Propagation Characteristics of Boolean Functions

Abstract

The relation between the Walsh-Hadamard transform and the autocorrelation function of Boolean functions is used to study propagation characteristics of these functions. The Strict Avalanche Criterion and the Perfect Nonlinearity Criterion are generalized in a Propagation Criterion of degree k. New properties and constructions for Boolean bent functions are given and also the extension of the definition to odd values of n is discussed. New properties of functions satisfying higher order SAC are derived. Finally a general framework is established to classify functions according to their propagation characteristics if a number of bits is kept constant. 'NFWO aspirant navorser, sponsored by the National Fund for Scientific Research (Belgium).

Propagation Characteristics and Correlation-Immunity of Highly Nonlinear Boolean Functions

“…Some applications require that the output difference of a function be uniformly distributed for lowweight input differences. This property, referred as propagation criterion [22], is notably important when the function is used in a hash function or in a block cipher.…”

Cryptographic Criteria on Vector Boolean Functions

“…Let f ∈ F n , then f satisfies the propagation criterion of degree l, PC(l)(1 ≤ l ≤ n), if f (x) changes with a probability of 1/2 whenever i(1 ≤ i ≤ t) bits of x are complemented Preneel et al (2006). …”

Cryptographic properties of Boolean functions defining elementary cellular automata

“…As a consequence, the design of this cryptographic functions needs to consider various characteristics simultaneously. In this sense, the most important cryptographic criteria for Boolean functions are the following [13,16,19]: algebraic degree, the balancedness, the resiliency, the nonlinearity, the propagation criterion (PC) and the non-existence of non-zero linear structures. Obviously, all of these char acteristics cannot be optimum at the same time and consequently trade-offs must be taken into account.…”

“…This criterion was introduced by Preneel et al [16] and it is based on the properties of the derivatives of Boolean functions that gives the behaviour of such functions when some variables of the input are complemented. The n-variable Boolean function f satisfies the PC with respect to B ⊂ F2 if for every b ∈ B the derivative function, Dbf, is balanced.…”