# Enhancing Time-Frequency Analysis with Zero-Mean Preprocessing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Correlation Power Analysis

#### 2.2. Masking Countermeasure

Algorithm 1: First-order masked AES-128 |

Input: Plaintext $PT\left[16\right]$, Master key $MK\left[16\right]$, AES S-box $S\left[256\right]$Output: Ciphertext $CT\left[16\right]$Initialization1: Choose masking values ${r}_{in}$, ${r}_{out}$, ${m}_{1}$, ${m}_{2}$, ${m}_{3}$, ${m}_{4}$ uniformly at random from ${\mathbb{z}}_{256}$ 2: Initialize $MS$ 3: for $i\in 0$ up to $255$ do4: $MS\left[i\oplus {r}_{in}\right]=S\left[i\right]\oplus {r}_{out}$ 5: end for6: ${m}_{1}^{\prime},{m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime}\leftarrow MixColumns\left({m}_{1},{m}_{2},{m}_{3},{m}_{4}\right)$ 7: $state\leftarrow Remask\left(PT,0,{m}_{1}^{\prime},{m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime}\right)$ 8: $rk\leftarrow KeySchedule\left(MK,{r}_{in},{r}_{out},{m}_{1}^{\prime},{m}_{2}^{\prime},{m}_{3}^{\prime},{m}_{4}^{\prime}\right)$ Encryption9: $state\leftarrow AddRoundKey\left(state,rk\right)$ 10: for $i\in 1$ up to 9 do11: $state\leftarrow SubBytes\left(state,MS\right)$ 12: $state\leftarrow ShiftRows\left(state\right)$ 13: $state\leftarrow Remask\left(state,{r}_{out},{m}_{1},{m}_{2},{m}_{3},{m}_{4}\right)$ 14: $state\leftarrow MixColumns\left(state\right)$ 15: $state\leftarrow AddRoundKey\left(state,rk\right)$ 16: end for17: $state\leftarrow SubBytes\left(state,MS\right)$ 18: $state\leftarrow ShiftRows\left(state\right)$ 19: $CT\leftarrow AddRoundKey\left(state,rk\right)$ 20: Return $CT$ |

#### 2.3. Hiding Countermeasure

#### 2.4. Second-Order Correlation Power Analysis

#### 2.5. Time-Frequency Analysis for Second-Order Side-Channel Analysis

**Definition**

**1.**

## 3. Mathematical Analysis on Time-Frequency Analysis with Zero-Mean Preprocessing

#### 3.1. Mathematical Analysis

#### 3.2. Zero-Mean Preprocessing

**Definition**

**2.**

**Definition**

**3.**

## 4. Experimental Results

**TFA**represents Belgarric et al.’s time-frequency analysis without any zero-mean preprocessing.**TFAwPS**represents a time-frequency analysis with standardization.**TFAwPN**represents a time-frequency analysis with min-max normalization.

#### 4.1. Introduction to the AES-M Datasets

#### 4.2. Experimental Results on AES-M-Sync

#### 4.3. Experimental Results on AES-M-Desync50

#### 4.4. Experimental Results on AES-M-Desync100

#### 4.5. Experimental Results on ASCAD Dataset

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The results of FO-CPA without preprocessing, TFA, and TFA with zero-mean preprocessing against AES-M and AES-RD. O/X represents either success or failure for key guesses.

AES-M-sync | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Byte | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

No preprocessing | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X |

TFA | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |

TFAwPS | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |

TFAwPN | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |

AES-M-desync30 | ||||||||||||||||

Byte | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

No preprocessing | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X |

TFA | O | O | O | O | O | O | O | O | X | O | O | O | O | O | O | O |

TFAwPS | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O | O |

TFAwPN | O | X | X | X | X | X | X | X | X | X | X | X | O | X | X | X |

AES-M-desync50 | ||||||||||||||||

Byte | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

No preprocessing | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X |

TFA | X | X | X | O | O | O | O | O | X | O | O | O | O | O | O | O |

TFAwPS | O | O | O | O | X | O | X | O | X | O | O | O | O | O | O | O |

TFAwPN | O | X | X | O | O | X | X | X | O | X | X | X | X | X | X | X |

AES-M-desync100 | ||||||||||||||||

Byte | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

No preprocessing | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X | X |

TFA | X | X | X | X | O | X | X | X | X | X | O | O | O | X | X | X |

TFAwPS | X | X | X | O | X | X | X | X | X | X | O | O | O | X | X | X |

TFAwPN | O | X | O | X | X | X | X | O | X | X | X | O | X | X | X | X |

AES-RD ^{1} | ||||||||||||||||

Byte | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

No preprocessing | X | X | X | X | X | O | X | X | X | X | X | X | X | X | X | X |

TFA | O | X | X | X | O | O | X | O | O | O | X | X | O | O | X | O |

TFAwPS | O | X | X | O | O | O | O | O | O | O | X | O | O | O | X | O |

TFAwPN | O | O | X | O | O | O | X | O | O | O | X | X | O | O | X | X |

^{1}AES-RD dataset https://github.com/ikizhvatov/randomdelays-traces (accessed on 8 November 2021).

