Algorithms
Extended Euclidean Algorithm (a, b)
•s0= 1, s1= 0, t0= 0, t1= 1
•Compute ri−2=ri−1qi−1+riand si=si−2−qi−1si−1,ti=ti−2−qi−1ti−1
as long as ri>0.
Solovay-Strassen(n)
•Select a random integer awith 1 ≤a≤n−1 and compute x:= a
n
(How?)
•If x= 0 then return ”composite” (Why is this step correct?)
•Compute y:= a(n−1)/2mod n(How?)
•Check if x≡ymod n. If yes then return ”prime” else return ”compos-
ite”.
Miller-Rabin(n)
•First find m, k so that n−1 = 2kmand mis odd.
•Choose a random integer awith 1 ≤a≤n−1 and compute b:=
ammod n(How?)
•If b≡1 mod nreturn ”prime”
•Iterate ktimes: (a) if b≡ −1 mod nreturn ”prime” (b)b:= b2mod n
•return ”composite”
Pollard algorithm(n, B)
•Compute a:= 2B!mod n(How?)
•Find d=gcd(a−1, n) and if 1 < d < n return delse return ”fail”
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