Robert Eisele
Engineer, Systems Architect and DBA

Modular multiplicative inverse calculator

The inverse of an element \(x\) is another element \(y\) such that \(x\circ y = e\), where \(e\) is the neutral element. For example:

\[ \begin{array}{rl} x + y &= 0, y = -x\\ x \cdot y &= 1, y = x^{-1}\\ \end{array} \]

In number theory and encryption often the inverse is needed under a modular ring. Less formal spoken, how can one divide a number under a modular relation? Here the multiplicative inverse comes in. The multiplicative inverse of an integer \(a\) modulo \(m\) is an integer \(x\) such that

\[a x\equiv 1 \pmod{m}\]

Dividing both sides by \(a\) gives

\[x\equiv a^{-1} \pmod{m}\]

The solution can be found with the euclidean algorithm, which is used for the calculator here: