Question 1

For many cryptographic algorithms, it is necessary to select one or more very large prime numbers.

Accepted Answer

Many cryptographic algorithms, especially those based on public-key cryptography such as RSA, rely on the selection of large prime numbers as a foundational aspect of their security. The difficulty of factoring large numbers that are the product of two primes is what makes these algorithms secure.

Question 2

One of the useful features of the Chinese remainder theorem is that it provides a way to manipulate potentially very large numbers mod M in terms of tuples of smaller numbers.

Accepted Answer

The Chinese remainder theorem allows for the representation and manipulation of integers in a modular arithmetic system by breaking down calculations into a series of operations on smaller numbers, each operation modulo a different prime divisor of M. This can simplify calculations involving very large numbers.

Question 3

the algorithm credited to Euclid for easily finding the greatest&#10;common divisor of two integers has broad significance in cryptography.

Accepted Answer

Euclid's algorithm for finding the greatest common divisor (GCD) of two integers is fundamental in number theory, which is a key underpinning of many cryptographic algorithms. The GCD is crucial for the mathematical operations involved in public key cryptography, such as RSA, making Euclid's algorithm significant in the field of cryptography.

Question 4

Unlike ordinary addition, there is not an additive inverse to each&#10;integer in modular arithmetic.

Accepted Answer

In modular arithmetic, every integer has an additive inverse such that their sum is congruent to zero modulo the modulus.

Question 5

If b|a we say that b is a divisor of A.

Accepted Answer

In mathematics, if b divides a (denoted as b|a), it means that a is divisible by b without leaving a remainder, making b a divisor of a.

Question 6

the first assertion of the CRt, concerning arithmetic operations,&#10;follows from the rules for modular arithmetic.

Accepted Answer

The first assertion of the Chinese Remainder Theorem (CRT) concerning arithmetic operations indeed follows from the rules for modular arithmetic, as it allows for the determination of a number modulo several pairwise coprime integers based on its remainders modulo these integers.

Question 7

the scheme where you can find the greatest common divisor of&#10;two integers by repetitive application of the division algorithm is&#10;known as the Brady algorithm.

Accepted Answer

The scheme is known as the Euclidean algorithm, not the Brady algorithm.

Question 8

the Chinese Remainder theorem is believed to have been&#10;discovered by the Chinese mathematician Agrawal in 100 A.D.

Accepted Answer

The answer of the Chinese Remainder theorem is believed to...

Question 9

All integers have primitive roots.

Accepted Answer

Only integers of the form $2^k, 4, p^k$, and $2p^k$ (where $p$ is an odd prime and $k$ is a positive integer) have primitive roots.

Question 10

the primitive roots for the prime number 19 are 2, 3, 10, 13, 14&#10;and 15.

Accepted Answer

The answer of the primitive roots for the prime number...

Question 11

two integers a and b are said to be congruent modulo n, if&#10;(a mod n) = (b mod n).

Accepted Answer

The answer of two integers a and b are said...

Question 12

Basic concepts from number theory that are needed for understanding finite fields include divisibility, the Euclidian algorithm, and modular arithmetic.

Accepted Answer

The answer of Basic concepts from number theory that are...

Question 13

the rules for ordinary arithmetic involving addition, subtraction,
and multiplication carry over into modular arithmetic.
9.two theorems that play important roles in public-key cryptography are Fermat's theorem and Euler's theorem.

Accepted Answer

The answer of the rules for ordinary arithmetic involving addition,...

Deck 2: Introduction to Number Theory