory and cryptography.

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