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CS595-Cryptography and Network Security
Cryptography and Network Security
Number Theory
Xiang-Yang Li
CS595-Cryptography and Network Security
Introduction to Number Theory
Divisors
b|a if a=mb for an integer m b|a and c|b then c|a b|g and b|h then b|(mg+nh) for any int. m,n
Prime number
P has only positive divisors 1 and p
Relatively prime numbers
No common divisors for p and q except 1
CS595-Cryptography and Network Security
GCD
Greatest common divisor gcd(a,b)
The largest number that divides both a and b
Euclid's algorithm
Find the GCD of two numbers a and b, a<b
Use fact if a and b have divisor d so does a-
b, a-2b …
CS595-Cryptography and Network Security
Cont.
GCD (a,b) is given by:
let g 0 =b g 1 =a gi+1 = gi-1 mod gi when g (^) i =0 then gcd(a,b) = gi-
The algorithm terminates in O( log b) rounds
Why?
CS595-Cryptography and Network Security
Properties
For any two integers a and b
Exist integers m and n : gcd(a,b) = ma + bn Example: a=2, b=3; we choose m=-1, n=1 so –2+3= a=6, b=11; we choose m=2, n=-1 so 2*6-11= Simple proof?
Integer a can be factored as
a = p 1 a 1 p2a 2 p 3 a3…. pn an where p (^) i is prime number CS595-Cryptography and Network Security
Modular Arithmetic
Congruence
a ≡ b mod n says when divided by n that a and b have the same remainder It defines a relationship between all integers a ≡ a a ≡ b then b ≡ a a ≡ b, b ≡ c then a ≡ c
CS595-Cryptography and Network Security
Cont.
addition
(a+b) mod n ≡(a mod n) + (b mod n)
subtraction
a-b mod n ≡ a+(-b) mod n
multiplication
ab mod n derived from repeated addition Possible: ab ≡ 0 where neither a, b ≡ 0 mod n CS595-Cryptography and Network Security
Cont.
Division
a/b mod n multiplied by inverse of b: a/b = ab -1^ mod n b-1b ≡ 1 mod n 3 -1^ ≡7 mod 10 because 3*7 ≡ 1 mod 10 Inverse does not always exist! Only when gcd(b,n)=
CS595-Cryptography and Network Security
Addition and Multiplication
Integers modulo n with addition and
multiplication form a commutative ring
with the laws of
Associativity (a+b)+c ≡ a+(b+c) mod n Commutativity a+b ≡ b+a mod n Distributivity (a+b)c ≡ (ac)+(b*c) mod n
CS595-Cryptography and Network Security
Galois Field
If n is constrained to be a prime number p
then this forms a Galois field modulo p
denoted GF(p) and all the normal laws
associated with integer arithmetic work
Exponentiation
b = a e^ mod p
Discrete Logarithms
find x where ax^ = b mod p
CS595-Cryptography and Network Security
Inverses and Euclid's Extended
GCD Routine
If (a,n)=1 then the inverse always exists
Can extend Euclid's algorithm to find
inverse by keeping track of gi = u i.n + vi.a
Extended Euclid's (or binary GCD)
algorithm to find inverse of a number a mod
n (where (a,n)=1) is:
CS595-Cryptography and Network Security
Inverse
Inverse(a,n) is given by:
X=(x1,x2,x3)=(1,0,n); Y=(y1,y2,y3)=(0,1,a) If y3=0 return x3=gcd(a,n); no inverse If y3=1 return y3=gcd(a,n); y2=a -1^ mod n Q=[x3/y3] T=X-Q*Y X=Y; Y=T Goto 2 nd^ step
CS595-Cryptography and Network Security
Simple Fact
Equation x^2 ≡ 1 mod p has only solutions 1,-
If p is prime number Simple proof: (x+1)(x-1) ≡ 0 mod p So if we find another solution, then p can not be prime number! Miller and Rabin 1975, Randomly chosen integer a If a 2 ≡ 1 mod p then p is not prime number Integer a is called the witness Otherwise p maybe, or maybe not a prime number CS595-Cryptography and Network Security
Witness Algorithm
Witness(a,n) Let b (^) kb (^) k-1…b 1 b 0 be the binary code of n- Let d= For i=k downto 0 x=d; d=d*d mod n If d=1 and x ≠ 1, and x ≠ n- return TRUE If b (^) i =1 then d=d*a mod n Endfor If d ≠ 1 then return TRUE Return FALSE
CS595-Cryptography and Network Security
Facts
Analysis the result of witness If returns TRUE, then n is not prime number Find other solutions for x 2 ≡ 1 mod n Otherwise, n maybe prime number Given odd n and random a Witness fails with probability less than 0. Run witness algorithm s times If one time, it is TRUE Then n is not prime number Otherwise, Pr( n is prime)>1-2 -s CS595-Cryptography and Network Security
Randomized Methods
Las Vegas Method Always produces correct results Runs in expected polynomial time Monte Carlo Method Runs in polynomial time May produce incorrect results with bounded probability No-Biased Monte Carlo Method Answer yes is always correct, but the answer no may be wrong Yes-biased Monte Carlo Method Answer no is always correct, but the answer yes may be wrong
CS595-Cryptography and Network Security
Witness Algorithm
Witness Algorithm is based on Monte Carlo
Method
It actually test compositeness, not primality When it reports yes, the number is always composite When it reports no, input may be composite, prime Probability Result Pr(input=composite | ans=composite)= 1 Pr(ans=no | input=composite)<1/ Pr(input=composite | ans=no) ≤ 1/ CS595-Cryptography and Network Security
Time Complexity
Each round of witness cost O(log n )
Unit: integer multiplication and modular arithmetic
So the primality testing cost O( s log n )
The confidence is 1-2 -s^ if report prime The confidence is 1 if report non-prime
CS595-Cryptography and Network Security
Primitive Root
Order of integer
The order of a modulo n is the smallest positive k such that a k≡1 mod n
Primitive Root
Integer a is a primitive root of n if the order of a modulo n is φ(n) Not all integers have primitive root Example n=pq for primes p and q Prime p has φ(p-1) primitive roots CS595-Cryptography and Network Security
Discrete Logarithms
Y ≡ gx^ mod p
Compute x Time complexity O(e(ln p)1/3(ln ln p)2/3)
CS595-Cryptography and Network Security
Quadratic Residue
Quadratic Residue
Integer b is a quadratic residue of integer n if
and only if x^2 ≡ b mod n has a solution for x
Otherwise b is called quadratic nonresidue
Given odd prime p,
b is quadratic residue, iff b (p-1)/2 ≡ 1 mod n
b is quadratic nonresidue, iff b (p-1)/2 ≡ -1 mod n
CS595-Cryptography and Network Security
Complexity Theory
The input length of a problem is the
number n of symbols used to characterize it
Function f(n) is order O(g(n)) if
f(n)<=c*|g(n)|, for all n>=N 0 , for some c
Polynomial time algorithm ( P )
solves any instance of a particular problem with input length n in time O(p(n)), where p is a polynomial
CS595-Cryptography and Network Security
Cont.
N on-deterministic polynomial time algorithm ( NP ) - is one for which any guess at the solution of an instance of the problem may be checked for validity in polynomial time.
NP-complete problems - are a subclass of NP problems for which it is known that if any such problem has a polynomial time solution, then all NP problems have polynomial solutions.
Co-NP : the complements of NP problems.
