Posted: November 23rd, 2022

# Large Prime Numbers and Cybersecurity(Cryptograpy course)

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Cryptography and Network Security

Seventh Edition

by William Stallings

1
Lecture slides prepared for “Cryptography and Network Security”, 7/e, by William Stallings, Chapter 2 – “Introduction to Number Theory”.

Chapter 2
Introduction to Number Theory

Number theory is pervasive in cryptographic algorithms. This chapter provides
sufficient breadth and depth of coverage of relevant number theory topics for understanding
the wide range of applications in cryptography. The reader familiar with these
topics can safely skip this chapter.
The first three sections introduce basic concepts from number theory that are
needed for understanding finite fields; these include divisibility, the Euclidian algorithm,
and modular arithmetic. The reader may study these sections now or wait until
ready to tackle Chapter 5 on finite fields.
Sections 2.4 through 2.8 discuss aspects of number theory related to prime numbers
and discrete logarithms. These topics are fundamental to the design of asymmetric
(public-key) cryptographic algorithms. The reader may study these sections now or
The concepts and techniques of number theory are quite abstract, and it is often
difficult to grasp them intuitively without examples. Accordingly, this chapter includes
a number of examples, each of which is highlighted in a shaded box.
2

Divisibility
We say that 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
The notation b | a is commonly used to mean b divides a
If b | a we say that b is a divisor of a
The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24
13 | 182; – 5 | 30; 17 | 289; – 3 | 33; 17 | 0

3
We say that a nonzero b divides a if a=mb for some m, where a, b, and m are integers. That is, b divides a if there is no remainder on division.
The notation b | a is commonly used to mean b divides a . Also, if b | a , we say that b is a divisor of a .

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
11 | 66 and 66 | 198 = 11 | 198

Subsequently, we will need some simple properties of divisibility for integers, which are as follows:
• 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.
4

Properties of Divisibility
To see this last point, note that:
If b | g , then g is of the form g = b * g1 for some integer g1
If b | h , then h is of the form h = b * h1 for some integer h1
So:
mg + nh = mbg1 + nbh1 = b * (mg1 + nh1 )
and therefore b divides mg + nh
b = 7; g = 14; h = 63; m = 3; n = 2
7 | 14 and 7 | 63.
To show 7 (3 * 14 + 2 * 63),
we have (3 * 14 + 2 * 63) = 7(3 * 2 + 2 * 9),
and it is obvious that 7 | (7(3 * 2 + 2 * 9)).

To see this last point, note that
• If b | g , then g is of the form g = b * g1 for some integer g1 .
• If b | h , then h is of the form h = b * h1 for some integer h1 .
So
mg + nh = mbg1 + nbh1 = b * (mg1 + nh1 )
and therefore b divides mg + nh .
5

Division Algorithm
Given any positive integer n and any nonnegative integer a, if we divide a by n we get an integer quotient q and an integer remainder r that obey the following relationship:
a = qn + r 0 ≤ r < n; q = [a/n] © 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. Given any positive integer n and any nonnegative integer a, if we divide a by n, we get an integer quotient q and an integer remainder r that obey the following relationship: a = qn + r, 0 ≤ r < n; q = [a/n] which is referred to as the division algorithm. 6 © 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. Figure 2.1a demonstrates that, given a and positive n, it is always possible to find q and r that satisfy the preceding relationship. Represent the integers on the number line; a will fall somewhere on that line (positive a is shown, a similar demonstration can be made for negative a). Starting at 0, proceed to n, 2n, up to qn such that qn ≤ a and (q + 1)n > a. The distance from qn to a is r, and we have found the unique values of q and r. The remainder r is often referred to as a residue .
For example:
a = 11; n = 7; 11 = 1 x 7 + 4; r = 4 q = 1
a = –11; n = 7; –11 = (–2) x 7 + 3; r = 3 q = –2
Figure 4.1b provides another example.
7

Euclidean Algorithm
One of the basic techniques of number theory
Procedure for determining the greatest common divisor of two positive integers
Two integers are relatively prime if their only common positive integer factor is 1

8
One of the basic techniques of number theory is the Euclidean algorithm, which
is a simple procedure for determining the greatest common divisor of two positive
integers. First, we need a simple definition: Two integers are relatively prime if their
only common positive integer factor is 1.

