[Paper Review] Further Results of the Cryptographic Properties on the Butterfly Structure.

Shihui Fu, Xiutao Feng|arXiv (Cornell University)|Jul 28, 2016

Coding theory and cryptography7 references3 citations

TL;DR

This paper proves the optimality of nonlinearity for butterfly structures with exponent $ e = 2^i + 1 $, resolving an open problem in cryptography. It further establishes that butterflies with trivial coefficients are bijective and achieve optimal nonlinearity, confirming their suitability as cryptographic primitives.

ABSTRACT

Recently, a new structure called butterfly introduced by Perrin et at. is attractive for that it has very good cryptographic properties: the differential uniformity is at most equal to 4 and algebraic degree is also very high when exponent $e=3$. It is conjecture that the nonlinearity is also optimal for every odd $k$, which was proposed as a open problem. In this paper, we further study the butterfly structures and show that these structure with exponent $e=2^i+1$ have also very good cryptographic properties. More importantly, we prove in theory the nonlinearity is optimal for every odd $k$, which completely solve the open problem. Finally, we study the butter structures with trivial coefficient and show these butterflies have also optimal nonlinearity. Furthermore, we show that the closed butterflies with trivial coefficient are bijective as well, which also can be used to serve as a cryptographic primitive.

Motivation & Objective

• To resolve the open problem regarding the optimality of nonlinearity in butterfly structures for odd $ k $.
• To investigate the cryptographic properties of butterfly structures with exponent $ e = 2^i + 1 $, particularly nonlinearity and differential uniformity.
• To analyze butterfly structures with trivial coefficients and determine their bijectivity and nonlinearity properties.
• To confirm the suitability of these structures as cryptographic primitives based on their theoretical properties.

Proposed method

• Theoretical analysis of the algebraic and differential properties of butterfly structures with exponent $ e = 2^i + 1 $, focusing on nonlinearity and differential uniformity.
• Proof of nonlinearity optimality for all odd $ k $, using algebraic and combinatorial techniques to analyze the Walsh spectrum.
• Study of butterfly structures with trivial coefficients, proving they are bijective via structural and algebraic analysis.
• Derivation of conditions under which the closed-form butterfly structures with trivial coefficients achieve optimal nonlinearity.
• Use of finite field properties and symmetric structure analysis to establish cryptographic strength.
• Application of known results on bent functions and vectorial Boolean functions to validate nonlinearity bounds.

Experimental results

Research questions

• RQ1 Is the nonlinearity of butterfly structures with exponent $ e = 2^i + 1 $ optimal for all odd $ k $?
• RQ2 Do butterfly structures with trivial coefficients maintain optimal nonlinearity and bijectivity?
• RQ3 Can the nonlinearity of butterfly structures be proven optimal in theory, resolving the open problem?
• RQ4 What are the cryptographic implications of using butterfly structures with trivial coefficients as primitives?

Key findings

• The nonlinearity of butterfly structures with exponent $ e = 2^i + 1 $ is proven to be optimal for all odd $ k $, fully resolving the open problem.
• Butterfly structures with trivial coefficients are shown to be bijective, confirming their structural suitability for cryptographic use.
• These structures achieve optimal nonlinearity when coefficients are trivial, enhancing their resistance to linear cryptanalysis.
• The differential uniformity of butterfly structures with $ e = 2^i + 1 $ remains at most 4, preserving strong resistance to differential attacks.
• The combination of optimal nonlinearity, low differential uniformity, and bijectivity confirms the cryptographic robustness of these structures.
• The theoretical results validate that butterfly structures with trivial coefficients can serve as secure cryptographic primitives.