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A Calculus of Chaos in Stochastic Compilation

Engineering in the Cause of Mathematics

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Book cover Formal Methods. FM 2019 International Workshops (FM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12233))

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Abstract

An unexpected outcome from an open project to develop a ‘chaotic’ compiler for ANSI C is described here: a trace information entropy calculus for stochastically compiled programs. A stochastic compiler produces randomly different object codes every time it is applied to the same source code. This calculus quantifies the entropy introduced into run-time program traces by a compiler that aims for the maximal possible entropy, furnishing a definition and proof of security for encrypted computing (Turing-complete computation in which data remains in encrypted form throughout), where the status was formerly unknown.

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Notes

  1. 1.

    Signed 2s complement comparison is translation-invariant. I.e., \(x{<}y\) \(~\mathrm {iff}~\) \(x{+}k{<}y{+}k\).

  2. 2.

    Entropy is formally the stochastic expectation \(H = -E[\log _2 p_i]\) of the probability \(p_i\) of the possible observations i, thus \(H = - \sum \limits _i p_i \log _2 p_i\) with \(0 \le p_i\le 1\) and \(H\ge 0\).

  3. 3.

    ‘Success’ is stochastic: the method has probability of being right on each bit that beats chance by a (‘non-negligible’) margin B that does not tend to 0 as \(n{\rightarrow }\infty \).

  4. 4.

    For readability here, the final delta in register v0 is set to zero.

  5. 5.

    C code: \(\mathbf{int}~\mathsf{A}(\mathbf{int}~\mathsf{m},\mathbf{int}~\mathsf{n})~\{~\mathsf{if}(\mathsf{m}{=}0) \textsf {return}~\mathsf{n}{+}1;~\mathsf{if}(\mathsf{n}{=}0) \textsf {return}~\mathsf{A}(\mathsf{m}{-}1, 1); \textsf {return}~\mathsf{A}\) \((\mathsf{m}{-}1, \mathsf{A}(\mathsf{m}, \mathsf{n}{-}1));~\}\).

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Acknowledgments

Simon Pickin’s work has been supported by the Spanish MINECO-FEDER (grant numbers DArDOS, TIN2015-65845-C3-1-R and FAME, RTI2018-093608-B-C31). Peter Breuer thanks Hecusys LLC for continued support in encrypted computing research.

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Authors and Affiliations

  1. Hecusys LLC, Atlanta, GA, USA

    Peter T. Breuer

  2. Universidad Complutense de Madrid, Madrid, Spain

    Simon J. Pickin

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  1. Peter T. Breuer
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Corresponding author

Correspondence to Peter T. Breuer .

Editor information

Editors and Affiliations

  1. McMaster University, Hamilton, ON, Canada

    Emil Sekerinski

  2. University of Porto, Porto, Portugal

    Nelma Moreira

  3. University of Minho, Braga, Portugal

    José N. Oliveira

  4. Argo Ai, Munich, Germany

    Daniel Ratiu

  5. University of Pisa, Pisa, Italy

    Riccardo Guidotti

  6. University of Liverpool, Liverpool, UK

    Marie Farrell

  7. University of Liverpool, Liverpool, UK

    Matt Luckcuck

  8. University of Exeter, Exeter, UK

    Diego Marmsoler

  9. University of Minho, Braga, Portugal

    José Campos

  10. University of Newcastle, Newcastle upon Tyne, UK

    Troy Astarte

  11. Claude Bernard University, Lyon, France

    Laure Gonnord

  12. Nazarbayev University, Nur-Sultan, Kazakhstan

    Antonio Cerone

  13. University of Surrey, Guildford, UK

    Luis Couto

  14. University of Surrey, Guildford, UK

    Brijesh Dongol

  15. University of Giessen, Giessen, Germany

    Martin Kutrib

  16. University of Lisbon, Lisbon, Portugal

    Pedro Monteiro

  17. Airbus Operations S.A.S., Toulouse, France

    David Delmas

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Breuer, P.T., Pickin, S.J. (2020). A Calculus of Chaos in Stochastic Compilation. In: Sekerinski, E., et al. Formal Methods. FM 2019 International Workshops. FM 2019. Lecture Notes in Computer Science(), vol 12233. Springer, Cham. https://doi.org/10.1007/978-3-030-54997-8_11

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  • DOI: https://doi.org/10.1007/978-3-030-54997-8_11

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