Cryptanalysis of RSA Factorization - Library(ISI Kolkata) - Indian ...

17 2.2 RSA Cryptosystem
istic primality test called the AKS algorithm. The time complexity of AKS was
of O(lp
12+ǫ ), and in 2005, Pomerance and Lenstra [101] improved the time complexity
up to O(lp 6+ǫ ). However, Miller-Rabin algorithm is much faster than AKS
in practice and hence it is the one used in the implementation of RSA. A formal
description of the Miller-Rabin test is presented in Algorithm 2.
1
2
3
4
5
6
7
8
9
10
11
Input: n > 3, an odd integer to be tested for primality
Output: ‘composite’ if n is composite, otherwise ‘probably prime’
Write n−1 = 2 s ·d with d odd;
Pick random a ∈ [1,n−1];
Set x ← a d mod n;
if x = 1 then
Return (‘probably prime’);
end
for i = 0 to s−1 do
if x ≡ −1 (mod n) then
Return (‘probably prime’);
end
else
x ← x 2 mod n;
end
end
Return (‘composite’);
Algorithm 2: Miller-Rabin Primality Test [126].
Product of Two Integers
Multiplication is fast. Even if one uses a trivial schoolbook multiplication algorithm
to find N and φ(N) in Step 2 of Algorithm 1, it can be done in polynomial
number (l p ×l q = O(l 2 p), as l p ≈ l q ) of single digit multiplications. However, with
large sized integers like p and q, it may be useful to try the Karatsuba multiplication
algorithm [87], which reduces the complexity to 3·l log 2
p 3 ≈ 3·lp
1.585 number of
single digit multiplications.
Extended Euclidean Algorithm (EEA)
In Step 3 of Algorithm 1, we choose random integers e and test if gcd(e,φ(N)) = 1.
To find the GCD, one can use the Euclidean algorithm as follows.