**Table A2.**The results of FO-CPA without preprocessing, TFA, and TFA with zero-mean preprocessing against ASCAD. O/X represents either success or failure for key guesses.

ASCAD-sync | ASCAD-desync50 | ASCAD-desync100 | |
---|---|---|---|

No preprocessing | X | X | X |

TFA | X | X | O |

TFAwPS | O | O | O |

TFAwPN | O | X | X |

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**Figure 1.**The power consumption traces for the original AES-M-sync dataset in which all traces are synchronized. (

**a**) 50 random traces during AES encryptions; (

**b**) 50 random traces zoomed on the middle area. (

**c**) Single trace zoomed on the front portion, including precomputation, key schedule, and one round with two green boxes. The green boxes represent time intervals for target operations corresponding to lines 4 and 12, respectively, in Algorithm 1.

**Figure 3.**(

**a**) The best guessing entropy of TFA results at the 14th byte. (

**b**) The best guessing entropy of TFAwPS results at the 15th byte.

**Figure 4.**(

**a**) The worst guessing entropy of TFA results at 3rd byte. (

**b**) Worst guessing entropy of TFAwPS result at 10th byte.

**Figure 5.**(

**a**) The power consumption traces for AES-M-desync50. (

**b-up**) The transformed waveforms by TFA. (

**b-down**) The transformed waveforms by TFAwPS.

**Figure 6.**(

**a**) The best guessing entropy of TFA results at 12th byte. (

**b**) The best guessing entropy of TFAwPS results at 2nd byte.

**Figure 7.**(

**a**) the worst guessing entropy of TFA results at 6th byte. (

**b**) The worst guessing entropy of TFAwPS results at 9th byte.

**Figure 8.**(

**a**) The power consumption traces for AES-M-desync100. (

**b-up**) The transformed waveforms by TFA. (

**b-down**) The transformed waveforms by TFAwPS.

**Figure 10.**(

**a**) The worst guessing entropy of TFA results at 6th byte. (

**b**) The worst guessing entropy of TFAwPS result at the 11th byte.

**Figure 11.**The power consumption traces for (

**a**) ASCAD-sync, (

**b**) ASCAD-desync50, and (

**c**) ASCAD-desync100.

TFA | TFAwPS | |
---|---|---|

Average max peak of absolute correlation coefficients | 0.036785 | 0.042720 |

Average confidence (1st max peak/2nd max peak) | 1.591291 | 1.840527 |

The number of found key bytes | 16 | 16 |

TFA | TFAwPS | |
---|---|---|

Average max peak of absolute correlation coefficients | 0.028241 | 0.027701 |

Average confidence (1st max peak/2nd max peak) | 1.234980 | 1.200778 |

The number of found key bytes | 12 | 13 |

TFA | TFAwPS | |
---|---|---|

Average max peak of absolute correlation coefficients | 0.024567 | 0.025794 |

Average confidence (1st max peak/2nd max peak) | 1.083455 | 1.123166 |

The number of found key bytes | 4 | 4 |

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## Share and Cite

**MDPI and ACS Style**

Jin, S.; Johansson, P.; Kim, H.; Hong, S.
Enhancing Time-Frequency Analysis with Zero-Mean Preprocessing. *Sensors* **2022**, *22*, 2477.
https://doi.org/10.3390/s22072477

**AMA Style**

Jin S, Johansson P, Kim H, Hong S.
Enhancing Time-Frequency Analysis with Zero-Mean Preprocessing. *Sensors*. 2022; 22(7):2477.
https://doi.org/10.3390/s22072477

**Chicago/Turabian Style**

Jin, Sunghyun, Philip Johansson, HeeSeok Kim, and Seokhie Hong.
2022. "Enhancing Time-Frequency Analysis with Zero-Mean Preprocessing" *Sensors* 22, no. 7: 2477.
https://doi.org/10.3390/s22072477