Primality Proving via One Round in ECPP and One Iteration in AKS

Primality Proving via One round in Ecpp and One Iteration

On August 2002, Agrawal, Kayal and Saxena announced the first deterministic and polynomial time primality testing algorithm. For an input n, the AKS algorithm runs in heuristic time Õ(log n). Verification takes roughly the same amount of time. On the other hand, the Elliptic Curve Primality Proving algorithm (ECPP) runs in random heuristic time Õ(log n) ( Õ(log n) if the fast multiplication is ...

Deterministic Primality Testing - understanding the AKS algorithm

Prime numbers play a very vital role in modern cryptography and especially the difficulties involved in factoring numbers composed of product of two large prime numbers have been put to use in many modern cryptographic designs. Thus, the problem of distinguishing prime numbers from the rest is vital and therefore there is a need to have efficient primality testing algorithms. Although there had...

An Introduction to the AKS Primality Test

Input: An integer n > 1. 0: if n is a power then output composite fi; 1: r := 2; 2: while (r < n) do 3: if gcd(r, n) 6= 1 then output composite fi; 4: if r is prime then 5: q := largest prime factor of r − 1; 6: if (q ≥ 4√r log n) and (n(r−1)/q 6≡ 1 mod r) then break fi; 7: fi; 8: r := r + 1; 9: od; 10: for a = 1 to 2 √ r log n do 11: if (x− a)n 6≡ (xn − a) mod (xr − 1, n) then output composite...

Pseudopowers and primality proving

The so-called pseudosquares can be employed in very powerful machinery for the primality testing of integers N . In fact, assuming reasonable heuristics (which have been confirmed for numbers to 2) they can be used to provide a deterministic primality test in time O(log N), which some believe to be best possible. In the 1980s D.H. Lehmer posed a question tantamount to whether this could be exte...

Proving Primality in Essentially Quartic