## Abstract

We have shown that the application of double random phase encoding (DRPE) to biometrics enables the use of biometrics as cipher keys for binary data encryption. However, DRPE is reported to be vulnerable to known-plaintext attacks (KPAs) using a phase recovery algorithm. In this study, we investigated the vulnerability of DRPE using fingerprints as cipher keys to the KPAs. By means of computational experiments, we estimated the encryption key and restored the fingerprint image using the estimated key. Further, we propose a method for avoiding the KPA on the DRPE that employs the phase retrieval algorithm. The proposed method makes the amplitude component of the encrypted image constant in order to prevent the amplitude component of the encrypted image from being used as a clue for phase retrieval. Computational experiments showed that the proposed method not only avoids revealing the cipher key and the fingerprint but also serves as a sufficiently accurate verification system.

© 2010 OSA

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### Equations (10)

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(1)
$${f}_{m}\left(x,y\right)=f\left(x,y\right)\mathrm{exp}\left\{jr\left(x,y\right)\right\}.$$
(2)
$$\mathrm{\Phi}(u,v)={F}_{m}\left(u,v\right)\mathrm{exp}\left\{j{K}_{E}\left(u,v\right)\right\}.$$
(3)
$$\begin{array}{c}{\widehat{f}}_{m}\left({x}_{d},{y}_{d}\right)=\Im \left[{F}_{m}{}^{*}\left(u,v\right)N\left(u,v\right)\right]\\ \text{\hspace{0.17em}}={f}_{m}{}^{*}\left({x}_{d},{y}_{d}\right)*n\left({x}_{d},{y}_{d}\right),\end{array}$$
(4)
$$n\left({x}_{d},{y}_{d}\right)\cong \{\begin{array}{c}\delta \left({x}_{d}-\alpha ,{y}_{d}-\beta \right)\\ random\text{\hspace{0.17em}}sequence\end{array}\begin{array}{c}(correct\text{ \hspace{0.17em} \hspace{0.17em}}fingerpri\text{}nt)\\ \text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}(incorrect\text{\hspace{0.17em} \hspace{0.17em}}fingerpri\text{}nt)\end{array}\phantom{\rule{.5em}{0ex}}\text{,}$$
(5)
$$\left|\mathrm{\Phi}(u,v)\right|=\left|{F}_{m}\left(u,v\right)\right|$$
(6)
$$\mathrm{exp}\left\{j{\widehat{K}}_{E}\left(u,v\right)\right\}=\frac{\mathrm{\Phi}\left(u,v\right)}{{\widehat{F}}_{m}\left(u,v\right)}=\frac{\mathrm{\Phi}\left(u,v\right)}{\Im \left[f\left(x,y\right)\mathrm{exp}\left\{j\widehat{r}\left(x,y\right)\right\}\right]}.$$
(7)
$${\mathrm{\Phi}}_{p}\left(u,v\right)=\frac{\mathrm{\Phi}\left(u,v\right)}{\left|\mathrm{\Phi}\left(u,v\right)\right|}.$$
(8)
$$\begin{array}{c}{\widehat{f}}_{p}\left({x}_{d},{y}_{d}\right)=\Im \left[\frac{{F}_{m}{}^{*}\left(u,v\right)}{\left|{F}_{m}\left(u,v\right)\right|}N\left(u,v\right)\right]\\ \text{\hspace{0.17em}}={f}_{p}\left({x}_{d},{y}_{d}\right)*n\left({x}_{d},{y}_{d}\right),\end{array}$$
(9)
$$SSE=10\mathrm{log}\frac{{\displaystyle \sum [f-|{f}_{m}{}^{(n)}|]}}{{\displaystyle \sum {f}^{2}}},$$
(10)
$$BER=\frac{{N}_{error}}{{N}_{total}},$$