Number theory is a branch of mathematics, which mainly studies the properties and structure of integers.

Number theory covers a series of profound theorems and conjectures about integers, as well as a large number of high-skill problems. Number theory originated from the exploration of basic problems of arithmetic and mathematics, and its origin can be traced back to the study of integer properties by ancient mathematicians.

The research content of number theory is very extensive, including but not limited to the following aspects: prime number and its distribution, factorization and congruence equation, Fermat's last theorem, continued fraction and Diophantine equation, algebraic number theory, analytic number theory, arithmetic algebra and so on. Number theory is widely used in mathematics, physics, computer science and other fields, such as cryptography, data structure and algorithm design in computer science.

The reason why number theory is considered as a relatively independent branch is because the problems and skills involved have unique properties. Problems in number theory often require highly skilled proof and calculation, and often involve some very special mathematical concepts and methods. This makes number theory a challenging and attractive branch of mathematics.

Application of number theory in cryptography;

1, prime number and its distribution: prime number is a basic concept in number theory, which refers to a positive integer that can only be divisible by 1 and itself. In cryptography, prime numbers are widely used in the design of encryption algorithms and digital signature schemes. For example, both RSA algorithm and DSA digital signature scheme use prime numbers as basic elements. The distribution of prime numbers also has important applications in cryptography, such as random number generation and password cracking.

2. Congruence equation: Congruence equation is a basic tool in number theory, which means that the remainder obtained after the integer or integer set modulo a given modulus satisfies a certain equality relationship. In cryptography, congruence equations are widely used in the design of encryption and decryption algorithms, such as RSA algorithm and AES encryption algorithm. Congruence equation is also used to design digital signature schemes, such as RSA digital signature scheme and DSA digital signature scheme.

3. Discrete logarithm problem: Discrete logarithm problem is a classic problem in number theory, which means that given a prime number P and an integer A, finding the integer X makes ax≡x(modp). In cryptography, discrete logarithm problem is widely used in the design of public key encryption and digital signature schemes, such as ElGamal encryption algorithm and Schnorr digital signature scheme. The discrete logarithm problem is also used to design hash functions, such as SHA-256 hash function.