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Cryptography and Network Security

Seventh Edition

by William Stallings

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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

wait until ready to read Part Three.

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

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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)).

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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]

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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.

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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

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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)

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.

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 = qn + r 0 ≤ r < n; q = [a/ n]
a = [a/ n] * n + ( a mod n)
11 mod 7 = 4; - 11 mod 7 = 3
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
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 = qn + r 0 ≤ r < n; q = [ a/ n]
a = [a/ n] * n + ( a mod n)
14
Modular Arithmetic
Congruent modulo n
Two integers a and b are said to be congruent modulo n if (a mod n) = (b mod n)
This is written as a = b(mod n)2
Note that if a = 0(mod n), then n | a
73 = 4 (mod 23); 21 = - 9 (mod 10)
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
Two integers a and b are said to be congruent modulo n , if (a mod n ) =
(b mod n ). This is written as a K b (mod n ).2
Note that if a = 0 (mod n ), then n | a .
15
Properties of Congruences
Congruences have the following properties:
1. a = b (mod n) if n (a – b)
2. a = b (mod n) implies b = a (mod n)
3. a = b (mod n) and b = c (mod n) imply a = c (mod n)
To demonstrate the first point, if n (a - b), then (a - b) = kn for some k
So we can write a = b + kn
Therefore, (a mod n) = (remainder when b + kn is divided by n) = (remainder when b is divided by n) = (b mod n)
23 = 8 (mod 5) because 23 - 8 = 15 = 5 * 3
- 11 = 5 (mod 8) because - 11 - 5 = - 16 = 8 * (- 2)
81 = 0 (mod 27) because 81 - 0 = 81 = 27 * 3
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
Congruences have the following properties:
1. a = b (mod n) if n (a – b)
2. a = b (mod n) implies b = a (mod n)
3. a = b (mod n) and b = c (mod n) imply a = c (mod n)
To demonstrate the first point, if n (a - b), then (a - b) = kn for some k
So we can write a = b + kn
Therefore, (a mod n) = (remainder when b + kn is divided by n) = (remainder when b is divided by n) = (b mod n)
16
Modular Arithmetic
Modular arithmetic exhibits the following properties:
1. [(a mod n) + (b mod n)] mod n = (a + b) mod n
2. [(a mod n) - (b mod n)] mod n = (a - b) mod n
3. [(a mod n) * (b mod n)] mod n = (a * b) mod n
We demonstrate the first property:
Define (a mod n) = ra and (b mod n) = rb. Then we can write a = ra + jn for some integer j and b = rb + kn for some integer k
Then:
(a + b) mod n = (ra + jn + rb + kn) mod n
= (ra + rb + (k + j)n) mod n
= (ra + rb) mod n
= [(a mod n) + (b mod n)] mod n
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
Modular arithmetic exhibits the following properties:
1. [(a mod n) + (b mod n)] mod n = (a + b) mod n
2. [(a mod n) - (b mod n)] mod n = (a - b) mod n
3. [(a mod n) * (b mod n)] mod n = (a * b) mod n
We demonstrate the first property:
Define (a mod n) = ra and (b mod n) = rb. Then we can write a = ra + jn for some integer j and b = rb + kn for some integer k.
Then:
(a + b) mod n = (ra + jn + rb + kn) mod n
= (ra + rb + (k + j)n) mod n
= (ra + rb) mod n
= [(a mod n) + (b mod n)] mod n
17
Remaining Properties:
Examples of the three remaining properties:
11 mod 8 = 3; 15 mod 8 = 7
[(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2
(11 + 15) mod 8 = 26 mod 8 = 2
[(11 mod 8) - (15 mod 8)] mod 8 = - 4 mod 8 = 4
(11 - 15) mod 8 = - 4 mod 8 = 4
[(11 mod 8) * (15 mod 8)] mod 8 = 21 mod 8 = 5
(11 * 15) mod 8 = 165 mod 8 = 5
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The remaining properties are proven as easily. Here are examples of the three
properties.
18
Table 2.2(a)
Arithmetic Modulo 8
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(This table can be found on page 37 in the textbook)
Table 2.2a and Table 2.2b provide an illustration of modular addition and multiplication
modulo 8. Looking at addition, the results are straightforward, and there is a regular
pattern to the matrix. Both matrices are symmetric about the main diagonal
in conformance to the commutative property of addition and multiplication.
19
Table 2.2(b)
Multiplication Modulo 8
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(This table can be found on page 37 in the textbook)
Similarly, the entries in the multiplication table are straightforward. In ordinary
arithmetic, there is a multiplicative inverse, or reciprocal, to each integer. In modular
arithmetic mod 8, the multiplicative inverse of x is the integer y such that
(x * y ) mod 8 = 1 mod 8. Now, to find the multiplicative inverse of an integer
from the multiplication table, scan across the matrix in the row for that integer to
find the value 1; the integer at the top of that column is the multiplicative inverse;
thus, (3 * 3) mod 8 = 1. Note that not all integers mod 8 have a multiplicative
inverse; more about that later.
20
Table 2.2(c)
Additive
and
Multiplicative Inverse
Modulo 8
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(This table can be found on page 37 in the textbook)
As in ordinary addition, there is an additive inverse, or negative, to each integer in
modular arithmetic. In this case, the negative of an integer x is the integer y such
that (x + y ) mod 8 = 0. To find the additive inverse of an integer in the left-hand
column, scan across the corresponding row of the matrix to find the value 0; the
integer at the top of that column is the additive inverse; thus, (2 + 6) mod 8 = 0.
21
Table 2.3
Properties of Modular Arithmetic for Integers in Zn
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(This table can be found on page 38 in the textbook)
If we perform modular arithmetic within Zn, the properties shown in Table 2.3 hold for integers in Zn We show in the next section that this implies that Zn is a commutative ring with a multiplicative identity element.
In general, an integer has a multiplicative inverse in Zn if that integer is relatively prime to n. Table 2.2c in the text shows that the integers 1, 3, 5, and 7 have a multiplicative inverse in Z 8, but 2, 4, and 6 do not.
22
Table 2.4
Extended Euclidean Algorithm Example
Result: d = 1; x = –111; y = 355
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(This table can be found on page 43 in the textbook)
Example of the Extended Euclidean Algorithm. (See pages 97 – 99 in textbook for details.)
23
Prime Numbers
Prime numbers only have divisors of 1 and itself
They cannot be written as a product of other numbers
Prime numbers are central to number theory
Any integer a > 1 can be factored in a unique way as

