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

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Chapter 7 introduction to number theory

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

  • 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

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

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

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 n

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 n1

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

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

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

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

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 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

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, +, ×)


Chapter 7 introduction to number theory

  • 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.


Chapter 7 introduction to number theory

  • 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.


Chapter 7 introduction to number theory

  • 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 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]


Chapter 7 introduction to number theory

  • 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

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

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.


Chapter 7 introduction to number theory

  • 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

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.


Chapter 7 introduction to number theory

  • 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.


Chapter 7 introduction to number theory

  • 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

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.


Chapter 7 introduction to number theory

  • 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

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

  • 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

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

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 theorem1

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:


Example1

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:


Chapter 7 introduction to number theory

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

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

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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