Single-Trace Side-Channel Attacks on the Toom-Cook: The Case Study of Saber

Abstract

The Toom-Cook method is a well-known strategy for building algorithms to multiply polynomials efficiently. Along with NTT-based polynomial multiplication, Toom-Cook-based or Karatsuba-based polynomial multiplication algorithms still have regained attention since the start of the NIST’s post-quantum standardization procedure. Compared to the comprehensive analysis done for NTT, the leakage characteristics of Toom-Cook have not been discussed. We analyze the vulnerabilities of Toom-Cook in the reference implementation of Saber, a third round finalist of NIST’s post-quantum standardization process. In this work, we present the first single-trace attack based on the soft-analytical side-channel attack (SASCA) targeting the Toom-Cook. The deep learning-based power analysis is combined with SASCA to decrease the number of templates since there are a large number of similar operations in the Toom-Cook. Moreover, we describe the optimized factor graph and improved belief propagation to make the attack more practical. The feasibility of the attack is verified by evaluation experiments. We also discuss the possible countermeasures to prevent the attack.

High- and Half-Degree Quantum Multiplication for Post-Quantum Security Evaluation

“…Later, in Marco Bodrato's advanced research in [6], several strategies TABLE 1: Toom based multiplication related research. We extracted information from [4]- [8], [12], [13] and [11] to elucidate on their particular Toom-Cook-based-related task in a classical-quantum environment. and techniques for achieving a high degree of precision in the implementation of Toom-Cook algorithms, specifically for both balanced and unbalanced scenarios, are clarified in classical hardware implementation.…”

“…There has been a significant surge in the adoption of Toom-Cook and number theoretic transform (NTT)-based polynomial multiplication as a result of their incorporation as essential elements in the post-quantum standardization initiative [3], [4]. In addition to classical and quantum multiplication designs, i.e., [5], [6], [7], [8], several studies [3], [4], [9] have put forth an interesting investigation concerning Toom-Cook multiplication. The current investigation, which reveals the vulnerabilities of the Toom-Cook multiplication algorithm, was carried out by [4], [10].…”

“…In addition to classical and quantum multiplication designs, i.e., [5], [6], [7], [8], several studies [3], [4], [9] have put forth an interesting investigation concerning Toom-Cook multiplication. The current investigation, which reveals the vulnerabilities of the Toom-Cook multiplication algorithm, was carried out by [4], [10]. Hence, the research described in [11] gives an insight into increasing the number of iterations to mitigate the vulnerability to peak analysis attacks in the correlation power analysis (CPA) technique to lower degree Toom-Cook, as noted by [4].…”

“…The current investigation, which reveals the vulnerabilities of the Toom-Cook multiplication algorithm, was carried out by [4], [10]. Hence, the research described in [11] gives an insight into increasing the number of iterations to mitigate the vulnerability to peak analysis attacks in the correlation power analysis (CPA) technique to lower degree Toom-Cook, as noted by [4].…”

“…This work presents a novel design that significantly improves the efficiency of the multiplication algorithm based on the substantial strategy for Toom-Cook high-degree algorithm from several prior works (e.g., [4]- [8], [12], [13] and [11]). To the best of our understanding, this is the first work to integrate high-degree and half-degree multiplication in classical and quantum design while maintaining an architecture degree of up to 20.5-way for Toom-Cook-based multiplication.…”