On the Entropy of Fingerprints

A biometric is just a long password, that is easy to remember and easy to enter (with the right hardware support). But just how long a password? Can we measure and compare the “something you are” against the “something you know” authentication criteria? I went looking on the web and yes there are some answers.

In An Analysis of Minutiae Matching Strength three IBM researchers outline how to measure the entropy of fingerprints and their resistance to brute force attacks as compared to passwords. The authors state that sampled biometrics are much longer than passwords (several hundred bytes to over a megabyte) and typically have a high information content. A password of equivalent length would be difficult to remember.

The authors use two models to arrive at these conclusions. In both models they assume that an extracted fingerprint sample can be represented as an image of 300 x 300 pixels, which can be divided into 400 non-overlapping sites of 20 x 20 pixels. Each site holds a minutia detailing a ridge and valley pattern of a fingerprint, and each minutia point has an angle of orientation represented by d = 4, 8 or 16 values. A sample fingerprint is considered a match against a template if a minimum number of N sites match where N is 10, 12, 14, 16 or 18.

So this is like saying that you have a password of length 400 where each character takes on at least d values and you accept a candidate password as correct if it matches the true password in at least N positions. Letting N = 10 and d = 4 yields just over 2^85 possible fingerprint configurations. So attempting to randomly guess a correct fingerprint template in this model only succeeds with one chance in 2^{-85}. This is very low indeed and corresponds to a random length 13 password based on the 94 printable ASCII characters.

What we have described is called the simple model by the authors, which does not account for certain minutia dependencies. A more complex model is proposed to compensate which also shows that the entropy is still as high as 80 bits with additional matches. Even with the complex model there were quite a few caveats, and a revised model was reported in the excellent 2008 survey paper Biometrics: A Tool for Information Security.

In section V.A of the survey paper the amount of discriminating information in a fingerprint is discussed. The revised model is somewhat more conservative in its comparisons to passwords. The authors now state that randomly matching on at least 20 from 36 minutia is at least as difficult as guessing a length 6 case-sensitive alphanumeric password (about 10^{11} in total).

The revised model was motivated by the desire to quantify the uniqueness of fingerprints due to their importance in determining guilt in court cases. And just like DNA tests, the assumed power of fingerprints to uniquely discriminate between individuals is being downgraded.

So in summary a biometric is just a long password, that is easy to remember and easy to enter (with the right hardware support). But you need to check the parameters of the matching algorithm and its assumptions to determine how strong your fingerprint as compared to a password.

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