Greatest Common Divisor (GCD)
The greatest common divisor of a and b is the largest integer that divides both a and b
We can use the notation gcd(a,b) to mean the greatest common divisor of a and b
We also define gcd(0,0) = 0
Positive integer c is said to be the gcd of a and b if:
c is a divisor of a and b
Any divisor of a and b is a divisor of c
An equivalent definition is:
gcd(a,b) = max[k, such that k | a and k | b]

Recall that nonzero b is defined to be a divisor of a if a = mb for some m , where a , b , and
m are integers. We will use the notation gcd(a , b ) to mean the greatest common divisor
of a and b . The greatest common divisor of a and b is the largest integer that divides
both a and b . We also define gcd(0, 0) = 0.
More formally, the positive integer c is said to be the greatest common divisor
of a and b if
1. c is a divisor of a and of b .
2. Any divisor of a and b is a divisor of c .
An equivalent definition is the following:
gcd(a , b ) = max[k , such that k | a and k | b ]
9

GCD
Because we require that the greatest common divisor be positive, gcd(a,b) = gcd(a,-b) = gcd(-a,b) = gcd(-a,-b)
In general, gcd(a,b) = gcd(| a |, | b |)
Also, because all nonzero integers divide 0, we have gcd(a,0) = | a |
We stated that two integers a and b are relatively prime if their only common positive integer factor is 1; this is equivalent to saying that a and b are relatively prime if gcd(a,b) = 1
gcd(60, 24) = gcd(60, – 24) = 12
8 and 15 are relatively prime because the positive divisors of 8 are 1, 2, 4, and 8, and the positive divisors of 15 are 1, 3, 5, and 15. So 1 is the only integer on both lists.

Because we require that the greatest common divisor be positive, gcd(a , b ) =
gcd(a , -b ) = gcd(-a , b ) = gcd(-a ,-b ). In general, gcd(a , b ) = gcd( | a | , | b | ).
Also, because all nonzero integers divide 0, we have gcd(a , 0) = a .
We stated that two integers a and b are relatively prime if their only common
positive integer factor is 1. This is equivalent to saying that a and b are relatively
prime if gcd(a , b ) = 1.
10

We now describe an algorithm credited to Euclid for easily finding the greatest
common divisor of two integers (Figure 2.2). This algorithm has broad significance
in cryptography.
11

We can find the greatest common divisor of two integers by repetitive application
of the division algorithm. This scheme is known as the Euclidean algorithm.
Figure 2.3 illustrates a simple example.
12

Table 2.1
Euclidean Algorithm Example
(This table can be found on page 34 in the textbook)

In this example, we begin by dividing 1160718174 by 316258250, which gives 3
with a remainder of 211943424. Next we take 316258250 and divide it by 211943424.
The process continues until we get a remainder of 0, yielding a result of 1078.
13

Modular Arithmetic
The modulus
If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n; the integer n is called the modulus
Thus, for any integer a:
a = p1 a1 * p2 a2 * . . . * pp1 a1
where p1 < p2 < . . . < pt are prime numbers and where each ai is a positive integer This is known as the fundamental theorem of arithmetic © 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. 24 An integer p > 1 is a prime number if and only if its only divisors are ±1
and ± p . Prime numbers play a critical role in number theory and in the techniques
discussed in this chapter.
Any integer a > 1 can be factored in a unique way as
a = p1 a1 * p2 a2 * . . . * pp1 a1

r > 0 ?
Swap
a and b
Replace
b with r
Replace
a with b
Divide a byb,
calling the
remainder r
GCD is
the final
value of b
START
END

Figure 2.3 Euclidean Algorithm Example: gcd(710, 310)
710 = 2 x 310 + 90
310 = 3 x 90 + 40
90 = 2 x 40 + 10
40 = 4 x 10
GCDGCD
Same GCD

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