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Chapter 7 – Introduction to Number Theory

Chapter 7 – Introduction to Number Theory. Instructor: 孫宏民 hmsun@cs.nthu.edu.tw Room: EECS 6402, Tel:03-5742968, Fax : 886-3-572-3694. Prime Numbers. prime numbers only have divisors of 1 and self they cannot be written as a product of other numbers

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Chapter 7 – Introduction to Number Theory

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  1. Chapter 7– Introduction to Number Theory Instructor: 孫宏民 hmsun@cs.nthu.edu.twRoom: EECS 6402, Tel:03-5742968, Fax : 886-3-572-3694

  2. Prime Numbers • prime numbers only have divisors of 1 and self • they cannot be written as a product of other numbers • note: 1 is prime, but is generally not of interest • eg. 2,3,5,7 are prime, 4,6,8,9,10 are not • prime numbers are central to number theory • list of prime number less than 200 is: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

  3. Prime Factorisation • to factor a number n is to write it as a product of other numbers: n=a × b × c • note that factoring a number is relatively hard compared to multiplying the factors together to generate the number • the prime factorisation of a number n is when its written as a product of primes • eg. 91=7×13 ; 3600=24×32×52

  4. Relatively Prime Numbers & GCD • two numbers a, b are relatively prime if have no common divisors apart from 1 • eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common factor • conversely can determine the greatest common divisor by comparing their prime factorizations and using least powers • eg. 300=21×31×52 18=21×32hence GCD(18,300)=21×31×50=6

  5. Fermat's Theorem • ap-1 mod p = 1 • where p is prime and gcd(a,p)=1 • also known as Fermat’s Little Theorem • useful in public key and primality testing

  6. Euler Totient Function ø(n) • when doing arithmetic modulo n • complete set of residues is: 0..n-1 • reduced set of residues is those numbers (residues) which are relatively prime to n • eg for n=10, • complete set of residues is {0,1,2,3,4,5,6,7,8,9} • reduced set of residues is {1,3,7,9} • number of elements in reduced set of residues is called the Euler Totient Function ø(n)

  7. Euler Totient Function ø(n) • to compute ø(n) need to count number of elements to be excluded • in general need prime factorization, but • for p (p prime) ø(p) = p-1 • for p.q (p,q prime) ø(p.q) = (p-1)(q-1) • eg. • ø(37) = 36 • ø(21) = (3–1)×(7–1) = 2×6 = 12

  8. Euler's Theorem • a generalisation of Fermat's Theorem • aø(n)mod N = 1 • where gcd(a,N)=1 • eg. • a=3;n=10; ø(10)=4; • hence 34 = 81 = 1 mod 10 • a=2;n=11; ø(11)=10; • hence 210 = 1024 = 1 mod 11

  9. Primality Testing • often need to find large prime numbers • traditionally sieve using trial division • ie. divide by all numbers (primes) in turn less than the square root of the number • only works for small numbers • alternatively can use statistical primality tests based on properties of primes • for which all primes numbers satisfy property • but some composite numbers, called pseudo-primes, also satisfy the property

  10. Miller Rabin Algorithm • a test based on Fermat’s Theorem • algorithm is: TEST (n) is: 1. Find integers k, q, k > 0, q odd, so that (n–1)=2kq 2. Select a random integer a, 1<a<n–1 3. if aqmod n = 1then return (“maybe prime"); 4. for j = 0 to k – 1 do 5. if (a2jqmod n = n-1) then return(" maybe prime ") 6. return ("composite")

  11. Probabilistic Considerations • if Miller-Rabin returns “composite” the number is definitely not prime • otherwise is a prime or a pseudo-prime • chance it detects a pseudo-prime is < ¼ • hence if repeat test with different random a then chance n is prime after t tests is: • Pr(n prime after t tests) = 1-4-t • eg. for t=10 this probability is > 0.99999

  12. Prime Distribution • prime number theorem states that primes occur roughly every (ln n) integers • since can immediately ignore evens and multiples of 5, in practice only need test 0.4 ln(n) numbers of size n before locate a prime • note this is only the “average” sometimes primes are close together, at other times are quite far apart

