  This paper provides 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" $L^{2}$-signal to a ``bad"
$L^{1}$-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. Finally, the feasibility of the new strategy
is verified by simulation and audio examples.
