Abstract
We demonstrate that XOR Arbiter PUFs with an even number of arbiter chains have inherently biased responses, even if all arbiter chains are perfectly unbiased. This rebukes the believe that XOR Arbiter PUFs are, like Arbiter PUFs, unbiased when ideally implemented and proves that independently manufactured Arbiter PUFs are not statistically independent.
As an immediate result of this work, we suggest to use XOR Arbiter PUFs with odd numbers of arbiter chains whenever possible. Furthermore, our analysis technique can be applied to future types of PUF designs and can hence be used to identify design weaknesses, in particular when using Arbiter PUFs as building blocks and when developing designs with challenge pre-processing. We support our theoretical findings through simulations of prominent PUF designs. Finally, we discuss consequences for the parameter recommendations of the Interpose PUF.
Investigating the reason of the systematic bias of XOR Arbiter PUF, we exhibit that Arbiter PUFs suffer from a systematic uniqueness weakness.
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Notes
- 1.
The sgn function returns the sign of the argument. In our setting, \({\text {sgn}}0\) will only occur with probability zero; for completeness we define \({\text {sgn}}0=1\).
- 2.
Notice that when using \(-1\) and 1 to represent bit values, the standard product of bit values corresponds to the logical XOR operation.
- 3.
In fact, some parameters have different variances [2], but this is immaterial to the discussion in this paper.
- 4.
An approximation of the bias \(\text {E}_{\textit{\textbf{c}}}\left[ r(\textit{\textbf{c}})\right] \) in dependence of the threshold value can be obtained using the Berry-Esseen-Theorem to approximate \(\sum _{i,j}w_{1,i}w_{2,j}x_{1}x_{2}\) for \(i\ne j\) as a Gaussian random variable with variance \(\sigma ^{2}\) over uniformly chosen random challenges, resulting in \(\text {E}_{\textit{\textbf{c}}}\left[ r(\textit{\textbf{c}})\right] \approx \text {erf}\left( \frac{\sum _{i=1}^{n}w_{1,i}w_{2,i}}{\sigma \sqrt{2}}\right) ;\)the value \(\sum _{i=1}^{n}w_{1,i}w_{2,i}\) in turn follows (in the manufacturing random process) a distribution composed of the sum of product-normal distributions, which has increasing variance for increasing n. Extending the setting, for higher (but even) k the distribution narrows as the variance of the product-normal distribution narrows. The later effect can be observed in our simulations, cf. Fig. 2.
- 5.
The software used for simulation and analysis publicly available as free software at https://github.com/nils-wisiol/pypuf/tree/2020-systematic-bias.
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Wisiol, N., Pirnay, N. (2020). Short Paper: XOR Arbiter PUFs Have Systematic Response Bias. In: Bonneau, J., Heninger, N. (eds) Financial Cryptography and Data Security. FC 2020. Lecture Notes in Computer Science(), vol 12059. Springer, Cham. https://doi.org/10.1007/978-3-030-51280-4_4
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