Explicit Values of the Upper Boomerang Connectivity Table, Lower Boomerang Connectivity Table, and Double Boomerang Connectivity Table of the Inverse Function

Core Concepts

The paper provides the explicit values of the Upper Boomerang Connectivity Table (UBCT), Lower Boomerang Connectivity Table (LBCT), and Double Boomerang Connectivity Table (DBCT) of the inverse function F(x) = x^(2^n-2) over F_2^n for arbitrary n. It also determines the double boomerang uniformity of the inverse function.

Abstract

The paper investigates the properties of the inverse function F(x) = x^(2^n-2) over the finite field F_2^n, focusing on the explicit values of the UBCT, LBCT, and DBCT.

Key highlights:

1. The authors provide the explicit values of the UBCT and LBCT of the inverse function F(x) over F_2^n for arbitrary n.
2. They determine that the inverse function F(x) is "hard" when n is odd, meaning the set of nonzero entries in the DBCT always satisfies b = c.
3. The authors completely compute all entries of the DBCT of F(x) over F_2^n for arbitrary n, and provide the precise number of elements with a given entry by means of Kloosterman sums.
4. They determine the double boomerang uniformity of the inverse function F(x) over F_2^n for arbitrary n.
5. The in-depth analysis of the DBCT of the inverse function contributes to a better evaluation of the S-box's resistance against boomerang attacks.

Stats

The paper provides the following key figures and metrics:

Explicit values of the UBCT and LBCT of the inverse function F(x) = x^(2^n-2) over F_2^n for arbitrary n.
Conditions under which the inverse function F(x) is "hard" when n is odd.
Explicit values of the DBCT of the inverse function F(x) = x^(2^n-2) over F_2^n for arbitrary n.
Precise number of elements with a given entry in the DBCT, expressed in terms of Kloosterman sums.
Double boomerang uniformity of the inverse function F(x) = x^(2^n-2) over F_2^n for arbitrary n.

Quotes

"The explicit values of the entries of DBCT and their cardinalities are crucial tool to test the resistance of block ciphers based on variants of the function against cryptanalytics such as differential and boomerang attacks."
"Our in-depth analysis of the DBCT of the inverse function contributes to a better evaluation of the S-box's resistance against boomerang attacks."

Key Insights Distilled From

The Explicit values of the UBCT, the LBCT and the DBCT of the inverse function

by Yuying Man,N... at arxiv.org 04-19-2024

https://arxiv.org/pdf/2404.12208.pdf

Deeper Inquiries

What other cryptographic properties of the inverse function could be explored to further understand its suitability for use in block ciphers

One cryptographic property of the inverse function that could be explored further is its non-linearity. Non-linearity is a crucial property in block ciphers as it helps in achieving confusion and diffusion, which are essential for cryptographic security. Analyzing the non-linearity of the inverse function can provide insights into its resistance against linear cryptanalysis, differential cryptanalysis, and other attacks. By studying the non-linearity of the inverse function, researchers can assess its suitability for use in block ciphers and understand how it contributes to the overall security of the cryptographic system.

How could the techniques developed in this paper be applied to analyze the UBCT, LBCT, and DBCT of other important functions used in symmetric cryptography

The techniques developed in this paper for analyzing the UBCT, LBCT, and DBCT of the inverse function can be applied to analyze other important functions used in symmetric cryptography. By applying similar methodologies to different functions, researchers can evaluate their resistance against specific attacks, such as differential and boomerang attacks. This approach can help in identifying the cryptographic properties of various functions, understanding their vulnerabilities, and enhancing their security. By extending the analysis to other functions, researchers can gain a comprehensive understanding of their behavior and suitability for cryptographic applications.

What are the implications of the "hardness" property of the inverse function when n is odd, and how could this be leveraged in the design of secure block ciphers

The "hardness" property of the inverse function when n is odd implies that the function exhibits specific characteristics that make it resistant to certain types of attacks, particularly boomerang attacks. Leveraging this property in the design of secure block ciphers can enhance the overall security of the cryptographic system. By incorporating the inverse function with this hardness property, cipher designers can create robust encryption schemes that are less susceptible to boomerang attacks, thereby increasing the overall security of the block cipher. This property can be utilized to strengthen the cryptographic properties of the block cipher and improve its resistance against advanced cryptanalytic techniques.