Formal Analysis of PUF Instances Leveraging Correlation-Spectra in Boolean Functions

  • Durba ChatterjeeEmail author
  • Aritra Hazra
  • Debdeep Mukhopadhyay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11947)


In this paper, we present a novel formal analysis scheme considering that the fabrication of a batch of \(N > 1\) PUFs is equivalent to drawing random instances of Boolean mappings. We model PUFs as black-box Boolean functions of dimension \(m \times 1\) and show combinatorially that random designs of such \(m \times 1\) functions exhibit correlation-spectra which can be used to characterize random and thus good designs of PUFs. We first develop theoretical results to quantize the correlation values and subsequently find the expected number of pairs of such Boolean functions which should belong in different regions of the spectra. We extend the concept of correlation to PUFs and theoretically prove that a randomly chosen sample of PUFs and Boolean functions follow the same distribution. In addition to this, we show through extensive experimental results that a randomly chosen sample of such PUFs also resembles the correlation-spectra property of the overall PUF population. We finally propose a formal analysis tool for evaluation of PUFs by observing the correlation-spectra of the PUF instances under test. We show through experimental results on 50 FPGAs that when the PUFs are infected by faults the usual randomness tests for the PUF outputs such as uniformity, fail to detect any aberration. However, the spectral-pattern is clearly shown to get affected, which we demonstrate by standard statistical measure like KL Divergence.


