Minimising the Cipher Length of Elliptic Curve Cryptography Basing on Chebyshev Polynomial

Introduction: Public Key Cryptography (PKC) is one of the key components of contemporary cryptographic systems, mainly employed for securing digital communications using protocols such as SSL and TLS. PKC relies on trapdoor functions, which are computationally easy in one direction but impossible to invert without access to a secret key. RSA and the Diffie-Hellman key exchange protocol are classical algorithms that have been extensively employed for secure data transfer. More recently, Elliptic Curve Cryptography (ECC) has been in the limelight because of its high security with compact key sizes and fast computation, based on the algebraic structure of elliptic curves. Objectives: This study will seek to alleviate one of the most significant weaknesses of ECC-based encryption—namely the expansion of ciphertext size in ElGamal encryption. The goal is to create a revised ECC-based encryption scheme that is highly secure while reducing ciphertext size. Furthermore, the proposed approach will seek to include digital signature functionality for improved data authentication and integrity. Methods: The new encryption scheme is based on the ElGamal encryption scheme but with an addition involving Chebyshev polynomials of the first kind as a random value generator. This inclusion enables secure randomization during encryption. The new scheme guarantees that the ciphertext size is equal to the plaintext size, essentially compressing the output without loss of security. The scheme also includes digital signature functionality to ensure authenticity and non-repudiation. Results: The revised ECC-based encryption method proves to be an enormous advancement compared to conventional ElGamal encryption in terms of ciphertext length. Without compromising the strong security attributes of ECC, the new method attains ciphertext compression, maintaining the encrypted message size equal to the plaintext size. Additionally, the scheme efficiently accommodates digital signature operations, providing improved message integrity and verification. Conclusions: The improved encryption scheme overcomes a significant obstacle to ECC-based cryptography by decreasing ciphertext length at no cost in security. Combining Chebyshev polynomials and digital signature capability is a step towards better secure and efficient cryptographic protocols. The scheme has special application value for any scenarios involving light encryption with tight security assurances, e.g., secure messaging, cloud data encryption, and IoT transmission.