An Efficient ElGamal Encryption Scheme Based on Polynomial Modular Arithmetic in $\text{F}_{2}^{\text{n}}$

Abstract

The ElGamal cryptosystem was originally proposed by Taher ElGamal in 1985, in which its security level is based on the Discrete Logarithm Problem (DLP). ElGamal cryptosystem is relatively an expensive algorithm. For security guarantees, ElGamal cryptosystem requires modulo operation of large prime integer whose size range approximately from 1,024 to 4,096 bits. As a consequence of such requirement, the application of ElGamal cryptosystem is limited for securing only small messages such as secret keys. This paper aims to propose an efficient variant of ElGamal cryptosystem. The proposed scheme is designed based on quotient ring of polynomial, $Z_{2} [x]/{ < }f (x ) { > }$, where $f\left( x \right)$ is an irreducible polynomial. The decryption algorithm was further optimized with the use of the multiplicative inverse of the generator g(x), which only generated once during the key generation algorithm, thus leading to a simpler and faster decryption process. The proposed scheme is as secure as the original ElGamal scheme, since both schemes are based on the DLP. The preliminary result shows that the proposed scheme minimizes complex arithmetic operations and achieves very practical performance compared to the classic ElGamal algorithm and its variants. The proposed $F_{2}^{n}$ based ElGamal scheme outperforms the $F_{p}$ based scheme by significantly reducing 69.74% of the numbers of required logic gates in the case study of VLSI implementation.