Design of two cryptographic functions from bose-type Sidon sets with differential 2-uniformity and near-optimal linearity

Sidon sets, defined by the property that every sum of any two elements is distinct, play a significant role in areas such as coding theory and cryptography. In this paper, we study two cryptographic functions, fq and gq , constructed from Bose-type Sidon sets. These functions exhibit three key cryptographic properties: low differential uniformity, uniform output distribution, and low linearity. The first ensures consistent sensitivity to input changes, offering resistance against differential cryptanalysis, as both functions achieve differential 2 -uniformity, the lowest non-trivial value possible, thus guaranteeing maximum theoretical resistance. The second maximizes entropy by ensuring a uniform output distribution, thereby avoiding statistical biases that attackers could exploit. The third property exhibits different behavior between the two functions: while fₚ demonstrates linear growth L(fₚ) ≈ 0.637p , the function gₚ achieves remarkably low linearity, empirically characterized by L(gₚ) ≈ 1.89√p , which is close to the optimal bound √p , providing superior resistance to linear cryptanalysis. This linearity result is supported by extensive computational evidence and is presented as a well-founded conjecture, whose analytical proof remains an open problem. We provide theoretical characterizations of these properties and validate them through comprehensive statistical testing, including lattice tests, Q-Q plots and chi-square tests, demonstrating their suitability for symmetric-key cryptography applications.