Researcher(s)
- Zhixiang Chen, Mathematics, University of Delaware
Faculty Mentor(s)
- Shuxing Li, Mathematical Sciences, University of Delaware
Abstract
Vectorial Boolean functions, which map finite sets to themselves, are essential in cryptography and coding theory due to their ability to encode and secure information. These functions have been critical to cryptosystems like the Data Encryption Standard (DES), an important encryption scheme in the evolution of modern cryptography. However, ensuring the security of DES requires these functions to meet specific criteria. In 1990, researchers Biham and Shamir identified that the derivative of the vectorial Boolean functions in DES must be balanced to prevent vulnerabilities to differential cryptanalysis attacks.
Building on this foundation, researchers have investigated the planarity of these functions, focusing on achieving maximum balance in their derivatives. Recently, our faculty supervisor and a team of researchers proposed a combinatorial perspective on function planarity, introducing the concept of vanishing flats. These combinatorial configurations provide a new way to analyze and understand the planarity of vectorial Boolean functions.
After analyzing all these information, our research aims to explore this combinatorial viewpoint further by achieving two primary objectives. First, we will use computer algebra systems to calculate the vanishing flats for several well-known vectorial Boolean functions. Second, we will analyze the experimental data to identify patterns in vanishing flats that can lead to theoretical explanations of their numbers and explicit descriptions.
This research will not only enhance our understanding of the planarity of vectorial Boolean functions but also provide valuable insights into their application in cryptography. Additionally, the project offers an excellent opportunity for students to delve into advanced abstract algebra, including finite fields and polynomials over finite fields, and gain practical experience with computer programming involving these algebraic structures.
Through this project, we hope to contribute to the ongoing efforts in cryptographic research, ensuring the development of more secure cryptosystems by leveraging the intricate properties of vectorial Boolean functions and their combinatorial characteristics.