Abstract
Public-key cryptosystems are broadly employed to provide security for digital information. Improving the efficiency of public-key cryptosystem through speeding up calculation and using fewer resources are among the main goals of cryptography research. In this paper, we introduce new symbols extracted from binary representation of integers called Big-ones. We present a modified version of the classical multiplication and squaring algorithms based on the Big-ones to improve the efficiency of big integer multiplication and squaring in number theory based cryptosystems. Compared to the adopted classical and Karatsuba multiplication algorithms for squaring, the proposed squaring algorithm is 2 to 3.7 and 7.9 to 2.5 times faster for squaring 32-bit and 8-Kbit numbers, respectively. The proposed multiplication algorithm is also 2.3 to 3.9 and 7 to 2.4 times faster for multiplying 32-bit and 8-Kbit numbers, respectively. The number theory based cryptosystems, which are operating in the range of 1-Kbit to 4-Kbit integers, are directly benefited from the proposed method since multiplication and squaring are the main operations in most of these systems.
Highlights
The growth of digital technologies has an exponential trend and as a consequence the need of information security increases even more than before [1, 2]
To optimize the calculations based on CBONS, we can limit the length of maximum Big-ones to “w” (O(n,P)such that n ≤ w) which we identified in this paper as the maximum length of Big-ones [30,31,32]
To compute the Big-ones Hamming weight, 10,000 random numbers [33] were generated with different maximum lengths, w = 2, . . . , 10, and different number lengths ranging from 32 bits to 8 Kbits
Summary
The growth of digital technologies has an exponential trend and as a consequence the need of information security increases even more than before [1, 2]. Most of the public-key cryptosystems [3] use modular exponentiation in their calculation. Few years later in 1978, one of the most used public-key cryptosystems, RSA [3], is based on the modular exponentiation. This paper focuses on the second approach, by proposing a new number representation, which will improve the squaring and multiplication operations, two of the three main operations in calculating modular exponentiation [14]. The binary method requires only (k−1) squarings and k/2 multiplications (on average), where k = log2b (see Algorithm 1). Improving the multiplication and squaring operations (as found in algorithm such as the right-to-left algorithm and its variants [6,7,8]) will inherently improve the efficiency of the exponentiation calculation [7]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
