The main objective of this work is to build, based on the Euclidean algorithm, a "matrix of algorithms" φB:N mxn*→N*mxn, with φ B(A) = (φbij(αij)), where B = (bij)1≤i≤m1≤j≤n is a fixed matrix on N*mxn. The function 2fB is called the algorithmic matrix function. Hem we show its properties and some applications to Cryptography and nonlinear Diophantine equations. The case n = m = 1 has particular interest. On this way we show equivalences between φB and the Carl Friedrich Gauβ's congruence module p.