Arithmetic of Points on Elliptic Curve over -adic Field modulo in Projective Coordinates

Elliptic Curve Cryptosystems (ECC) have emerged as a powerful alternative to traditional public-key cryptosystems, offering equivalent security with significantly smaller key sizes. The efficiency of ECC, in terms of minimizing encryption time and enhancing computational performance, is strongly influenced by the number of point addition and doubling computations required in elliptic curve arithmetic. In this context, the study of arithmetic of points in elliptic curves plays a crucial role. This study emphasizes computations in projective coordinates of points on elliptic curves defined over the -adic field . A comparative study shows that arithmetic in projective coordinates reduces the number of operations required, thereby enhancing the efficiency relative to the affine coordinate system. The coordinate-level -adic expansions of the arithmetic may be obtained by employing -adic expansion techniques to the arithmetic of points on the elliptic curve in projective coordinates for = 2, 3, ... In this paper, coordinate-level -adic expansions of the arithmetic of points on in projective coordinates are formulated, an algorithm for the computations is given illustrating the step-by-step process for computing point addition and doubling in and its efficiency over the arithmetic of points on in affine coordinates is described. This provides a systematic framework for performing elliptic curve arithmetic efficiently.