# Tight and Scalable Side-Channel Attack Evaluations through Asymptotically Optimal Massey-like Inequalities on Guessing Entropy

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## Abstract

**:**

## 1. Introduction

- We demonstrate that a recent improvement on Massey’s inequality between Massey’ Guessing entropy and Shannon’s entropy (Rioul’s improved inequality) is asymptotically optimal (which is highly relevant to scalability).
- We provide a new improvement on Massey’s inequality that is even tighter than the above for all finite-size data distributions.
- We extend and prove the above results when dealing with multiple lists of probabilities (distributions), as is the case when dealing with the results of side-channel attacks on multiple key bytes (proving scalability).
- We apply our results on concrete side-channel attack datasets to demonstrate the improvements of the methods from this paper over the state of the art.

## 2. Preliminaries

## 3. The Asymptotically Optimal Massey-like Inequality

**Proof.**

## 4. Refinement for Finite Support Distributions

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

^{th}component of the sequence ${\mathbf{Q}}_{k}$, we define the terms of the list $\left\{{\mathbf{Q}}_{k}\right\}$ as follows. We let the support of the first term coincide with $\mathbf{p}$, i.e., ${\mathbf{Q}}_{0}=\left({p}_{0},\phantom{\rule{0.166667em}{0ex}}{p}_{1},\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}{p}_{n},\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}\dots \right)$, and we define the other terms by recurrence:

## 5. Scalability of Bounds

**Theorem**

**3.**

**Theorem**

**4.**

**Proofs.**

## 6. Evaluation on Side-Channel Attack Data

#### 6.1. Evaluation Data

- For each dataset (power traces), we run a Template Attack [23] using the set of power traces to determine the most likely value of each of the 16 bytes of the AES key. The result of this attack is a list of probabilities ${\mathbf{p}}^{k}=\{{p}_{1},{p}_{2},\dots ,{p}_{256}\}$ for each of the 16 bytes of the AES key ($K=[{k}_{1}{k}_{2}\dots {k}_{16}]$).
- Using the lists of probabilities ${\mathbf{p}}^{1},{\mathbf{p}}^{2},\dots ,{\mathbf{p}}^{16}$, we compute the bounds (those from this paper as well as those from CHES 2017) first for each byte individually and then for attacks on two or more key bytes. Please note that a direct computation of the guessing entropy through the computation of the cross-product of several lists of probabilities (e.g., for more than 8 key bytes) is not feasible as we would have to process lists of more than ${2}^{64}$ elements. Instead, the bounds from this paper (as well as those from CHES 2017) use directly and very efficiently the lists of probabilities for each key byte, without performing the cross-product, to derive security metrics for attacks on many target bytes.

#### 6.2. Evaluation on a Single Byte

#### 6.3. Evaluation on Two Bytes

#### 6.4. Evaluation on All 16 Bytes

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Bounds for the simulated (

**left**) and real (

**right**) datasets, when targeting a single subkey byte. These are averaged results over 100 experiments.

**Figure 2.**Bounds for the simulated (

**left**) and real (

**right**) datasets, when targeting two subkey bytes. These are averaged results over 100 experiments.

**Figure 3.**Bounds for the simulated (

**left**) and real (

**right**) datasets, when targeting all the 16 AES key bytes. These are averaged results over 100 experiments.

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**MDPI and ACS Style**

Tănăsescu, A.; Choudary, M.O.; Rioul, O.; Popescu, P.G. Tight and Scalable Side-Channel Attack Evaluations through Asymptotically Optimal Massey-like Inequalities on Guessing Entropy. *Entropy* **2021**, *23*, 1538.
https://doi.org/10.3390/e23111538

**AMA Style**

Tănăsescu A, Choudary MO, Rioul O, Popescu PG. Tight and Scalable Side-Channel Attack Evaluations through Asymptotically Optimal Massey-like Inequalities on Guessing Entropy. *Entropy*. 2021; 23(11):1538.
https://doi.org/10.3390/e23111538

**Chicago/Turabian Style**

Tănăsescu, Andrei, Marios O. Choudary, Olivier Rioul, and Pantelimon George Popescu. 2021. "Tight and Scalable Side-Channel Attack Evaluations through Asymptotically Optimal Massey-like Inequalities on Guessing Entropy" *Entropy* 23, no. 11: 1538.
https://doi.org/10.3390/e23111538