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

where p1 < p2 < . . . < pt are prime numbers and where each ai is a positive integer
This is known as the fundamental theorem of arithmetic; a proof can be found in any text on number theory.
Table 2.5
Primes Under 2000
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(This table can be found on page 44 in the textbook)
Table 2.5 shows the primes less than 2000. Note the way
the primes are distributed. In particular, note the number of primes in each range
of 100 numbers.
25
Fermat's Theorem
States the following:
If p is prime and a is a positive integer not divisible by p then
ap-1 = 1 (mod p)
An alternate form is:
If p is prime and a is a positive integer then
ap = a (mod p)
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
26
Two theorems that play important roles in public-key cryptography are Fermat’s
theorem and Euler’s theorem.
Fermat’s theorem states the following: If p is prime and a is a positive integer not
divisible by p, then
ap-1 = 1 (mod p)
An alternative form of Fermat’s theorem is also useful: If p is prime and a is a
positive integer, then
ap = a (mod p)
Table 2.6
Some Values of Euler’s Totient Function ø(n)
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(This table can be found on page 48 in the textbook)
27
Before presenting Euler’s theorem, we need to introduce an important quantity in
number theory, referred to as Euler’s totient function, written ø (n ), and defined as
the number of positive integers less than n and relatively prime to n . By convention,
ø(1) = 1.
Table 2.6 lists the first 30 values of ø (n ). The value ø(1) is without meaning
but is defined to have the value 1.
It should be clear that, for a prime number p ,
ø (p ) = p - 1
Euler's Theorem
States that for every a and n that are relatively prime:
aø(n) = 1(mod n)
An alternative form is:
aø(n)+1 = a(mod n)
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28
Euler’s theorem states that for every a and n that are relatively prime:
aø(n) = 1(mod n)
As is the case for Fermat’s theorem, an alternative form of the theorem is also
Useful:
aø(n)+1 = a(mod n)
Miller-Rabin Algorithm
Typically used to test a large number for primality
Algorithm is:
TEST (n)
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29
The algorithm due to Miller and Rabin [MILL75, RABI80] is typically used to test
a large number for primality.
The procedure TEST takes a candidate integer n as input and returns the result composite
if n is definitely not a prime, and the result inconclusive if n may or may not
be a prime.
How can we use the Miller-Rabin algorithm to determine with a high degree of confidence whether or not an integer
is prime? It can be shown [KNUT98] that given an odd number n that is not prime
and a randomly chosen integer, a with 1 < a < n - 1, the probability that TEST
will return inconclusive (i.e., fail to detect that n is not prime) is less than 1/4.
Thus, if t different values of a are chosen, the probability that all of them will pass
TEST (return inconclusive) for n is less than (1/4)t . For example, for t = 10, the
probability that a nonprime number will pass all ten tests is less than 10-6 . Thus,
for a sufficiently large value of t, we can be confident that n is prime if Miller’s test
always returns inconclusive .
This gives us a basis for determining whether an odd integer n is prime
with a reasonable degree of confidence. The procedure is as follows: Repeatedly
invoke TEST (n) using randomly chosen values for a . If, at any point, TEST returns
composite , then n is determined to be nonprime. If TEST continues to
return inconclusive for t tests, then for a sufficiently large value of t , assume
that n is prime.
1.
2.
3.
4.
5.
6.
Find integers k, q, with k > 0, q odd, so that (n – 1)=2kq ;

