1. Introduction to Number Theory
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CIS 5371 Cryptography
Introduction to Number Theory

2. Preview Number Theory Essentials
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Preview
Number Theory Essentials
Congruence classes, Modular arithmetic
Prime numbers challenges
Fermat’s Little theorem
The Totient function
Euler's Theorem
Quadratic residuocity
Foundation of RSA

3. Number Theory Essentials
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Number Theory Essentials
Fundamental theorem of arithmetic:
Every positive integer has a unique factorization that
is a product of prime powers.

4. 4/27/2017 8 7 5 6 4 3 2 1 9 10 11 12 13 14
……..
.. .
1  6  11 (mod 5)
……..

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Modular arithmetic

6. The integers modulo n  a,b,n  I, a  b mod n iff n | (a-b) *
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The integers modulo n
 a,b,n  I, a  b mod n iff n | (a-b) *
28  6 mod 11: (28-6)/11 = 2  I
219  49 mod 17: (219-49)/17 = 12  I
Symmetry: If a  b mod n then b  a mod n
Transitivity:
If a  b mod n and b  c mod n then a  c mod n
* n divides (a-b)

7. Modular arithmetic: notation
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Modular arithmetic: notation
Form: a  b mod n (congruence relation) a = b mod n (modulus operator)  indicates that the integers a and b fall into the same congruence class modulo n = means that integer a is the reminder of the division of integer b by integer n. Example: 14  2 mod 3 and 2 = 14 mod 3

8. Modular arithmetic & cryptography
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Modular computations can be utilized to scramble data.
Cryptographic systems utilize modular (or elliptic curve (EC)) arithmetic.
Several cryptographic systems use prime modulus arithmetic.
Transitivity, commutativity, associativity, distribution, reflexive

9. Prime Number Challenges
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Finding large prime numbers.
Recognizing large numbers as prime.

10. How Do We Find Large Prime Numbers?
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How Do We Find Large Prime Numbers?
Look them up ?
Compute them ?
Do they REALLY have to be prime?
Yes, they really have to be prime.

11. Finding large primes m

12. Fermat's Little Theorem
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Fermat's Little Theorem
For a = 2, p cannot be 2, 4, 6, 8, etc.
For a = 5, p cannot be 5 , 10, 15, etc.
Choosing p smaller than a produces unpredictable results

13. Let p = 5, pick values for a: a =2: 24 = 16 mod 5 = 1
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Let p = 5, pick values for a:
a =2: 24 = 16 mod 5 = 1
a =3: 34 = 81 mod 5 = 1
a =4: 44 = 256 mod 5 = 1
For a = 2, p cannot be 2, 4, 6, 8, etc.
For a = 5, p cannot be 5 , 10, 15, etc.
Choosing p smaller than a produces unpredictable results

14. Let p = 11, pick values a : a=3: 310 = 59049 mod 11 = 1
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Let p = 11, pick values a :
a=3: = mod 11 = 1
a=5: = mod 11 = 1
a=7: = mod 11 = 1
a=8: = mod 11 = 1
For a = 2, p cannot be 2, 4, 6, 8, etc.
For a = 5, p cannot be 5 , 10, 15, etc.
Choosing p smaller than a produces unpredictable results

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For a = 2, p cannot be 2, 4, 6, 8, etc.
For a = 5, p cannot be 5 , 10, 15, etc.
Choosing p smaller than a produces unpredictable results

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May also want to discuss STRONG primes (primes such that (p - 1)/2 is also prime).

18. Exponentiations 3811502 mod 751 = 3812 * 381750 * 381750 mod 751
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Exponentiations
mod 751 =
3812 * * mod 751
3812 mod 751 * 1 mod 751
mod 751
218

19. Exponentiations a p-1  1 mod p 713 mod 11  x
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Exponentiations
a p-1  1 mod p
713 mod 11  x
710 mod 11* 73 mod 11  x
1 mod 11 * 73 mod 11  x
73 mod 11  x
343 mod 11  2
May also want to discuss STRONG primes (primes such that (p - 1)/2 is also prime).

20. The totient function (n)
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The totient function (n)

21. Deriving (n) Primes: (p) = p-1
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Deriving (n)
Primes: (p) = p-1
Product of 2 primes: (pq) = (p-1)(q-1)
General case (i.e. for all integers x) = ?

22. Deriving (n) Product of 2 relatively prime numbers
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Deriving (n)
Product of 2 relatively prime numbers
if gcd (m,n) = 1, then: (mn) = (m) * (n)
15 = 3*5 and
Example: (15)=2*4=8

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24. Quadratic Residuosity
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Quadratic Residuosity
An integer a is a quadratic residue with respect to n if:
a is relatively prime to n and
there exists an integer b such that: a = b2 mod n
Quadratic Residues for n = 7: QR(7)={1, 2, 4}
a = 1: b = 1 (12 = 1 mod 7), 6, 8, 13, 15, 16, 20, 22, …
a = 2: b = 3 (32 = 2 mod 7), 4, 10, 11, 17, 18, 24, 25, …
a = 4: b = 5, 9, 12, 19, 23, 26, …
Notice that 2, 3, 5, and 6 are not QR mod 7.
QR(n) forms a group with respect to multiplication.
While we will not use QR immediately, it is frequently used in cryptography.

25. The Foundation of RSA x y mod n = x (y mod (n)) mod n
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The Foundation of RSA
x y mod n = x (y mod (n)) mod n
The proof of this follows from Euler's Theorem
If y mod (n) = 1,
then for any x : xy mod n = x mod n
If we can choose e and d such that
ed = 1 mod (n)
then we can encrypt by raising x to the e th power and decrypt by raising to the d th power.