College
College of Arts & Sciences
Mentor Information
Lukas Koelsch
Description
Bent Boolean functions have been studied extensively due to their applications in the construction of cryptographic hash functions and block ciphers. A function is homogeneous when all terms have the same number of variables, known as the degree. We show that the bound on the degree of homogeneous bent Boolean functions can be strengthened; for such a function $f: \mathbb{F}_2^n \to \mathbb{F}_2$, the maximum degree is $d \leq \frac{n}{4} + \frac{1}{4}\log_2(n+3) + \frac{3}{2} + \frac{1}{4}\log_2\left(\frac{\pi}{2}\right)$, improving the prior bound from a $\sqrt{n}$ factor to $\log_2(n)$, and explicitly calculating the constants. Using advanced techniques from the field of integer programming and coding theory, we offer more succinct proofs of known bounds on the degree, while also providing an exact characterization of the weight distributions for specific $n \in \{6,8,10,12\}$ and $d \in \{3,4,5\}$.
Improved Bounds on the Degree of Homogeneous Bent Boolean Functions
Bent Boolean functions have been studied extensively due to their applications in the construction of cryptographic hash functions and block ciphers. A function is homogeneous when all terms have the same number of variables, known as the degree. We show that the bound on the degree of homogeneous bent Boolean functions can be strengthened; for such a function $f: \mathbb{F}_2^n \to \mathbb{F}_2$, the maximum degree is $d \leq \frac{n}{4} + \frac{1}{4}\log_2(n+3) + \frac{3}{2} + \frac{1}{4}\log_2\left(\frac{\pi}{2}\right)$, improving the prior bound from a $\sqrt{n}$ factor to $\log_2(n)$, and explicitly calculating the constants. Using advanced techniques from the field of integer programming and coding theory, we offer more succinct proofs of known bounds on the degree, while also providing an exact characterization of the weight distributions for specific $n \in \{6,8,10,12\}$ and $d \in \{3,4,5\}$.
