Construction and equivalence for generalized boolean functions.

Area of Science:

• Number Theory
• Abstract Algebra
• Cryptography

Background:

• The study of function equivalences, specifically EA-equivalence and CCZ-equivalence, is crucial in understanding the properties of Boolean functions and their applications in cryptography.
• Previous work initiated the study of these equivalences for functions mapping vector spaces over finite fields F_p to cyclic groups Z_{p^k}, with specific findings for p=2 and k=2.

Purpose of the Study:

• To extend the results on EA-equivalence and CCZ-equivalence to broader classes of functions from V_n^{(p)} to Z_{p^k}.
• To develop constructions for generalized bent functions, particularly for odd p and for p=2 with even n.
• To present a method for constructing generalized bent functions for p=2 and odd n from functions with a smaller domain.

Main Methods:

• Extension of existing equivalence studies to larger classes of functions.
• Application of direct sum and semi-direct sum constructions.
• Utilization of a secondary bent function construction method.
• Development of a novel construction for generalized bent functions based on functions with reduced dimensions.

Main Results:

• The study successfully extends the analysis of EA-equivalence and CCZ-equivalence to more general function classes.
• New methods for constructing generalized bent functions are presented, creating large affine spaces of such functions.
• A specific construction is provided for generalized bent functions from V_n^{(2)} to Z_{2^k} (n odd) using generalized bent functions from V_{n-1}^{(2)} to Z_{2^{k-1}}.

Conclusions:

• The research deepens the understanding of function equivalences in finite fields and their relation to bent functions.
• The presented constructions offer valuable tools for generating families of generalized bent functions, important for cryptographic design.
• The findings contribute to the theory of bent functions and their practical construction in coding theory and cryptography.