GAC Properties Of A Class Of Cryptographic Functions

Boolean functions play an important role in the design of stream ciphers, block ciphers and Hash functions. Constructing a Boolean function with good properties is an important subject in cryptography. To use Boolean functions in cryptography, many cryptographic criteria, such as balancedness, high nonlinearity, correlation immunity, resiliency, algebraic immunity, algebraic degree, strict avalanche criterion (SAC), propagation criterion (PC), and global avalanche criterion (GAC), have been introduced. To construct cryptographic Boolean functions, it is necessary to satisfy a variety of cryptographic properties. However, these properties cannot be optimum simultaneously. And the trade-offs among them must be considered.In this thesis, the GAC properties of the balanced Boolean functions are studied. The GAC properties could overcome the shortcomings of the SAC or PC characteristic, and be able to measure the overall avalanche characteristic of a Boolean function. GAC properties include two indicators:the sum-of-squares indicator and the absolute indicator. The smaller the sum-of-squares indicator and the absolute indicator, the better GAC properties.In this thesis, we analyze the GAC properties of a class of Boolean functions. This kind of Boolean functions satisfy high nonlinearity, strict avalanche criterion (SAC) and m-resilient. We give the distribution of Walsh spectrum of the functions, and calculate the sum-of-squares indicator of the Boolean functions by using the relationship between the sum-of-squares indicator and the Walsh spectra. Based on the definition of the autocorrelation functions, a detailed discussion is presented on different cases, and an upper bound of the absolute indicator of the functions is obtained. Finally, when the object functions are balanced, their sum-of-squares indicator is σf= 22n+5.23n/2+1+2n/2+2k+3, and absolute indicator satisfies △f≤22k+2n/2, where，and n≥10 is even. Our results show that the GAC properties of this class of Boolean functions are good.