Number vs complexity of a round Number of rounds 50 o Triple des Serpent ars 30 RC6 20 DES o Twofish Rijndael 10 Complexity of a round
16 Number of rounds Complexity of a round Triple DES DES Serpent Rijndael Mars RC6 Twofish Number vs. complexity of a round 10 20 30 40 50
Mathematicians Security lexibilit oftware Hardware efficiency efficiency Computer Computer scientists Engineers 17
17 Mathematicians Computer scientists Computer Engineers Security Software efficiency Hardware efficiency Flexibility
Number theory a divides b if there existscst. b= ac (denoted by ab) properties for all a, b, cEZ -ala - if a b and blc, then alc if al b and a]c, then al (bx+cy)for all Xy E Z if alb and bla, then a=±b
18 Number theory • a divides b if there exists c s.t. b = ac (denoted by a|b) • properties for all a,b,c Z – a|a – if a|b and b|c, then a|c – if a|b and a|c, then a|(bx+cy) for all x,y Z – if a|b and b|a, then a = b
Division in z leta,b∈ Z with b≠0, then there exist q and rs t a=+r, where osr< b q is the quotient q-a div b and r is the remainder r= a mod b c is a common divisor of a and b if cla and cb 19
19 Division in Z • let a,b Z with b ≠ 0, then there exist q and r s.t. a = qb+r, where 0 r < |b| • q is the quotient q = a div b and r is the remainder r = a mod b • c is a common divisor of a and b if c|a and c|b
gcd and lcm gcd( a, b)is the largest positive integer that divides both a and b Icm(a, b) is the smallest positive integer divisible by bot oth a and b lcm(a, b)=a *b/gcd(a,b) a and b are said to be relatively prime or coprime if gcd(a, b)=1 p>2 is prime if its only positive divisors are 1 and p
20 gcd and lcm • gcd(a,b) is the largest positive integer that divides both a and b • lcm(a,b) is the smallest positive integer divisible by both a and b • lcm(a,b)=a*b/gcd(a,b) • a and b are said to be relatively prime or coprime if gcd(a,b)=1 • p ≥ 2 is prime if its only positive divisors are 1 and p