Select a random integer a, 1 < a < n – 1 ;
if aq mod n = 1 then return (“inconclusive") ;
for j = 0 to k – 1 do
if (a2jq mod n = n – 1) then return (“inconclusive") ;
return (“composite") ;
Deterministic Primality Algorithm
Prior to 2002 there was no known method of efficiently proving the primality of very large numbers
All of the algorithms in use produced a probabilistic result
In 2002 Agrawal, Kayal, and Saxena developed an algorithm that efficiently determines whether a given large number is prime
Known as the AKS algorithm
Does not appear to be as efficient as the Miller-Rabin algorithm
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
Prior to 2002, there was no known method of efficiently proving the primality of very
large numbers. All of the algorithms in use, including the most popular (Miller-Rabin),
produced a probabilistic result. In 2002 (announced in 2002, published in 2004),
Agrawal, Kayal, and Saxena [AGRA04] developed a relatively simple deterministic
algorithm that efficiently determines whether a given large number is a prime. The algorithm,
known as the AKS algorithm, does not appear to be as efficient as the Miller-
Rabin algorithm. Thus far, it has not supplanted this older, probabilistic technique.
30
Chinese Remainder Theorem (CRT)
Believed to have been discovered by the Chinese mathematician Sun-Tsu in around 100 A.D.
One of the most useful results of number theory
Says it is possible to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli
Can be stated in several ways
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
31
One of the most useful results of number theory is the Chinese remainder theorem
(CRT). In essence, the CRT says 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. We present here a formulation that is most
useful from the point of view of this text. An alternative formulation is explored in
Problem 2.33.
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. This can be useful when M is 150 digits or more. However, note
that it is necessary to know beforehand the factorization of M .
Provides a way to manipulate (potentially very large) numbers mod M in terms of tuples of smaller numbers
This can be useful when M is 150 digits or more
However, it is necessary to know beforehand the factorization of M
Table 2.7
Powers of Integers, Modulo 19
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
(This table can be found on page 57 in the textbook)
Table 2.7 shows all the powers of a, modulo 19 for all positive a <19. The
length of the sequence for each base value is indicated by shading. Note the
following:
1. All sequences end in 1. This is consistent with the reasoning of the preceding
few paragraphs.
2. The length of a sequence divides ø (19) = 18. That is, an integral number of
sequences occur in each row of the table.
3. Some of the sequences are of length 18. In this case, it is said that the base
integer a generates (via powers) the set of nonzero integers modulo 19. Each
such integer is called a primitive root of the modulus 19.
More generally, we can say that the highest possible exponent to which a number
can belong (mod n ) is ø (n ). If a number is of this order, it is referred to as a
primitive root of n . The importance of this notion is that if a is a primitive root of n ,
then its powers
a , a2 ,. . . , aø(n)
are distinct (mod n ) and are all relatively prime to n . In particular, for a prime number
p , if a is a primitive root of p , then
a , a2 ,. . . , ap-1
are distinct (mod p ). For the prime number 19, its primitive roots are 2, 3, 10, 13, 14,
and 15.
Not all integers have primitive roots. In fact, the only integers with primitive
roots are those of the form 2, 4, pa , and 2pa , where p is any odd prime and a is a
positive integer. The proof is not simple but can be found in many number theory
books, including [ORE76].
32
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
Table 2.8
Tables of Discrete Logarithms, Modulo 19
(This table can be found on page 60 in the textbook)
Table 2.8, which is directly derived from Table 2.7, shows the sets of discrete
logarithms that can be defined for modulus 19.
33
Summary
Divisibility and the division algorithm
The Euclidean algorithm
Greatest Common Divisor
Finding the Greatest Common Divisor
Modular arithmetic
The modulus
Properties of congruences
Modular arithmetic operations
Properties of modular arithmetic
Euclidean algorithm revisited
The extended Euclidean algorithm
Prime numbers
Fermat’s Theorem
Euler’s totient function
Euler’s Theorem
Testing for primality
Miller-Rabin algorithm
A deterministic primality algorithm
Distribution of primes
The Chinese Remainder Theorem
Discrete logarithms
Powers of an integer, modulo n
Logarithms for modular arithmetic
Calculation of discrete logarithms
© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved.
34
Chapter 2 summary.
0
n 2n 3n qn (q + 1)na
n
r
Figure 2.1 The Relationship a = qn + r; 0 ≤ r < n
(a) General relationship
0 15
15
10
30
= 2 15
70
(b) Example: 70 = (4 15) + 10
45
= 3 15
60
= 4 15
75
= 5 15
Figure 2.2 Euclidean Algorithm
No
No Yes
a > b ?

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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