International Association
for Cryptologic Research

An RSA generalization using complex integers was introduced by Elkamchouchi, Elshenawy, and Shaban in 2002. This scheme was further extended by Cotan and Teșeleanu to Galois fields of order $n \geq 1$. In this generalized framework, the key equation is $ed - k (p^n-1)(q^n-1) = 1$, where $p$ and $q$ are prime numbers. Note that, the classical RSA, and the Elkamchouchi \emph{et al.} key equations are special cases, namely $n=1$ and $n=2$. In addition to introducing this generic family, Cotan and Teșeleanu describe a continued fractions attack capable of recovering the secret key $d$ if $d < N^{0.25n}$. This bound was later improved by Teșeleanu using a lattice based method. In this paper, we explore other lattice attacks that could lead to factoring the modulus $N = pq$. Namely, we propose a series of partial exposure attacks that can aid an adversary in breaking this family of cryptosystems if certain conditions hold.