Cryptology chaoter 5 finite fields

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

True

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

False

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.

False

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

True

Cryptographic algorithms do not rely on properties of finite fields.

False

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

True

The Advanced Encryption Standard uses infinite fields.

False

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

True

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

True

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

True

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

False

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

True

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

True

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

False

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

True

The greatest common divisor of two integers is the largest positive integer that exactly _ both integers.

divides

Two integers are _ if their only common positive integer factor is 1.

relatively prime

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

greatest common divisor

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

commutative

A _ is a set of elements on which two arithmetic operations have been defined and which has the properties of ordinary arithmetic, such as closure, associativity, commutativity, distributivity, and having both additive and multiplicative inverses.

field

A _ is a field with a finite number of elements.

finite field

If b|a, we say that b is a _ of a.

divisor

For given integers a and b, the extended _ algorithm not only calculates the greatest common divisor d but also two additional integers x and y.

Euclidean

A group is said to be _ if it satisfies the condition a b = b a for all a, b in G.

abelian

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 _ .

indeterminate

With the understanding that remainders are allowed, we can say that polynomial division is possible if the coefficient set is a _ .

field

By analogy to integers, an irreducible polynomial is also called a _ .

prime polynomial

The congruence relation is used to define _ .

residue classes

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

congruence

The order of a finite field must be of the form pⁿ where p is a prime and n is a _ .

positive integer