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

19 2.2 RSA Cryptosystem
Afterfindinganewhichisrelativelyprimetoφ(N), oneneedstofinditsinverse
modulo φ(N) in Step 4 of the algorithm. That is, one needs to find an integer
d such that ed ≡ 1 (mod φ(N)). For that one can use the Extended Euclidean
algorithm, as presented in Algorithm 4. The time complexity of this algorithm is
the same as that of the Euclidean algorithm, i.e, O(l 2 N ) for a,b with bitlength l N.
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Input: Two positive integers a,b
Output: r,s,t with r = sa+tb where r = gcd(a,b)
Initialize a 0 = a,b 0 = b,t 0 = 0,t = 1,s 0 = 1,s = 0;
q = ⌊ a 0
b 0
⌋;
r = a 0 −qb 0 ;
while r > 0 do
t 1 = t 0 −qt;
t 0 = t;
t = t 1 ;
t 1 = s 0 −qs;
s 0 = s;
a 0 = b 0 ;
b 0 = r;
q = ⌊ a 0
b 0
⌋;
r = a 0 −qb 0 ;
end
r = b 0 ;
return r,s,t.
Algorithm 4: The Extended Euclidean Algorithm [126].
Square and Multiply Algorithm
For Steps 5 and 6 in Algorithm 1, one has to perform modular exponentiations.
For this purpose, one may use the Square and Multiply Algorithm [126]. The
time complexity of a single modular exponentiation x y mod N using the square
and multiply algorithm is O(l y lN 2 ), which is O(l3 N ) as the exponents e,d are both
less than N. For a quick reference, we present the famous square and multiply
algorithm for modular exponentiation in Algorithm 5.