Physically Unclonable Functions Formal analysis Boolean functions 


  1. 1.
    Bolotnyy, L., Robins, G.: Physically unclonable function-based security and privacy in RFID systems. In: Fifth Annual IEEE International Conference on Pervasive Computing and Communications, PerCom 2007, pp. 211–220. IEEE (2007)Google Scholar
  2. 2.
    Chatterjee, U., Sahoo, D.P., Mukhopadhyay, D., Chakraborty, R.S.: Trustworthy proofs for sensor data using FPGA based physically unclonable functions. In: Design, Automation and Test in Europe Conference and Exhibition (DATE), pp. 1504–1507. IEEE (2018)Google Scholar
  3. 3.
    Delvaux, J., Peeters, R., Gu, D., Verbauwhede, I.: A survey on lightweight entity authentication with strong PUFs. ACM Comput. Surv. (CSUR) 48(2), 26 (2015)CrossRefGoogle Scholar
  4. 4.
    Delvaux, J., Verbauwhede, I.: Fault injection modeling attacks on 65 nm arbiter and RO sum PUFs via environmental changes. IEEE Trans. Circuits Syst. I Regul. Pap. 61(6), 1701–1713 (2014)CrossRefGoogle Scholar
  5. 5.
    Ganji, F.: On the Learnability of Physically Unclonable Functions. Springer, Cham (2018). Scholar
  6. 6.
    Ganji, F., Tajik, S., Fäßler, F., Seifert, J.-P.: Strong machine learning attack against PUFs with no mathematical model. In: Gierlichs, B., Poschmann, A.Y. (eds.) CHES 2016. LNCS, vol. 9813, pp. 391–411. Springer, Heidelberg (2016). Scholar
  7. 7.
    Herder, C., Yu, M., Koushanfar, F., Devadas, S.: Physical unclonable functions and applications: a tutorial. Proc. IEEE 102(8), 1126–1141 (2014)CrossRefGoogle Scholar
  8. 8.
    Hori, Y., Yoshida, T., Katashita, T., Satoh, A.: Quantitative and statistical performance evaluation of arbiter physical unclonable functions on FPGAs. In: International Conference on Reconfigurable Computing and FPGAs, pp. 298–303, December 2010Google Scholar
  9. 9.
    Immler, V., Hiller, M., Obermaier, J., Sigl, G.: Take a moment and have some t: hypothesis testing on raw PUF data. In: 2017 IEEE International Symposium on Hardware Oriented Security and Trust (HOST), pp. 128–129. IEEE (2017)Google Scholar
  10. 10.
    Khalafalla, M., Gebotys, C.: PUFs deep attacks: enhanced modeling attacks using deep learning techniques to break the security of double arbiter PUFs. In: 2019 Design, Automation and Test in Europe Conference and Exhibition (DATE), pp. 204–209. IEEE (2019)Google Scholar
  11. 11.
    Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Machida, T., Yamamoto, D., Iwamoto, M., Sakiyama, K.: A new mode of operation for arbiter PUF to improve uniqueness on FPGA. In: 2014 Federated Conference on Computer Science and Information Systems, pp. 871–878. IEEE (2014)Google Scholar
  13. 13.
    Maes, R., Van Herrewege, A., Verbauwhede, I.: PUFKY: a fully functional PUF-based cryptographic key generator. In: Prouff, E., Schaumont, P. (eds.) CHES 2012. LNCS, vol. 7428, pp. 302–319. Springer, Heidelberg (2012). Scholar
  14. 14.
    Maiti, A., Gunreddy, V., Schaumont, P.: A systematic method to evaluate and compare the performance of physical unclonable functions. In: Athanas, P., Pnevmatikatos, D., Sklavos, N. (eds.) Embedded Systems Design with FPGAs, pp. 245–267. Springer, New York (2013). Scholar
  15. 15.
    O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, New York (2014)CrossRefGoogle Scholar
  16. 16.
    Ravikanth, P.S.: Physical one-way functions. Ph.D. thesis, Massachusetts (2001)Google Scholar
  17. 17.
    Rioul, O., Solé, P., Guilley, S., Danger, J.: On the entropy of physically unclonable functions. In: IEEE International Symposium on Information Theory (ISIT), pp. 2928–2932, July 2016Google Scholar
  18. 18.
    Rührmair, U.: Oblivious transfer based on physical unclonable functions. In: Acquisti, A., Smith, S.W., Sadeghi, A.-R. (eds.) Trust 2010. LNCS, vol. 6101, pp. 430–440. Springer, Heidelberg (2010). Scholar
  19. 19.
    Rührmair, U., Busch, H., Katzenbeisser, S.: Strong PUFs: models, constructions, and security proofs. In: Sadeghi, A.R., Naccache, D. (eds.) Towards Hardware-Intrinsic Security, pp. 79–96. Springer, Heidelberg (2010). Scholar
  20. 20.
    Sahoo, D.P., Bag, A., Patranabis, S., Mukhopadhyay, D., Chakraborty, R.S.: Fault-tolerant implementations of physically unclonable functions on FPGA. In: Chakraborty, R., Mathew, J., Vasilakos, A. (eds.) Security and Fault Tolerance in Internet of Things, pp. 129–153. Springer, Cham (2019). Scholar
  21. 21.
    Sarkar, P., Maitra, S.: Cross-correlation analysis of cryptographically useful boolean functions and S-boxes. Theory Comput. Syst. 35(1), 39–57 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Suh, G.E., Devadas, S.: Physical unclonable functions for device authentication and secret key generation. In: 44th ACM/IEEE Design Automation Conference, pp. 9–14, June 2007Google Scholar
  23. 23.
    Tajik, S., Lohrke, H., Ganji, F., Seifert, J.P., Boit, C.: Laser fault attack on physically unclonable functions. In: 2015 Workshop on Fault Diagnosis and Tolerance in Cryptography (FDTC), pp. 85–96. IEEE (2015)Google Scholar
  24. 24.
    Wang, Y., Wang, C., Gu, C., Cui, Y., O’Neill, M., Liu, W.: Theoretical analysis of delay-based PUFs and design strategies for improvement. In: 2019 IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1–5. IEEE (2019)Google Scholar
  25. 25.
    Zhang, B., Srihari, S.N.: Properties of binary vector dissimilarity measures. In: Proceedings of JCIS International Conference on Computer Vision, Pattern Recognition, and Image Processing, vol. 1 (2003)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Indian Institute of Technology KharagpurKharagpurIndia

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