Question 1

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

Accepted Answer

A ring is commutative if it satisfies the condition ab = ba for all a,b in R, which is the condition given in the question. The other options are not applicable as cyclic rings are those generated by a single element, abelian rings are those where addition is commutative, and infinite rings are those with infinitely many elements.

Question 2

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

Accepted Answer

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.

Accepted Answer

Polynomial division requires the use of a field coefficient set, which satisfies certain properties necessary for division. If these properties are not met by the coefficient set, then division is not always defined. Therefore, the statement is true.

Question 4

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

Accepted Answer

In modular arithmetic, every integer has an additive inverse, which is the integer that when added to the original integer, results in a multiple of the modulus. For example, in mod 5 arithmetic, the additive inverse of 2 is 3, because 2 + 3 = 5, which is a multiple of 5 (the modulus).

Question 5

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

Accepted Answer

This is true. A finite field of order p is essentially a set with p elements and an arithmetic structure defined on it, which is based on modular arithmetic mod p.

Question 6

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

Accepted Answer

Polynomial arithmetic includes addition, subtraction, and multiplication operations. Division is also possible, but it may involve fraction or radical expressions, so it is not considered a primary operation.

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.

Accepted Answer

The answer of Let S be the set of integers,positive,negative,and...

Question 8

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

Accepted Answer

Finite fields are used in various cryptographic applications, including elliptic curve cryptography, symmetric-key cryptography, and hash functions. They offer several cryptographic advantages like providing a unique structure and efficient arithmetic operations.

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.

Accepted Answer

The scheme where you can find the greatest common divisor of two integers by repetitive application of the division algorithm is known as the Euclidean algorithm, not the Brady algorithm.

Question 10

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

Accepted Answer

The term "greatest common divisor" or GCD is used to describe the largest integer that divides two or more numbers without a remainder. Therefore, option A is the correct choice. Option B, "prime polynomial", is not relevant to this concept. Option C, "lowest common divisor", is not a commonly used term, and it is not accurate to describe the largest integer that divides two numbers. Option D, "integral divisor", is not a commonly used term and is also not accurate to describe the largest integer that divides two numbers.

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.

Accepted Answer

This statement is incorrect. In fact, for small values of p it is practical to construct a multiplication table and find the multiplicative inverse of an element. This is because the table will have a smaller number of entries to compute and can be easily done by hand or with a simple computer program. The impracticality arises when p becomes very large and the table becomes too large to easily compute.

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 __________ .

Accepted Answer

In the context of abstract algebra, we are interested in studying the algebraic properties of polynomials, rather than evaluating them for specific values. Therefore, the variable x is called an "indeterminate" to emphasize the fact that it is a formal symbol representing a mathematical object, rather than a specific numerical value.

Question 13

Cryptographic algorithms do not rely on properties of finite fields.

Accepted Answer

Many popular cryptographic algorithms, such as AES and RSA, rely on finite fields or groups for their mathematical operations. For example, AES uses finite field arithmetic to perform the S-box substitution and Galois field multiplication in its rounds, while RSA uses modular arithmetic over a finite field of integers modulo a prime number.

Question 14

The Advanced Encryption Standard uses infinite fields.

Accepted Answer

The Advanced Encryption Standard (AES) uses a finite field, specifically a Galois field, to perform its encryption and decryption operations.

Question 15

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

Accepted Answer

The Euclidean algorithm can be adapted to find the multiplicative inverse of a polynomial in a polynomial ring modulo another polynomial, using the extended Euclidean algorithm.

Question 16

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

Accepted Answer

This is the definition of congruence modulo n.

Question 17

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

Accepted Answer

This is the definition of congruence modulo a given modulus. Two integers are congruent modulo m if they leave the same remainder when divided by m. Therefore, mod expresses that two arguments have the same remainder with respect to a given modulus, making the statement true.

Question 18

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

Accepted Answer

This is true. In modular arithmetic, addition, subtraction, and multiplication follows the same rules as in ordinary arithmetic, but with the additional rule that all operations must be performed modulo some base number.

Question 19

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

Accepted Answer

This is the definition of a field.

Question 20

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

Accepted Answer

The answer of The remainder r in the division algorithm...

Quiz 4: Basic Concepts in Number Theory and Finite Fields