Evaluate the history of cryptography from its origins. Analyze how cryptography was used and describe how it grew within history. The writing assignment requires 2 pages to evaluate the history. You must use a minimum of three scholarly articles to complete the assignment. The assignment must be properly APA formatted with a separate title and reference page.

Cryptography and Network Security:

Principles and Practice

Eighth Edition

Chapter 2

Introduction to Number Theory

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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

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Properties of Divisibility (1 of 2)

• 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

11 | 66 and 66 | 198 = 11 | 198

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

and n

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Properties of Divisibility (2 of 2)

• 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)).

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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]

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Figure 2.1 The Relationship a = qn +

r; 0 ≤ r < n

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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

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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]

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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 |)

gcd(60, 24) = gcd(60, − 24) = 12

• 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

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.

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Figure 2.2 Euclidean Algorithm

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Figure 2.3 Euclidean Algorithm

Example

: gcd(710, 310)

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Table 2.1 Euclidean Algorithm

Example

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Modular Arithmetic (1 of 3)

• 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

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Modular Arithmetic (2 of 3)

• 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)

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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

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Modular Arithmetic (3 of 3)

• 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

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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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Table 2.2 (a) Arithmetic Modulo 8

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Table 2.2 (b) Multiplication Modulo 8

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Table 2.2 (c) Additive and

Multiplicative Inverse Modulo 8

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Table 2.3 Properties of Modular

Arithmetic for Integers in Zn

Property Expression

Commutative Laws (w + x) mod n = (x + w) mod n

(w × x) mod n = (x × w) mod n

Associative Laws [(w + x) + y] mod n = [w + (x + y)] mod n

[(w × x) × y] mod n = [w × (x × y)] mod n

Distributive Law [w × (x + y)] mod n = [(w × x) + (w × y)] mod n

Identities (0 + w) mod n = w mod n

(1 × w) mod n = w mod n

Additive Inverse (−w) For each w Zn, there exists a z such that w + z 0 mod n

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Table 2.4 Extended Euclidean

Algorithm Example

i ri qi xi yi

−1 1759 Blank 1 0

0 550 Blank 0 1

1 109 3 1 −3

2 5 5 −5 16

3 4 21 106 −339

4 1 1 −111 355

5 0 4 Blank Blank

Result: d = 1; x = −111; y = 355

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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

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Table 2.5 Primes Under 2000

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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)

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Table 2.6 Some Values of Euler’s

Totient Function ø(n)

n ɸ (n)

1 1

2 1

3 2

4 2

5 4

6 2

7 6

8 4

9 6

10 4

n ɸ (n)

11 10

12 4

13 12

14 6

15 8

16 8

17 16

18 6

19 18

20 8

n ɸ (n)

21 12

22 10

23 22

24 8

25 20

26 12

27 18

28 12

29 28

30 8

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Euler’s Theorem

• States that for every a and n that are relatively prime:

aø(n) = 1(mod n)

• An alternate form is:

aø(n)+1 = a(mod n)

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Miller-Rabin Algorithm

• Typically used to test a large number for primality

• Algorithm is:

TEST (n)

1. Find integers k, q, with k > 0, q odd, so that (n – 1)=2kq ;

2. Select a random integer a, 1 < a < n – 1 ;

3. if aq mod n = 1 then return (“inconclusive”) ;

4. for j = 0 to k – 1 do

5. if (a2jq mod n = n – 1) then return (“inconclusive”) ;

6. return (“composite”) ;

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Deterministic Primality Algorithm

• Prior to 2002 there was no known method of efficiently

proving the primality of very large numbers

• All of the

algorithm

s 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

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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

• 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

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Table 2.7 Powers of Integers, Modulo 19

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Table 2.8 Tables of Discrete

Logarithms, Modulo 19 (1 of 2)

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Table 2.8 Tables of Discrete

Logarithms, Modulo 19 (2 of 2)

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Summary

• Understand the concept of

divisibility and the division algorithm

• Understand how to use the

Euclidean algorithm to find the

greatest common divisor

• Present an overview of the

concepts of modular arithmetic

• Explain the operation of the

extended Euclidean algorithm

• Discuss key concepts relating to

prime numbers

• Understand Fermat’s theorem

• Understand Euler’s theorem

• Define Euler’s totient function

• Make a presentation on the topic of

testing for primality

• Explain the Chinese remainder

theorem

• Define discrete logarithms

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