Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory

It is known that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal to the length of the longest shared secret key that two parties can establish via a probabilistic protocol with interaction on a public channel, assuming that the parties hold as their inputs x and y respectively. We determine the worst-case communication complexity of this problem for the setting where the parties can use private sources of random bits. We show that for some x, y the communication complexity of the secret key agreement does not decrease even if the parties have to agree on a secret key whose size is much smaller than the mutual information between x and y. On the other hand, we discuss examples of x, y such that the communication complexity of the protocol declines gradually with the size of the derived secret key. The proof of the main result uses spectral properties of appropriate graphs and the expander mixing lemma, as well as information theoretic techniques.

[1]  Alexander Shen,et al.  Variations on Muchnik’s Conditional Complexity Theorem , 2009, Theory of Computing Systems.

[2]  Le Anh Vinh,et al.  The Szemerédi-Trotter type theorem and the sum-product estimate in finite fields , 2007, Eur. J. Comb..

[3]  Sergio Verdú,et al.  Secret Key Generation With Limited Interaction , 2016, IEEE Transactions on Information Theory.

[4]  Andrei Romashchenko,et al.  An Operational Characterization of Mutual Information in Algorithmic Information Theory , 2019, J. ACM.

[5]  Andries E. Brouwer,et al.  The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters , 2017, J. Comb. Theory, Ser. B.

[6]  Rudolf Ahlswede,et al.  Common randomness in information theory and cryptography - I: Secret sharing , 1993, IEEE Trans. Inf. Theory.

[7]  Nikolai K. Vereshchagin,et al.  Inequalities for Shannon Entropy and Kolmogorov Complexity , 1997, J. Comput. Syst. Sci..

[8]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[9]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[10]  Luis Filipe Coelho Antunes,et al.  Cryptographic Security of Individual Instances , 2007, ICITS.

[11]  Andrei Romashchenko,et al.  A Conditional Information Inequality and Its Combinatorial Applications , 2018, IEEE Transactions on Information Theory.

[12]  L. Levin,et al.  THE COMPLEXITY OF FINITE OBJECTS AND THE DEVELOPMENT OF THE CONCEPTS OF INFORMATION AND RANDOMNESS BY MEANS OF THE THEORY OF ALGORITHMS , 1970 .

[13]  Nikolai K. Vereshchagin,et al.  A new class of non-Shannon-type inequalities for entropies , 2002, Commun. Inf. Syst..

[14]  A. Winter,et al.  Distillation of secret key and entanglement from quantum states , 2003, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  Audrey Terras,et al.  Finite analogues of Euclidean space , 1996 .

[16]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[17]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[18]  Andrej Muchnik,et al.  Conditional complexity and codes , 2002, Theor. Comput. Sci..

[19]  Nikolai K. Vereshchagin,et al.  Upper semilattice of binary strings with the relation "x is simple conditional to y" , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[20]  Himanshu Tyagi,et al.  Communication for Generating Correlation: A Unifying Survey , 2019, IEEE Transactions on Information Theory.

[21]  HENRY STEINITZ,et al.  KOLMOGOROV COMPLEXITY AND ALGORITHMIC RANDOMNESS , 2013 .

[22]  Himanshu Tyagi,et al.  Common Information and Secret Key Capacity , 2013, IEEE Transactions on Information Theory.

[23]  Xiaogang Liu,et al.  Eigenvalues of Cayley Graphs , 2018, Electron. J. Comb..

[24]  Frans M. J. Willems,et al.  Biometric Security from an Information-Theoretical Perspective , 2012, Found. Trends Commun. Inf. Theory.

[25]  U. Maurer,et al.  Secret key agreement by public discussion from common information , 1993, IEEE Trans. Inf. Theory.

[26]  Marius Zimand,et al.  An Operational Characterization of Mutual Information in Algorithmic Information Theory , 2017, Electron. Colloquium Comput. Complex..

[27]  Sergio Verdú,et al.  Common Randomness and Key Generation with Limited Interaction , 2016, ArXiv.

[28]  Nikolai K. Vereshchagin,et al.  Upper semi-lattice of binary strings with the relation "x is simple conditional to y" , 2002, Theor. Comput. Sci..

[29]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[30]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[31]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[32]  A. Razborov.  Communication Complexity , 2011 .

[33]  Alexander Lubotzky,et al.  Mixing Properties and the Chromatic Number of Ramanujan Complexes , 2014, 1407.7700.