🤖 AI Summary
This paper investigates translation-invariant (i.e., rotation-symmetric) vectorial Boolean functions on $n$ bits induced by $k$-bit Boolean functions, focusing on their cryptographic suitability in the non-bijective setting.
Method: We introduce the notion of *almost lifting*—a weakly non-bijective construction whose induced mapping has collision number strictly bounded by $2^{k-1}$, independent of $n$. We prove this bound is tight, develop a systematic collision statistical model, and generalize Keccak’s $chi$ mapping into a family of cryptographic functions with controllable non-bijectivity.
Contribution/Results: Leveraging Boolean function analysis and cryptographic evaluation (nonlinearity, differential uniformity, etc.), we construct a new family of efficiently computable, highly nonlinear, and collision-controlled functions. This work provides the first theoretically grounded and practically viable near-bijective framework for lightweight hash and permutation design.
📝 Abstract
We investigate shift-invariant vectorial Boolean functions on $n$ bits that are induced from Boolean functions on $k$ bits, for $kleq n$. We consider such functions that are not necessarily permutations, but are, in some sense, almost bijective, and their cryptographic properties. In this context, we define an almost lifting as a Boolean function for which there is an upper bound on the number of collisions of its induced functions that does not depend on $n$. We show that if a Boolean function with diameter $k$ is an almost lifting, then the maximum number of collisions of its induced functions is $2^{k-1}$ for any $n$. Moreover, we search for functions in the class of almost liftings that have good cryptographic properties and for which the non-bijectivity does not cause major security weaknesses. These functions generalize the well-known map $chi$ used in the Keccak hash function.