Deck 4: Basic Concepts in Number Theory and Finite Fields

Question 1

A ring is said to be _________ if it satisfies the condition ab = ba for all a,b in R.

A)cyclic
B)commutative
C)abelian
D)infinite

Answer 1

B

Question 2

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

Answer 2

False

Question 3

If we attempt to perform polynomial division over a coefficient set that is not a field,we find that division is not always defined.

Answer 3

True

Question 4

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

Question 5

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

Question 6

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

Question 7

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

Question 8

Finite fields play a crucial role in several areas of cryptography.

Question 9

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.

Question 10

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

A)greatest common divisor
B)prime polynomial
C)lowest common divisor
D)integral divisor

Question 11

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.

Question 12

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 __________ .

A)monic
B)constant
C)indeterminate
D)coefficient

Question 13

Cryptographic algorithms do not rely on properties of finite fields.

Question 14

The Advanced Encryption Standard uses infinite fields.

Question 15

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

Question 16

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

Question 17

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

Question 18

The rules for ordinary arithmetic involving addition,subtraction,and multiplication carry over into modular arithmetic.

Question 19

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

Question 20

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

Question 21

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).

Question 22

If a is an integer and n is a nonzero integer,we define a mod n to be the remainder when a is divided by n.The integer n is called the __________ and the remainder is called the residue.

Question 23

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

Question 24

A zero-degree polynomial is called a __________ polynomial and is simply an element of the set of coefficients.

Question 25

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).

Question 26

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.