Optimizing Scalar Selection in Elliptic Curve Cryptography Using Differential Evolution for Enhanced Security

Abstract: Elliptic Curve Cryptography (ECC) is a fundamental component of modern public-key cryptosystems that enable efficient and secure digital signatures, key exchanges, and encryption. Its core operation, scalar multiplication, denoted as $k \cdot P$, where $P$ is a base point and $k$ is a private scalar, relies heavily on the secrecy and unpredictability of $k$. Conventionally, $k$ is selected using user input or pseudorandom number generators. However, in resource-constrained environments with weak entropy sources, these approaches may yield low-entropy or biased scalars, increasing susceptibility to side-channel and key recovery attacks. To mitigate these vulnerabilities, we introduce an optimization-driven scalar generation method that explicitly maximizes bit-level entropy. Our approach uses differential evolution (DE), a population-based metaheuristic algorithm, to search for scalars whose binary representations exhibit maximal entropy, defined by an even and statistically uniform distribution of ones and zeros. This reformulation of scalar selection as an entropy-optimization problem enhances resistance to entropy-based cryptanalytic techniques and improves overall unpredictability. Experimental results demonstrate that DE-optimized scalars achieve entropy significantly higher than conventionally generated scalars. The proposed method can be integrated into existing ECC-based protocols, offering a deterministic, tunable alternative to traditional randomness, ideal for applications in blockchain, secure messaging, IoT, and other resource-constrained environments.