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Random-Energy Secret Sharing via Extreme Synergy (2309.14047v1)

Published 25 Sep 2023 in cond-mat.dis-nn, cond-mat.stat-mech, cs.CR, cs.IT, and math.IT

Abstract: The random-energy model (REM), a solvable spin-glass model, has impacted an incredibly diverse set of problems, from protein folding to combinatorial optimization to many-body localization. Here, we explore a new connection to secret sharing. We formulate a secret-sharing scheme, based on the REM, and analyze its information-theoretic properties. Our analyses reveal that the correlations between subsystems of the REM are highly synergistic and form the basis for secure secret-sharing schemes. We derive the ranges of temperatures and secret lengths over which the REM satisfies the requirement of secure secret sharing. We show further that a special point in the phase diagram exists at which the REM-based scheme is optimal in its information encoding. Our analytical results for the thermodynamic limit are in good qualitative agreement with numerical simulations of finite systems, for which the strict security requirement is replaced by a tradeoff between secrecy and recoverability. Our work offers a further example of information theory as a unifying concept, connecting problems in statistical physics to those in computation.

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References (22)
  1. A. Shamir, How to share a secret, Commun. ACM 22, 612 (1979).
  2. G. R. Blakley, Safeguarding cryptographic keys, in 1979 International Workshop on Managing Requirements Knowledge (MARK) (1979) pp. 313–318.
  3. B. Derrida, Random-energy model: Limit of a family of disordered models, Phys. Rev. Lett. 45, 79 (1980).
  4. B. Derrida, Random-energy model: An exactly solvable model of disordered systems, Phys. Rev. B 24, 2613 (1981).
  5. J. D. Bryngelson and P. G. Wolynes, Spin glasses and the statistical mechanics of protein folding., Proc. Natl. Acad. Sci. USA 84, 7524 (1987).
  6. E. I. Shakhnovich and A. M. Gutin, Implications of thermodynamics of protein folding for evolution of primary sequences, Nature 346, 773 (1990).
  7. S. Hormoz and M. P. Brenner, Design principles for self-assembly with short-range interactions, Proc. Natl. Acad. Sci. USA 108, 5193 (2011).
  8. C. K. Fisher and P. Mehta, The transition between the niche and neutral regimes in ecology, Proc. Natl. Acad. Sci. USA 111, 13111 (2014).
  9. C. R. Laumann, A. Pal, and A. Scardicchio, Many-body mobility edge in a mean-field quantum spin glass, Phys. Rev. Lett. 113, 200405 (2014).
  10. A. Lazarides, A. Das, and R. Moessner, Fate of many-body localization under periodic driving, Phys. Rev. Lett. 115, 030402 (2015).
  11. N. Sourlas, Spin-glass models as error-correcting codes, Nature 339, 693 (1989).
  12. S. Mertens, Random costs in combinatorial optimization, Phys. Rev. Lett. 84, 1347 (2000).
  13. C. Borgs, J. Chayes, and B. Pittel, Phase transition and finite-size scaling for the integer partitioning problem, Random Struct. Algorithms 19, 247 (2001).
  14. We slightly abuse the notation and use σ𝜎\sigmaitalic_σ to denote both random variables and their realizations.
  15. The mutual information between two equal halves of the REM [29] is a special case of our result, with h=0ℎ0h\!=\!0italic_h = 0 and m=v=1/2𝑚𝑣12m\!=\!v\!=\!1/2italic_m = italic_v = 1 / 2.
  16. P. Grassberger, Toward a quantitative theory of self-generated complexity, Int. J. Theor. Phys. 25, 907 (1986).
  17. O. Cohen, V. Rittenberg, and T. Sadhu, Shared information in classical mean-field models, J. Phys. A 48, 055002 (2015).
  18. V. Ngampruetikorn, I. Nemenman, and D. J. Schwab, Extrinsic vs Intrinsic Criticality in Systems with Many Components (2023), in preparation.
  19. W. Bialek, S. E. Palmer, and D. J. Schwab, What makes it possible to learn probability distributions in the natural world? (2020), arXiv:2008.12279 [cond-mat.stat-mech] .
  20. Note that secret decoding can be computationally difficult even if it is information-theoretically allowed. Here, we focus on the information-theoretic properties of secret codes rather than the existence of efficient algorithms.
  21. We need only consider the condition for k−1𝑘1k\!-\!1italic_k - 1 shares since the data processing inequality implies that fewer shares cannot increase information. If I⁢(σm;τr)=0𝐼superscript𝜎𝑚superscript𝜏𝑟0I(\sigma^{m};\tau^{r})\!=\!0italic_I ( italic_σ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ; italic_τ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) = 0 and r>0𝑟0r\!>\!0italic_r > 0, then I⁢(σm;τr−1)=0𝐼superscript𝜎𝑚superscript𝜏𝑟10I(\sigma^{m};\tau^{r-1})\!=\!0italic_I ( italic_σ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ; italic_τ start_POSTSUPERSCRIPT italic_r - 1 end_POSTSUPERSCRIPT ) = 0 due to nonnegativity of mutual information.
  22. The condition for k𝑘kitalic_k shares suffices since the data processing inequality implies that more shares cannot decrease information. If I⁢(σm;τr)=S⁢(σm)𝐼superscript𝜎𝑚superscript𝜏𝑟𝑆superscript𝜎𝑚I(\sigma^{m};\tau^{r})\!=\!S(\sigma^{m})italic_I ( italic_σ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ; italic_τ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) = italic_S ( italic_σ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) and r<n𝑟𝑛r\!<\!nitalic_r < italic_n, then I⁢(σm;τr+1)=S⁢(σm)𝐼superscript𝜎𝑚superscript𝜏𝑟1𝑆superscript𝜎𝑚I(\sigma^{m};\tau^{r+1})\!=\!S(\sigma^{m})italic_I ( italic_σ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ; italic_τ start_POSTSUPERSCRIPT italic_r + 1 end_POSTSUPERSCRIPT ) = italic_S ( italic_σ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) as the secret entropy bounds the information from above.
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