A New Septupling Point 7P Arithmetic Formula for LD Coordinate and Affine Over Binary Elliptic Curve Cryptosystem

Abstract

The Elliptic Curve Cryptography (ECC) is one of the most prominent Asymmetric-based cryptosystems as it affords a higher level of security with small keys. According to National Institute of Standards and Technology (NIST), ECC gains the smallest secure key over the binary curve. In literature, the best field over binary curves is Lopez-Dahab (LD) and Affine coordinates, and it considered fit for lightweight cryptography in resource-constrained devices such as Internet of Things (IoTs). ECC consists of three operational levels; scalar multiplication, point arithmetic and field arithmetic. This research focuses on point arithmetic precomputation useful in scalar multiplication and then for field arithmetic. There is no existing formula for Septupling point  over binary curves in LD and Affine coordinates. A new precomputed Septupling point  is introduced in this paper using LD and non-supersingular affine coordinate over the binary field . This paper uses the form , consisting of Doubling point, Tripling point and the point addition. Also, the form  means the Sixtupling point is also proposed. Results show that the  is characterized by a cost of , while the cost of Sixtupling point  is . The point is mathematically proved as valid. The proposed point can be implemented for different scalar multiplication such as  for , and multi-based scalar such as {2, 3, 7}-based scalar multiplication method.