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The concept of entropy is important for identifying the biometric modalities suitable for BE. Entropy is de ned as a measure of the average information content. In the context of BE, the entropy of a biometric is the upper limit for the size of the key that can be securely bound to the biometric. In other words, if one tries to bind a 128-bit key to a biometric with only 40 bits of entropy, then the security strength of such a system will not be more than 40 bits. Unlike passwords, biometrics must provide some error tolerance, since there are no two identical biometric samples. How, then, can we estimate biometric entropy The dif culty is illustrated by the following example: If a binary 320 320 ngerprint image has a size of 320 320 = 102,400 bits, does it mean that its entropy is 102,400 Technically, the answer could be Yes, if all the bits were statistically independent. However, a legitimate user will never be able to obtain a positive veri cation in such a system requiring that all 102,400 bits be kept unchanged. The system must be error-tolerant. In this case, the entropy will be much lower. In general, the biometric entropy is algorithm-dependent; and therefore, there is no such absolute thing as entropy of a ngerprint. While there are several de nitions of entropy, the notion of min-entropy, H (A), introduced by Dodis et al. [30] (see also reference 35), is most relevant for BE purposes: H (A) = log2 (maxa Pr[A = a]). (26.1)
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Here A is a random variable (i.e., a set of features in case of biometrics) that can take any value, a, with a probability Pr[A = a]. By taking the maximum probability, we assume that the attacker s best strategy would be to guess the most likely value (for example, of a key). This de nition shows how many nearly uniform random bits can be extracted from the distribution.
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In case of two variables, an average min-entropy, H (A|B), of A given B is considered: H (A|B) = log2 Eb B max Pr[A = a|B = b] a = log2 Eb B 2 H (A|B=b) . (26.2)
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It can be interpreted for the purposes of BE in the following way: B is a helper data that is available to the attacker. By knowing B, the attacker can predict A with the maximum probability maxa Pr[A = a|B = b]. On average, the attacker s chance of success in predicting A is then Eb B [maxa Pr[A = a|B = b]], where Eb B is the average over B. It is logical to take average rather than maximum over B, since B is not under the attacker s control. The average min-entropy H (A|B) is essentially the minimum strength of the key that can be consistently extracted from A when B is known. The difference between H (A) and H (A|B), L = H (X) H (X|P), (26.3)
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is called the entropy loss, or the information leak, of a BE scheme. In practical terms, one can estimate the biometric discrimination entropy as minus binary logarithm of FAR at zero distance (or, more exactly, at maximum similarity score), H log2 (FAR(@d = 0)), (26.4)
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meaning that it corresponds to the point where exactly the same biometric sample is presented on enrollment and veri cation. In other words, this FAR(@ d = 0)) is a probability of nding two identical samples among the population. However, we require that the system be realistically designed; that is, it provides a proper error tolerance for legitimate users, meaning that those samples are identical only within the tolerance limits. Note that the point d = 0 is never used itself as an operating point in any biometric system (unlike a password management scheme). There is usually no impostor data available at d = 0, so that FAR(@ d = 0)) can be only approximately estimated by extrapolation. There are two basic approaches to estimating the biometric entropy: empirical estimation and theoretical modeling. Empirical estimations are usually based on John Daugman s paper on iris recognition [36]. It computes an inter-class (i.e., impostors ) distribution of Hamming distance for a large data sample. The histogram of the distribution is approximated with a normalized binomial distribution, and the total number of Bernoulli trials is called a number of degrees of freedom. It is estimated from the mean, p, and the standard deviation, : NDoF = p(1 p)/ 2 . For iris, Daugman obtained NDoF = 173 bits (later upgraded to 249 bits [37]). Those numbers have been extensively quoted in many publications. The number of degrees of freedom is an adequate estimate of the discrimination entropy, as for the binomial distribution, H log2 (FAR(@ d = 0)) = NDoF .
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