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CIS 5371 Cryptography

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

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CIS 5371 Cryptography

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  1. CIS 5371Cryptography Introduction to Number Theory

  2. 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 • Fundamental theorem of arithmetic: Every positive integer has a unique factorization that is a product of prime powers.

  4. .. . 10 5 1  6  11 (mod 5) 0 14 9 4 1 6 11 …….. …….. 3 2 8 7 12 13 …….. ……..

  5. Modular arithmetic

  6. 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 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 3and 2 = 14 mod 3

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

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

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

  11. Finding large primes m

  12. Fermat's Little Theorem 0

  13. 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

  14. Let p = 11, pick values a : • a=3: 310 = 59049 mod 11 = 1 • a=5: 510 = 9765625 mod 11 = 1 • a=7: 710 = 282475249 mod 11 = 1 • a=8: 810 = 1073741824 mod 11 = 1

  15. Exponentiations • 3811502 mod 751 = • 3812 *381750 * 381750 mod 751 • 3812 mod 751 * 1 mod 751 • 145161 mod 751 • 218

  16. Exponentiations • ap-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

  17. The totient function (n)

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

  19. 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

  20. 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 bsuch 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.

  21. 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 thpower and decrypt by raising to the d thpower.

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