Abstract--If a biometric template is compromised, the invasion of user privacy is inevitable. Since human biometric is irreplaceable and irrevocable, such an invasion implies a permanent loss of user identity. As a result, many transformation methods are proposed to convert the biometric template into non-invertible version of itself. However, if the transformation functions are not carefully designed, the “non-invertible” templates are susceptible to be partially or fully invertible. In this paper, we proposed a template inversion technique, namely small-angle approximation attack, which is less studied in the literature. We reveal that the angle-based many-to-one property of the non-linear functions is highly vulnerable to small-angle approximation attack. I. INTRODUCTION Biometric template possess the privacy of users, thus privacy invasion may occur when the biometric template is compromised. Furthermore, biometric is permanently associated with individual, which implies that once it is lost, the identity is potentially compromised forever. For these reasons, a biometric system with template protection is favorable in terms of preserving user privacy. Biometric template protection has four criteria, namely non-invertibility, revocability, diversity and performance. Among four criteria, non-invertibility is usually the most challenging task to achieve [1]. Non-invertibility in this paper refers to the computational hardness in recovering the fingerprint feature vector or minutiae from the generated bit-string and/or helper data. To analyze the non-invertibility property of a biometric template, the many-to-one property of the transformation functions has always been taken as a main criterion for evaluation; yet, the careless design of the many-to-one transformation function would inevitably compromise the non-invertibilitiy of the transformation function. A well-known instance proposed in literature is Rathas’ surface folding scheme [1]. It is shown that inappropriate parameters selected for minutia transformation lead to a small range of shifting for the transformed minutiae [2]. In fact, such transformations degenerate the many-to-one surface folding function to a linear function. Therefore, non-invertibility can be compromised via restoring the small shifted fingerprint minutiae. II. SMALL-ANGLE APPROXIMATION ATTACK ON BIOPHASOR In this paper, we propose a small-angle approximation attack to evaluate the non-invertibility of two angle-based fingerprint representations, i.e. inverse tangent based BioPhasor [3] and sinus (cosine) based Graph-based Hamming Embedding (GHE) [4]. The attack idea is inspired by the attack [2] on the surface folding in [1] and small-angle approximation that is approximately true in the limit where the angle of trigonometric functions approach zero [5]. A rigorous analysis is performed to demonstrate the feasibility of the proposed attack, which serves a caution on evaluating the non- invertibility of the angle-based biometric templates that currently is overestimated by the many-to-one functions. A. Background of BioPhasor BioPhasor is constructed based on the iterated mixing between the tokenised pseudo-random number (PRN) and the biometric feature. The construction of BioPhasor can be formulated as a three-step algorithm as follows: Let where x is the biometric feature vector derived from raw biometric image (e.g. multichannel Gabor Filter based FingerCode in [3]) and p is the feature dimensions: 1) Use token to generate a set of PRN, and then transform into an orthonormal projection matrix by applying Gram–Schmidt process. 2) Mix the x with iteratively to form the complex number, and calculate their complex arguments, . 3) Average the complex arguments , where and p . B. Small-angle approximation attacks on BioPhasor General speaking, BioPhasor can be re-formulated as follow: where and represent an ordered fixed-length feature vector and a set of random numbers respectively; and n and m denote the length of and respectively. If rad, then . This small- angle approximation can be observed in Fig. 1 (a). We analyze two cases separately, i.e. i) ; ii) [5]. i) For , the BioPhasor degenerates to: or if Small-Angle Approximation Attack on Angle-based Many-to- One Functions for Biometric Templates Zhe Jin 1 , Bok-Min, Goi 1 , Andrew Beng Jin Teoh 2 , Yong Haur Tay 1 , Pan Zheng 3 1 Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia 2 School of Electrical and Electronic Engineering, Yonsei University, South Korea 3 Faculty of Engineering, Computing and Science, Swinburne University of Technology, Malaysia. jin.zhe@1utar.my, {goibm, tayyh}@utar.edu.my, bjteoh@yonsei.ac.kr, panzheng@gmail.com This research is supported by Science Fund 01-02-11-SF0201 MOSTI, Malaysia and UTARRF, No: IPSR/RMC/UTARRF/2013-C2/G04. 2015 International Conference on Consumer Electronics-Taiwan (ICCE-TW) 978-1-4799-8745-0/15/$31.00©2015 IEEE 98