Explicit Evaluation Of Exponential Sums Of Some Functions

Boolean functions have important applications in cryptographic systems. The constructionand analysis of these functions having optimal cryptographic properties are two key problems ofthe cryptology all the time. The research about the exponential sums of Boolean functions mayshed some light on the difficulties of evaluating the nonlinearity of Boolean functions, findingthe correlation distribution for many similar m-sequences and the Hamming weight of cycliccode. In this thesis, it is firstly presented that the summary of all important results about theexponential sums, then a unique method is given to evaluate the exponential sums ofMulti-Orbits rotation symmetric Boolean functions (RSBFs). Finally we propose a new methodto simplify the evaluation of exponential sums of Gold-like function in polynomial form. Themain results of this thesis are as follows:(1) We translate the evaluation of the exponential sums of Multi-Orbits RSBFs into thecalculation of the nullity of the Multi-Orbits RSBFs and the symbol of these functions by firstdifference method. We get the results about the relationship of the Multi-Orbits RSBFs weightbetween having n variables and n/p variables. A recursive method is effectively given out toevaluate the exponential sums of these RSBFs. When having odd variables, explicit evaluation of2-orbits RSBFs is as an example to put it into practice. This unique method is not identical to thetraditional to calculate the one orbit RSBF, and it can be used to the general which could not beevaluated now.(2) With the property of trace function, the knowledge of bilinear functions, and also thespecial resolution of the characteristic polynomial, a new method is provided to simplify theevaluation of the exponential sums of Gold-like function. Also we describe the criterion of thesespecial resolutions when the Variable number is the power of2, and a sufficient and necessarycriterion is provided to distinguish the special resolutions when the Variable number is odd. Inaddition, a new method is also provided to evaluate the exponential sums of these quadraticfunctions. Not like to the present results, using this new method can resolve a large and moregeneral question of the exponential sums of these quadratic functions on finite fields.