We present PUFKY: a practical and modular design for a cryptographic key generator based on a Physically Unclonable Function (PUF). A fully functional reference implementation is developed and successfully evaluated on a substantial set of FPGA devices. It uses a highly optimized ring oscillator PUF (ROPUF) design, producing responses with up to 99% entropy. A very high key reliability is guaranteed by a syndrome construction secure sketch using an efficient and extremely low-overhead BCH decoder. This first complete implementation of a PUF-based key generator, including a PUF, a BCH decoder and a cryptographic entropy accumulator, utilizes merely 17% (1162slices) of the available resources on a low-end FPGA, of which 82% are occupied by the ROPUF and only 18% by the key generation logic. PUFKY is able to produce a cryptographically secure 128-bit key with a failure rate <10−9 in 5.62ms. The design's modularity allows for rapid and scalable adaptations for other PUF implementations or for alternative key requirements. The presented PUFKY core is immediately deployable in an embedded system, e.g. by connecting it to an embedded microcontroller through a convenient bus interface.

PUFKY: A Fully Functional PUF-based Cryptographic Key Generator

We explore properties of the entropy, Kullback-Leibler information, and mutual information for order statistics. The probability integral transformation plays a pivotal role in developing our results. We provide bounds for the entropy of order statistics and some results that relate entropy ordering of order statistics to other well-known orderings of random variables. We show that the discrimination information between order statistics and data distribution, the discrimination information among the order statistics, and the mutual information between order statistics are all distribution free and are computable using the distributions of the order statistics of the samples from the uniform distribution. We also discuss information properties of spacings for uniform and exponential samples and provide a large sample distribution-free result on the entropy of spacings. The results show interesting symmetries of information orderings among order statistics.

Information properties of order statistics and spacings

We express the joint entropy of progressively censored order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint entropy of progressively Type-II censored data. We then construct a goodness-of-fit test statistic based on Kullback-Leibler information with progressively Type-II censored data. Finally, by using Monte Carlo simulations, the power of the test is estimated, and compared against several alternatives under different progressive censoring schemes

Testing Exponentiality Based on Kullback-Leibler Information With Progressively Type-II Censored Data

Information-theoretic methodologies are increasingly being used in various disciplines. Fre- quently an information measure is adapted for a problem, yet the perspective of information as the unifying notion is overlooked. We set forth this perspective through presenting information- theoretic methodologies for a set of problems in probability and statistics. Our focal measures are Shannon entropy and Kullback-Leibler information. The background topics for these mea- sures include notions of uncertainty and information, their axiomaticfoundation, interpretations, properties, and generalizations. Topics with broad methodological applications include discrep- ancy between distributions, derivation of probability models, dependence between variables, and Bayesian analysis. More specific methodological topics include model selection, limiting distributions, optimal prior distribution and design of experiment, modeling duration variables, order statistics, data disclosure, and relative importance of predictors. Illustrations range from very basic to highly technical ones that draw attention to subtle points.

Information Measures in Perspective

/pdf/pufky-a-fully-functional-puf-based-cryptographic-key-4mifaro6eh.pdf

PUFKY: a fully functional PUF-based cryptographic key generator

The entropy of a sequence of random variables under order restrictions is examined. A theorem that shows the amount of entropy reduction when the sequence is ordered is presented. Upper and lower bounds to the entropy reduction and conditions under which they are achieved are derived. Some interesting properties of the entropy of the individual order statistics are also presented. It is shown that the difference between the average entropy of the individual order statistics and the entropy of a member of the original independent identically distributed (IID) population is a constant, regardless of the original distribution. Finally, the entropies of the individual order statistics are found to be symmetric about the median when the probability density function (PDF) of the original IID sequence is symmetric about its mean. >

S. Chen

Papers

The entropy of ordered sequences and order statistics