 In such a field with "m" numbers, every nonzero element "a" has a unique modular multiplicative inverse, "a"−1 such that This inverse can be found by solving the congruence equation "ax" ≡ 1 mod "m", or the equivalent linear Diophantine equation

This equation can be solved by the Euclidean algorithm, as described above.Finding multiplicative inverses is an essential step in the RSA algorithm, which is widely used in electronic commerce; specifically, the equation determines the integer used to decrypt the message.Although the RSA algorithm uses rings rather than fields, the Euclidean algorithm can still be used to find a multiplicative inverse where one exists.