New Single-Trace Side-Channel Attacks on a Specific Class of Elgamal Cryptosystem

Abstract

In 2005, Yen et al. proposed the first $N-1$ attack on the modular exponentiation algorithms such as BRIP and square-and-multiply-always methods. This attack makes use of the ciphertext $N-1$ as a distinguisher of low order to obtain a strong relation between side-channel leakages and secret exponent. The so-called $N-1$ attack is one of the most important order-2 element attacks, as it requires a non-adaptive chosen ciphertext which is considered as a more realistic attack model compared to adaptive chosen ciphertext scenario. To protect the implementation against $N-1$ attack, several literatures propose the simplest solution, i.e. \textquotedblleft block the special message $N-1$ . In this paper, we conduct an in-depth research on the $N-1$ attack based on the square-and-multiply-always (SMA) and Montgomery Ladder (ML) algorithms. We show that despite the unaccepted ciphertext $N-1$ countermeasure, other types of $N-1$ attacks is applicable to specific classes of Elgamal cryptosystems. We propose new chosen-message power-analysis attacks with order-4 elements which utilize a chosen ciphertext $c$ such that $c^2= -1 \bmod p$ where $p$ is the prime number used as a modulus in Elgamal. Such a ciphertext can be found simply when $p\equiv 1\mod 4$. We demonstrate that ML and SMA algorithms are subjected to our new $N-1$-type attack by utilizing a different ciphertext. We implement the proposed attacks on the TARGET Board of the ChipWhisperer CW1173 and our experiments validate the feasibility and effectiveness of the attacks by using only a single power trace