Chapter 7 – Introduction to Number Theory

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Chapter 7 – Introduction to Number Theory. Instructor: 孫宏民 [email protected] 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

Instructor: 孫宏民

[email protected]: 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
• 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

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
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
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
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)
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
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
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
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")

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
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
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, +, ×)
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.
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.
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 +.
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]
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].
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.
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.

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.
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.
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.
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.

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.

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).
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

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

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.
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
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:
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:

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.

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
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