Elementary Remarks on Some Quadratic Based Identity Based Encryption Schemes

In the design of an identity-based encryption (IBE) scheme, the primary security assumptions center around quadratic residues, bilinear mappings, and lattices. Among these approaches, one of the most intriguing is introduced by Clifford Cocks and is based on quadratic residues. However, this scheme has a significant drawback: a large ciphertext to plaintext ratio. A different approach is taken by Zhao et al., who design an IBE still based on quadratic residues, but with an encryption process reminiscent of the Goldwasser-Micali cryptosystem. In the following pages, we will introduce an elementary method to accelerate Cocks\u27 encryption process and adapt a space-efficient encryption technique for both Cocks\u27 and Zhao et al.\u27s cryptosystems

Continued Fractions Applied to a Family of RSA-like Cryptosystems

Let $N=pq$ be the product of two balanced prime numbers $p$ and $q$. Murru and Saettone presented in 2017 an interesting RSA-like cryptosystem that uses the key equation $ed - k (p^2+p+1)(q^2+q+1) = 1$, instead of the classical RSA key equation $ed - k (p-1)(q-1) = 1$. The authors claimed that their scheme is immune to Wiener\u27s continued fraction attack. Unfortunately, Nitaj \emph{et. al.} developed exactly such an attack. In this paper, we introduce a family of RSA-like encryption schemes that uses the key equation $ed - k [(p^n-1)(q^n-1)]/[(p-1)(q-1)] = 1$, where $n>1$ is an integer. Then, we show that regardless of the choice of $n$, there exists an attack based on continued fractions that recovers the secret exponent