For g=gcd(a,b) uses the Extended Euclidean Algorithm to write g=ka+mb for integers k and m with k>0.
RSA Decryption Key
For composite N=pq (p and q primes) and encryption exponent
e with gcd(e, \(\phi\)(N))=1, finds the smallest positive
decryption exponent d.
Modular Exponentiation
For \(b,n\in \Bbb Z\) and \(m\in\Bbb N\),
finds \(b^m\) (mod n)
Relatively Prime Integers
Given \(n\in\Bbb Z\) finds values in list
relatively prime to n.
For a given \(n\in\Bbb N\), you may find all
\(0\le a<n\) with \(gcd(a,n)=1\) or only those values
from a list that you enter.
Square Roots Modulo n
For natural number n and integer a, finds a's
modulo n square roots if it has any.
Cyclic Subgroup by Generated by
Element \(a\in(\Bbb Z/n\Bbb Z)^*\)
For integers a, n with gcd(a, n)=1,
finds the cyclic subgroup \(H=<a>\)
for \(a\in (\Bbb Z/n\Bbb Z)^*\).
Chinese Remainder Theorem
Solve simultaneous congruences using
the Chinese Remainder Theorem.
For \(x\equiv c_i (mod n_i)\), first row
enter \(c_i\)'s, second enter
corresponding \(n_i\)'s.
Pollard's Algorithm
For natural number n, the algorithm attempts to factor n.
It can fail to factor composities which have no relatively small
prime factors and, as it involves two random selections, will not
always produce the same output for a given imput.
Specify an elliptic curve
\( E:\ y^2=x^3+Ax+B\ (mod\ p) \).
For points \( P, Q\in E \), find \(P\oplus Q\). Enter
the point at infinity as (0,0).
Input \(P\)
Input \(Q\)
Specify an elliptic curve:
\( E:\ y^2=x^3+Ax+B \).
Click once to add P to the plot, a second time to add Q.