In block ciphers, the nonlinear components, also known as sub- stitution boxes (S-boxes), are used with the purpose of inducing confusion in cryptosystems. For the last decade, most of the work on designing S-boxes over the points of elliptic curves has been published. The main purpose of these studies is to hide data and improve the security levels of crypto algorithms. In this work, we design pair of nonlinear components of a block cipher over the residue class of Gaussian integers (GI). The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters. But the prime field dependent on the Elliptic curve (EC) provides one S-box at a time by fixing three parameters a, b, and p. The newly designed pair of S-boxes are assessed by various tests like nonlinearity, bit independence criterion, strict avalanche criterion, linear approximation probability, and differential approximation probability.