新分数阶算子及其在密码学上的应用

New Fractional Operator and Its Applications in Cryptography

利用周期区间上的Hilbert变换推导出一维情形下Laplace算子的积分形式,推导过程的难点在于圆上的Hilbert变换本身是一个震荡型的奇异积分,只有在取Cauchy主值意义下才有意义.利用这个积分形式进一步定义了一个在周期区间上的新分数阶算子.这个算子是定义在一个有限区间里,与分数阶Laplace算子相比,更容易进行数值实现,并证明新分数阶算子抛物型方程的解的适定性及反问题的不适定性.由反问题的不适定性构造出一个单向函数和一个数字签名方案.

An integral form of Laplace operator in one dimension is derived by using the Hilbert transform in the period interval.The difficulty of the derivation is that the Hilbert transformation in the period interval is an oscillatory singular integral,and it can make sense only when Cauchy's principal value is adopted.Using this integral form of Laplace operator,a new fractional operator on the periodic interval is further defined.Compared with the fractional Laplace operator,the new fractional operator is defined in a finite interval and much easier to realize numerical implementation.The well-posed of the new fractional operator parabolic equations and the ill-posed of the inverse problem are proven.At last,a one-way function based on ill-posed of the inverse problem is constructed and a signature scheme is constructed.