Efficient Implementation of Scalar Multiplication for Elliptic Curve Cryptography using Ancient Indian Vedic Mathematics over GF(p)

Abstract Modular multiplication is one of the most time-consuming operations that account for almost 80% of computational overhead in a scalar multiplication in elliptic curve cryptography. In this paper, we present a new speed record for modular multiplication over 192-bit NIST prime P-192 on 8-bit AVR ATmega microcontrollers. We propose a new integer representation named Range Shifted Representation (RSR) which enables an efficient merging of the reduction operation into the subtractive Karatsuba multiplication. This merging results in a dramatic optimization in the intermediate accumulation of modular multiplication by reducing a significant amount of unnecessary memory access as well as the number of addition operations. Our merged modular multiplication on RSR is designed to have two duplicated groups of 96-bit intermediate values during accumulation. Hence, only one accumulation of the group is required and the result can be used twice. Consequently, we significantly reduce the number of load/store instructions which are known to be one of the most time-consuming operations for modular multiplication on constrained devices. Our implementation requires only 2888 cycles for the modular multiplication of 192-bit integers and outperforms the previous best result for modular multiplication over P-192 by a factor of 17%. In addition, our modular multiplication is even faster than the Karatsuba multiplication (without reduction) which achieved a speed record for multiplication on AVR processor.

Download Full-text

Some Techniques of Ancient Indian Vedic Mathematics for Elliptic Curve Cryptography over the Ring A4

International Journal of Computer Sciences and Engineering ◽

10.26438/ijcse/v7i5.13301337 ◽

2019 ◽

Vol 7 (5) ◽

pp. 1330-1337

Author(s):

Manoj Kumar ◽

Ankur Kumar

Keyword(s):

Elliptic Curve ◽

Elliptic Curve Cryptography ◽

Vedic Mathematics

Download Full-text

Secure and Robust Telemedicine using ECC on Radix-8 with Formal Verification

Research Anthology on Telemedicine Efficacy, Adoption, and Impact on Healthcare Delivery ◽

10.4018/978-1-7998-8052-3.ch030 ◽

2021 ◽

pp. 573-590

Author(s):

Gautam Kumar ◽

Hemraj Saini

Keyword(s):

Computational Complexity ◽

Elliptic Curve ◽

Formal Verification ◽

Elliptic Curve Cryptography ◽

Clock Cycle ◽

Scalar Multiplication ◽

Efficient Technique ◽

Computation Efficiency ◽

Research Gap ◽

Hardware Performance

The scalar multiplication techniques used in Elliptic curve cryptography (ECC) are having the scope for gaining the computation efficiency. This is possible through the reduction of precomputed operations. Finding the more efficient technique compares to the most recent or efficient one is a research gap for all schemes. The manuscript presents an application oriented work for Telemedicine using ECC. It is based on robust application on reduced computational complexity. The methodology we apply for the same is Scalar Multiplication without precomputation on Radix-8. Introduced software and the hardware performance are reporting a big advantage over all the related proposed techniques. The reason to cover this problem is to provide a path on a fascinating area of ECC on a smaller key size be applicable for all applications on a same level of security strengths. The smaller length key gives the higher speed and shorter clock cycle to initiate the operation.

Download Full-text