Chapter 8. Introduction to Number Theory


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1 Chapter 8 Introduction to Number Theory CRYPTOGRAPHY AND NETWORK SECURITY 1
2 Index 1. Prime Numbers 2. Fermat`s and Euler`s Theorems 3. Testing for Primality 4. Discrete Logarithms 2
3 Prime Numbers 3
4 Prime Numbers Prime number is a central concern of number theory Integer p > 1 is a prime number if its only divisors are ±1 and ±p Any Integer a > 1 can be factored in a unique way as a = p 1 a1 p 2 a2 p 1 a1 p t at 91 = = =
5 Prime Numbers (cont`d) a = P a p p P where each a p 0 The integer 12 is represented by {a 2 = 2, a 3 = 1} The integer 18 is represented by {a 2 = 1, a 3 = 2} The integer 91 is represented by {a 7 = 1, a 13 = 1} a = P a p p P, b = P b p p P define k = ab, Then k = P k p p P It follows k p = a p + b p for all p P k = = = 216 k 2 = = 3; k 3 = = = =
6 Prime Numbers (cont`d) a = P a p p P, b = P b p p P If a b, then a p b p for all p a = 12; b = 36; = 2 2 3; 36 = a 2 = 2 = b 2 a 3 = 1 b 3 Thus, the inequality a p b p is satisfied for all prime numbers. 6
7 Prime Numbers (cont`d) It is easy to determine gcd of two positive integers if we express each integer as the product of primes. 300 = = gcd 18,300 = = 6 If k = gcd a, b, then k p = min a p, b p for all p Determining the prime factors of a large number is not easy task, so preceding relationship doesn`t not directly lead to a practical method of calculating gcd. *gcd: greatest common devisor 7
8 Fermat and Euler`s Theorems 8
9 Fermat`s Theorems These two theorems play important roles in publickey cryptography. Fermat`s Theorem If p is a prime and a is a positive integer not divisible by p, then a p 1 1(mod p) a = 7, p = = 49 = 11 mod = 121 = 7 mod = 49 = 11 mod = 121 = 7 mod 19 a p 1 = 7 18 = = 7 11 = 77 = 1(mod 19) 9
10 Fermat`s Theorems: Proof set p = 1,2, p 1 set X = a mod p, 2a mod p, p 1 a mod p Suppose ja = ka(mod p), where 1 j < k p 1, and a is relative prime with a, we can eliminate a. Then it resulting j = k(mod p), but it is impossible. Therefore every elements in set X will be different. a 2a p 1 a [(1 2 p 1 ](mod p) a p 1 p 1! = a p 1 = 1 mod p p 1! mod p 10
11 Fermat`s Theorems (Cont`d) Alternative form: a p = a mod p This form does not requires a and p be relative prime. p = 5, a = 3 a p = 3 5 = 243 = 3 mod5 = a modp p = 5, a = 10 a p = 10 5 = = 10 mod5 = 0 mod5 = a(modp) 11
12 Euler`s Totient Function Written φ(n), and defined as the number of positive integers less than n and relatively prime to n. Determine φ(37), 37 is prime, 1~36 are relatively prime to 37, Thus φ 37 = 36 For a prime number p, φ p = p 1 If we have two prime numbers p and q with p!= q, for n = pq. φ n = φ pq = φ p φ q = (p 1) (q 1) φ n = pq 1 q 1 + p 1 = pq p + q + 1 = p 1 q 1 = φ p φ q φ 21 = φ 3 φ 7 = = 2 6 = 12 {1,2,3,5,8,10,11,13,16,17,19,20} 12
13 Euler`s Theorem Every a and n that are relatively prime: a φ n = 1(mod n) it is true if n is prime, because of Fermat`s theorem exists. φ n = n 1 a n 1 = 1 mod n But this also holds for any integer n. a = 3; n = 10; φ 10 = 4, a φ n = 3 4 = 81 = 1 mod10 a = 2; n = 11; φ 11 = 10, a φ n = 2 10 = 1024 = 1 mod11 13
14 Euler`s Theorem: Proof R = x 1, x 2,, x φ n each x i is a unique positive integer less than n. now multiply by a, and modulo n; S = { ax 1 mod n, ax 2 mod n,, (ax φ n mod n)} From here, very similar with Fermat`s Theorem`s proof. 14
15 Testing for Primality 15
16 Testing for Primality MillerRabin Algorithm: First, Any positive odd integer n>=3 can be expressed as n 1 = 2 k q with k > 0, q odd And we need two properties of prime numbers that we will need. 1. if p is prime and a is a positive integer less than p, than a 2 mod p = 1 2. let p be a prime number greater than 2. we can write p 1 = 2 k q with k >, q odd. Let a be any integer in 1 < a < p 1. then one of two condition is true 1. a q is congruent to 1 mod p. that is a q mod p = 1 or a q = 1(mod p) 2. one of numbers a q, a 2q, a 4q,, a 2k 1 q is congruent to 1 mod p. that is j in range (1 j k) such that a 2j 1 q mod p = 1 modp = p 1 or a 2j 1q = 1(mod p) 16
