We provide a double encryption algorithm that uses the lack of invertibility of the fractional Fourier transform (FRFT) on $L^{1}$. One encryption key is a function, which maps a "good" $L^{2}(\mathbb{R})$ signal to a "bad" $L^{1}(\mathbb{R})$ signal. The FRFT parameter which describes the rotation associated with this operator on the time-frequency plane provides the other encryption key. With the help of approximate identities, such as of the Abel and Gauss means of the FRFT established in \cite{CFGW}, we recover the encrypted signal on the FRFT domain. This design of an encryption algorithm seems new even when using the classical Fourier transform.

中文翻译：

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基于分数阶傅里叶变换域变化的信号加密策略

我们提供了一种双重加密算法，该算法在 $L^{1}$ 上使用分数傅立叶变换 (FRFT) 的不可逆性。一个加密密钥是一个函数，它将“好”$L^{2}(\mathbb{R})$ 信号映射到“坏”$L^{1}(\mathbb{R})$ 信号。FRFT 参数描述了在时频平面上与该算子相关联的旋转，提供了另一个加密密钥。在近似恒等式的帮助下，例如在\cite{CFGW}中建立的FRFT的Abel和Gauss均值，我们在FRFT域上恢复了加密信号。即使在使用经典傅立叶变换时，这种加密算法的设计似乎也是新的。