Multiply the polynomials P1 = x5 +x2+ x) by P2 = (x7 + x4 +x3+x2 + x) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1). The result is

Basic Concepts in Number Theory and Finite Fields topics are: Number theory, group rings, fields, modular arithmetic and polynomial theory.   Number theory is a branch of mathematics that studies the properties and relationships of numbers. It plays a key role in cryptography, the practice of hiding information by converting it into unreadable text.  Number theory is used in cryptography in the following ways: Encryption algorithms: Concepts from number theory, such as Euler's totient function and the Chinese Remainder Theorem, are used in various encryption algorithms. Modular... Show more

Basic Concepts in Number Theory and Finite Fields topics are: Number theory, group rings, fields, modular arithmetic and polynomial theory.
 

Number theory is a branch of mathematics that studies the properties and relationships of numbers. It plays a key role in cryptography, the practice of hiding information by converting it into unreadable text. 

Number theory is used in cryptography in the following ways:
Encryption algorithms: Concepts from number theory, such as Euler's totient function and the Chinese Remainder Theorem, are used in various encryption algorithms.
Modular arithmetic: Modular arithmetic is a system of arithmetic for integers that allows for the creation of groups, rings, and fields. These are fundamental building blocks of most modern public-key cryptosystems.
RSA: RSA is a widely adopted and standardized number theory cryptographic algorithm. It is easy to understand and implement, making it accessible to many developers and users. 

In mathematics, a finite field is a set of objects that contains a finite number of elements. The elements of a finite field can be added, subtracted, multiplied, and divided. The order of a finite field is the number of elements in the field, which is either a prime number or a prime power. 

Finite fields are fundamental in many areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography, and coding theory. Finite fields are well-suited to computer calculations and are used in many modern cryptographic applications. 

Here are some basic concepts in number theory and finite fields:
Arithmetic modulo p: Arithmetic on the coefficients is performed modulo p.
Finite fields of order p: These fields can be defined using arithmetic mod p.
Finite fields of order pn: For n > 1, these fields can be defined using arithmetic over polynomials.

Field axioms: A field must satisfy the following axioms:
The elements of the field form an Abelian group under addition.
The nonzero elements of the field form an Abelian group under multiplication.
Multiplication distributes over addition. 

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