17 MillerRabin Algorithm(con`t) If n is prime, then either the first element in the list of residues, or remainders(a q, a 2q,, a 2k 1q, a 2kq ) mod n equals 1; or some elements is the list equals (n1); otherwise, n is not a prime. Also, if condition met, it don`t exactly mean that n is prime. n = 2047 = 23 89, than n 1 = and mod 2047 = 1, so condition mets, but is not prime. 17
18 MillerRabin Algorithm(con`t) In the TEST procedure, it takes a candidate integer n as input and returns the result composite if n is not a prime, and the result inconclusive if n may or may not be a prime. TEST(n) 1. Find integers k,q, with k>0, q odd, so that (n 1 = 2 k q ); 2. Select a random integer a, 1 < a < n 1; 3. If a q mod n = 1 then return ( inconclusive ); 4. For j=0 to k1 do 5. If a 2j q mod n = n 1 then return ( inconclusive ); 6. return ( composite ); 18
19 Discrete Logarithms 19
20 The Powers of an Integer, Modulo n a m = 1 mod n If a and n are relatively prime, then there is at least one integer m that satisfies this equation. Namely, M = φ(n). The least positive exponent m for this equation holds is referred to in several ways: The order of a(mod n) The exponent to which a belongs(mod n) The length of the period generated by a 20
21 21
22 The Powers of an Integer, Modulo n 1. All sequences end in The length of a sequence divides φ 19 = 18. That is, an integral number of sequences occur in each row of table. 3. Some of the sequences are of length 19. In this case, it is said that 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. If a number is of this φ n order it is referred as primitive root. If a is a primitive root of n, a, a 2,, a φ n are distinct (mod n) and are all relatively prime to n. Not all integers have primitive roots. The only integers of the form 2,4, p a, 2p a where p is any odd prime and a is a positive integer. 22
23 Logarithms for Modula Arithmetic Review the ordinary logarithm`s properties y = x log x(y) log x (1) = 0 log x x = 1 log x yz = log x y + log x (z) log x y r = r log x (y) Consider primitive root a for some prime number p. We know that the powers of a from 1 through (p1) produce each integer from 1 through (p1) exactly once. And any integer b satisfies b = r mod p for some r, where 0 r (p 1) b = a i mod p where 0 i (p 1) 23
24 Logarithms for Modula Arithmetic(cont`d) This exponent i is referred to as the discrete logarithm of the number b for the base a(mod p). We denote this as dlog a,p b. dlog a,p 1 = 0 because a 0 mod p = 1 mod p = 1 dlog a,p a = 1 because a 1 mod p = a 24
25 25
26 Logarithms for Modula Arithmetic(cont`d) Now consider x = a dlog a,p x mod p y = a dlog a,p y mod p xy = a dlog a,p xy mod p xy mod p = [ x mod p y mod p mod p a dlog a,p xy mod p = a dlog a,p x mod p a dlog a,p y mod p mod p =(a dlog a,p x +dlog a,p y )mod p 26
27 Logarithms for Modula Arithmetic(cont`d) But if consider Euler`s theorem, that every a and n that are relatively prime. a φ(n) = 1(mod n) Any positive integer z can be expressed in the form z = q + kφ n, with 0 q < φ(n). Therefore, by Euler`s theorem, a z = a q mod n if z = q mod φ(n) Apply this to the foregoing equality we have, dlog a,p xy And generalizing, dlog a,p y r = [dlog a,p x +dlog a,p y ](mod φ(p)) = [r dlog a,p y ](mod φ(p)) 27
28 28
29 Calculation of Discrete Logarithm Consider the equation y = g x mod p If g,x,p is given, it is straightforward to calculate y. At worst, we must perform x repeated multiplications, and algorithms exit for achieving grater efficiency. However, given y,g, and p it is in general, very difficult to calculate x. 29
30 Thank you for listening 30
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