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Chapter 7 – Introduction to Number Theory

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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 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

- 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

- 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

- 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

- 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)

- 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

- 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

- 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

- 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")

- 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 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

- Binary Operation: Given a nonempty set S and a function op : S×SS, 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), eS, aS, such that a op e= e op a= a.
- Example: For (Z, +), e=0, for (Z, ×), e= 1.
- Inverses: For (S, op), aS, bS, 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 cS.
- 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 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|nZ}, 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].

- 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.

- Given integers a, b, and n 0, a, is congruent to b modulo n, written
ab mod n

if and only if

ab = kn

for some integer k.

Ex. 41 93 mod13. 18 10 mod8.

- If ab 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 ari mod n.
- For any modulus n, the set of integers {0, 1,…, n1} forms a complete set of residues modulo n.

- Unlike ordinary integer arithmetic, modular arithmetic sometimes permits the computation of multiplicative inverse
- That is, given an integer a in the range [0, n1], it may be possible to find a unique integer x in the range [0, n1] 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, ..., n1) is a distinct residue mod n, and that the set {ai mod n}i=0, ..., n1
- 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:
30 mod 5 = 0

31 mod 5 = 3

32 mod 5 = 1

33 mod 5 = 4

34 mod 5 = 2.

- Ex. n = 4 and a = 2:
20 mod 4 = 0

21 mod 4 = 2

22 mod 4 = 0

23 mod 4 = 2.

- 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

axa(n) , or

x = a(n)1 mod n.

If n is prime, this is simply

x = a(n1)1 mod n = an2 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).

- 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)

g0n

g1a

i 1

whilegi 0

gi+1gi1modgi

ii + 1

returngi1

- Extended Euclid's algorithm for computing inverse (loop invariant: gi = uin + via):
- inv(a, n)
g0n; g1a;

u0 1; v0 0;

u1 0; v1 1;

i 1

whilegi 0

ygi1divgi

gi+1gi1ygi

ui+1ui1yui

vi+1vi1yvi

ii + 1

xvi1

ifx 0

returnx

else

returnx + n

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

- 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

- 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:

- 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

= (511 + 232) mod 10 = 7.

- 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

- 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