Cryptography & Network Security: Introduction to Number Theory

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Questions and Answers

What does the notation 'b | a' mean?

b is a remainder of a

b is a divisor of a (correct)

b is a multiple of a

b is a factor of a

Which of the following properties of divisibility is NOT mentioned in the text?

If b | g and b | h, then b | (mg + nh) for arbitrary integers m and n

If a | b and b | c, then a | c

If a | 1, then a = ±1

Any b ≠ 0 divides 0 (correct)

What is the purpose of the example with $b = 7, g = 14, h = 63, m = 3, n = 2$ in the text?

To demonstrate that 7 is a divisor of $3 * 14 + 2 * 63$ (correct)

To demonstrate that 7 is a factor of $3 * 14 + 2 * 63$

To demonstrate that 7 is a multiple of $3 * 14 + 2 * 63$

To demonstrate that 7 is a divisor of 14 and 63

What is the meaning of the statement '$13 | 182$' in the text?

13 is a divisor of 182 Signup and view all the answers

Which of the following statements about divisibility is NOT mentioned in the text?

If a | b and b | c, then a | (b + c) Signup and view all the answers

What is the significance of the statement '$17 | 0$' in the text?

17 is a divisor of 0 Signup and view all the answers

If 24 = 9 (mod n), what is the possible value of n?

3 Signup and view all the answers

Given 57 = 6 (mod m), what could be a possible value for m?

7 Signup and view all the answers

What is the value of m if 83 = -2 (mod m)?

6 Signup and view all the answers

If 90 = 6 (mod p), what are the potential values for p?

20 Signup and view all the answers

For which of the following equations does the given congruence hold? 25 = 4 (mod x)

$x = 6$ Signup and view all the answers

If $64 = -3$ (mod y), what is a possible value for y?

$11$ Signup and view all the answers

What is the relationship between a positive integer a and a nonnegative integer n when divided, according to the Division Algorithm?

a = qn + r, where 0 ≤ r < n Signup and view all the answers

What is the general relationship given in the Division Algorithm?

a = qn + r Signup and view all the answers

What is the Euclidean Algorithm primarily used for in number theory?

Determining the greatest common divisor of two positive integers Signup and view all the answers

When are two integers considered relatively prime?

When their only common positive integer factor is 1 Signup and view all the answers

How is the greatest common divisor (GCD) of two numbers defined?

The largest integer that divides both numbers Signup and view all the answers

What does gcd(0,0) equal according to the content provided?

0 Signup and view all the answers

What algorithm, developed in 2002, efficiently determines whether a given large number is prime?

AKS algorithm Signup and view all the answers

Who is believed to have discovered the Chinese Remainder Theorem (CRT) around 100 A.D.?

Sun-Tsu Signup and view all the answers

Which theorem provides a way to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli?

Chinese Remainder Theorem (CRT) Signup and view all the answers

What property must be known beforehand for the Chinese Remainder Theorem (CRT) to be applied when dealing with potentially very large numbers?

Factorization of M Signup and view all the answers

Which of the following algorithms produces a probabilistic result for proving the primality of very large numbers?

Fermat's Primality Test Signup and view all the answers

What theorem does not appear to be as efficient as the Miller-Rabin algorithm for determining whether a given large number is prime?

AKS algorithm Signup and view all the answers

What is the result of [(11 mod 8) + (15 mod 8)] mod 8?

2 Signup and view all the answers

Which property of modular arithmetic is demonstrated by the expression [(11 mod 8) - (15 mod 8)] mod 8?

Additive inverse Signup and view all the answers

What is the result of [(11 mod 8) * (15 mod 8)] mod 8?

5 Signup and view all the answers

According to the information provided, what is the unique way that any integer a > 1 can be factored?

a = p1^a1 * p2^a2 * ... * pn^an Signup and view all the answers

What is the result of the Extended Euclidean Algorithm example provided?

d = 1; x = -111; y = 355 Signup and view all the answers

Which property of prime numbers is mentioned in the text?

Prime numbers only have divisors of 1 and itself Signup and view all the answers

Study Notes

Divisibility

A nonzero b divides a if a = mb for some m, where a, b, and m are integers.
b divides a if there is no remainder on division.
Notation b | a means b divides a.
If b | a, then b is a divisor of a.
Example: 1, 2, 3, 4, 6, 8, 12, and 24 are divisors of 24.

Properties of Divisibility

If a | 1, then a = ±1.
If a | b and b | a, then a = ±b.
Any b ≠ 0 divides 0.
If a | b and b | c, then a | c.
If b | g and b | h, then b | (mg + nh) for arbitrary integers m and n.

Modular Arithmetic

Two integers a and b are congruent modulo n if (a mod n) = (b mod n).
Notation a = b (mod n) means a and b are congruent modulo n.
Example: 73 = 4 (mod 23); 21 = -9 (mod 10).

Properties of Congruences

If a = b (mod n), then n | (a - b).
If a = b (mod n), then b = a (mod n).
If a = b (mod n) and b = c (mod n), then a = c (mod n).

Division Algorithm

Given positive integer n and nonnegative integer a, dividing a by n gives quotient q and remainder r such that a = qn + r.
0 ≤ r < n.

Greatest Common Divisor (GCD)

The GCD of a and b is the largest integer that divides both a and b.
Notation gcd(a, b) represents the GCD of a and b.
gcd(a, b) is the largest c such that c divides a and b.
gcd(a, b) = max[k, such that k | a and k | b].

Chinese Remainder Theorem (CRT)

The CRT states that it is possible to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli.
The CRT can be stated in several ways.
It provides a way to manipulate large numbers modulo M in terms of tuples of smaller numbers.

Prime Numbers

Prime numbers only have divisors of 1 and themselves.
Prime numbers are central to number theory.
Any integer a > 1 can be factored in a unique way as a = p1^a1 * p2^a2 * ....

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Description

Test your knowledge on Chapter 2 'Introduction to Number Theory' from William Stallings' book 'Cryptography and Network Security Eighth Edition'. Explore concepts like divisibility and divisors in integer numbers.