Exam 4: Basic Concepts in Number Theory and Finite Fields

The scheme where you can find the greatest common divisor of two integers by repetitive application of the division algorithm is known as the Brady algorithm.

The remainder r in the division algorithm is often referred to as a __________ .

Cryptographic algorithms do not rely on properties of finite fields.

Polynomial arithmetic includes the operations of addition, subtraction and multiplication.

In the context of abstract algebra we are usually not interested in evaluating a polynomial for a particular value of x.To emphasize this point the variable x is sometimes referred to as the __________ .

The polynomial cx)is said to be the __________ of ax)and bx)if cx)divides both ax)and bx)and any divisor of ax)and bx)is a divisor of cx).

Two integers a and b are said to be congruent modulo n,if a mod n)= b mod n).

Unlike ordinary addition,there is not an additive inverse to each integer in modular arithmetic.

Let S be the set of integers,positive,negative,and 0,under the usual operations of addition and multiplication.S is an __________ domain.

A field is a set in which we can do addition,subtraction, multiplication and division without leaving the set.

A __________ g of a finite field F or order q is an element whose first q - 1 powers generate all the nonzero elements of F. 15.Consider a field F defined by a polynomial fx).An element b contained in F is called a __________ of the polynomial if fb)= 0.

A polynomial fx)over a field F is called __________ if and only if fx)cannot be expressed as a product of two polynomials,both over F,and both of degree lower than that of fx).

The euclidean algorithm cannot be adapted to find the multiplicative inverse of a polynomial.

It is easy to find the multiplicative inverse of an element in gp)for large values of p by constructing a multiplication table,however for small values of p this approach is not practical.

The Advanced Encryption Standard uses infinite fields.

A cyclic group is always commutative and may be finite or infinite.

As a congruence relation,mod expresses that two arguments have the same remainder with respect to a given modulus.

Finite fields of order p can be defined using arithmetic mod p.

The __________ of two numbers is the largest integer that divides both numbers.

GF stands for __________ field in honor of the mathematician who first studied finite fields.