  13. Algebraic System • Binary Operation: Given a nonempty set S and a function op : S×SS, then op is a binary operation on S. • Examples: S= N and op = × :the multiple of integer S= N and op = +: the addition of integer. • Algebratic Systems: (S, op1, op2, …, opn), where S is a nonenmpty set and there are at least one binary operation on S. • Examples: (R, +, ×) and (Z, +, ×)

  14. Properties of Algebratic Systems • Closure: a op b  S, where a and b S. • Associative: (a op b) op c= a op (b op c) for (S, op), where a, b, and c S. • Communicative: a op b = b op a for (S, op) for a and b S. • (Z, -) have no communicative property. • Identity: For (S, op), eS, aS, such that a op e= e op a= a. • Example: For (Z, +), e=0, for (Z, ×), e= 1. • Inverses: For (S, op), aS, bS, such that a op b = b op a = e. • Symbol : a-1 or -a.

  15. Example: For (Z, +), the inverse of a is -a. • Example: For (R/{0}, ×), the inverse of a is a-1 while, for (Z, ×), there is no inverse for any integer. • Distribution: For (S, +, *), a*(b+c)= a*b+a*c, where a, b, and cS. • Semigroup (G,*): An algebratic system (G, *) with the following properties: Closure, association, and an identity. • Theorem: For a semigroup (G, *), the identity is unique.

  16. Groups (G, *) : A semigroup (G, *) with inverses. • Examples:(Z, +), (R/{0}, *) are groups. • Abelian (Commutative) Groups: the group with communitative property • Theorem: For a group (G, *), the inverse of an element in G is unique. • Field (F, +, *): • (F, +) is a commutative group. • (F, *) is a semigroup and (F-{0}, *) is a commutative group, where 0 is the identity for the operation +.

  17. Finite Fields • Finite Group (G, *): A group (G, *) with finite elements in G. • Example: ({0, 1, …, N-1}, +N) is a finite group, where N is an integer. • Cyclic Group (G, *): For a group (G, *), there exists an element a such that G= {an|nZ}, where an =a*a* …*a (n-1 times). • a:primitive root (with the order n=|G|). • Example: ({1, …, 6}, *7) is a cyclic group with the primitive root 3. [{3, 2, 6, 4, 5, 1}, & order= 6]

  18. Generator with order m: am=1. • Finite Fields: A field (F, +, *) with finite elements in F. • Example:GF(P)= ({0, 1, …, P-1}, +P, *P) for a prime number P. [The first finite fields].

  19. Some Famous Finite Fields [P is a prime number] • GF(P) or ZP. • GF(Pn): Given an irreducible polynomial Q(x) of degree n over GF(P). • GF(2n) for P= 2. • Example: Q(x)= x3+x+1 over GF(2) • (x+1)+ (x)= 1. • (x+1)*x2= x2+x+1.

  20. Congruences • Given integers a, b, and n 0, a, is congruent to b modulo n, written ab mod n if and only if ab = kn for some integer k. Ex. 41  93 mod13. 18  10 mod8.

  21. If ab mod n, then b is called a residue of a modulo n (conversely, a is a residue of b modulo n). • A set of n integers {r1, …, rn} is called a complete set of residues modulo n if, for every integer a, there is exactly one ri in the set such that ari mod n. • For any modulus n, the set of integers {0, 1,…, n1} forms a complete set of residues modulo n.

  22. Computing Inverses • Unlike ordinary integer arithmetic, modular arithmetic sometimes permits the computation of multiplicative inverse • That is, given an integer a in the range [0, n1], it may be possible to find a unique integer x in the range [0, n1] such that ax mod n = 1. • Ex. 3 and 7 are multiplicative inverses mod 10 because 21 mod 10 = 1. • Thm. If gcd(a, n) = 1, then (ai mod n)  (aj mod n) for each i, j such that 0 i < j < n.

