与Walsh-Hadamard变换相关的新变换：理论、方法及密码学应用

Walsh-Hadamard变换是研究密码函数一个关键的理论工具，但它仅能考量一个函数与线性函数的相关性，导致这类函数在应用中可能存在潜在的攻击。本项目拟研究一种与Walsh-Hadamard变换相关的新变换，即q-变换，它考量的是一个函数与另一函数（包括线性与非线性）的相关性。项目主要研究：一是完善q-变换的数学理论。涉及q-变换的计算问题、q-变换的数学期望及其它的数学问题，为其应用奠定理论基础。二是系统探索q-变换的密码学应用。讨论基于q-变换所定义的密码学新指标，如q-bent函数、q-nearly bent函数、q-相关免疫函数、 q-弹性函数等新概念。在相关安全密码算法的设计中，q-变换的应用将有助于选择具有更高密码强度的密码函数，以更有效地抵御密码攻击，因此在理论研究和实际应用上都具有积极的意义。希望通过项目的研究，初步建立关于q-变换理论及其密码学应用的新体系。

Walsh-Hadamard transform is a key tool for studying cryptographic functions. However, it only measures the correlation between a function and a linear function, which leads to a possible attack on such functions and hence have certain restrictions in applications. The project will study a new form of Walsh-Hadamard transform, namely q-transform. The q-transform measures the correlation between a function and another (linear or nonlinear) function. So it generalizes the Walsh-Hadamard transform that only measures the correlation between a function and linear functions. There are two main research topics in the project. One topic is to improve the mathematical theory of q-transform, which includes the computation, the mathematical expectation of q-transform and other mathematical problems. The other is to investigate cryptographic applications of q-transform systematically. We will discuss new cryptographic measures based on q-transform, such as q-bent functions, q-nearly bent functions, q-correlation immune functions and q-resilient functions. The applications of q-transform is very useful for designing secure cryptographic algorithms and resisting attacks. It has positive significance to enhance cryptographic primitives. We hope that the research can build a new system of the theory of q-transform and its Cryptographic Applications.