Further Results of the Cryptographic Properties on the Butterfly Structures
This work addresses a specific open problem in cryptography, providing theoretical proofs for the optimality of butterfly structures, which is incremental but important for cryptographic primitive design.
The paper tackles the problem of proving optimal nonlinearity for butterfly structures in cryptography, showing that structures with exponent e=2^i+1 have good cryptographic properties and proving optimal nonlinearity for every odd k, solving an open problem.
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.