  23. This property implies that each ai mod n (i = 0, ..., n1) is a distinct residue mod n, and that the set {ai mod n}i=0, ..., n1 • is a permutation of the complete set of residues {0, ..., n 1}. • This property does not hold when a and n have a common factor. • If gcd(a, n) = 1, then there exists an integer x, 0 < x < n, such that ax mod n = 1.

  24. Ex. n = 5 and a = 3: 30 mod 5 = 0 31 mod 5 = 3 32 mod 5 = 1 33 mod 5 = 4 34 mod 5 = 2. • Ex. n = 4 and a = 2: 20 mod 4 = 0 21 mod 4 = 2 22 mod 4 = 0 23 mod 4 = 2.

  25. Solving for Inverse • Euler's generalization of Fermat's theorem gives us an algorithm for solving the equation ax mod n = 1, where gcd(a, n) = 1. Since a(n) mod n = 1, we may compute x as axa(n) , or x = a(n)1 mod n. If n is prime, this is simply x = a(n1)1 mod n = an2 mod n.

  26. Ex. Let a = 3 and n = 7. Then x = 35 mod 7 = 5. • Ex. Let a = 2 and n = 15. Then x = 27 mod 15 = 8. • With this approach, to compute x, you have to know (n).

  27. Another Approach • x can also be computed using an extension of Euclid's algorithm for computing the greatest common divisor. • This is more suitable for computers to do. Euclid's algorithm for computing greatest common divisor : gcd(a, n) g0n g1a i 1 whilegi 0 gi+1gi1modgi ii + 1 returngi1

  28. Extended Euclid's Algorithm • Extended Euclid's algorithm for computing inverse (loop invariant: gi = uin + via): • inv(a, n) g0n; g1a; u0 1; v0 0; u1 0; v1 1; i 1 whilegi 0 ygi1divgi gi+1gi1ygi ui+1ui1yui vi+1vi1yvi ii + 1 xvi1 ifx 0 returnx else returnx + n

  29. Example • Ex. To solve 3x mod 7 = 1 using the algorithm, we have • Because v2 = 2 is negative, the solution is x = 2 + 7 = 5.

  30. Chinese Remainder Theorem • used to speed up modulo computations • working modulo a product of numbers • eg. mod M = m1m2..mk • Chinese Remainder theorem lets us work in each moduli mi separately • since computational cost is proportional to size, this is faster than working in the full modulus M

  31. Chinese Remainder Theorem • can implement CRT in several ways • to compute (A mod M) can firstly compute all (ai mod mi) separately and then combine results to get answer using:

  32. Example • Ex. Solve the equation "3x mod 10 = 1". Since 10 = 2  5, d1 = 2 and d2 = 5. We first find solutions x1 and x2, as follows: 3x mod 2 = 1 mod 2 = 1 x1 = 1 3x mod 5 = 1 mod 5 = 1  x2 = 2 Then we apply the Chinese Remainder Theorem to find a common solution x to the equations:

  33. x mod 2 = x1 = 1 x mod 5 = x2 = 2 . We find y1 and y2 satisfying (10/2) y1 mod 2 = 1 y1 = 1 (10/5) y2 mod 5 = 1 y2 = 3 Thus, we have x = ((10/2) y1 x1 + (10/5) y2 x2) mod 10 = (511 + 232) mod 10 = 7.

  34. Primitive Roots • from Euler’s theorem have aø(n)mod n=1 • consider ammod n=1, GCD(a,n)=1 • must exist for m= ø(n) but may be smaller • once powers reach m, cycle will repeat • if smallest is m= ø(n) then a is called a primitive root • if p is prime, then successive powers of a "generate" the group mod p • these are useful but relatively hard to find

  35. Discrete Logarithms or Indices • the inverse problem to exponentiation is to find the discrete logarithm of a number modulo p • that is to find x where ax = b mod p • written as x=loga b mod p or x=inda,p(b) • if a is a primitive root then always exists, otherwise may not • x = log3 4 mod 13 (x st 3x = 4 mod 13) has no answer • x = log2 3 mod 13 = 4 by trying successive powers • whilst exponentiation is relatively easy, finding discrete logarithms is generally a hard problem

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