We study the connection between mixing properties for bipartite graphs and materialization of the mutual information in one-shot settings. We show that mixing properties of a graph imply impossibility to extract the mutual information shared by the ends of an edge randomly sampled in the graph. We apply these impossibility results to some questions motivated by information-theoretic cryptography. In particular, we show that communication complexity of a secret key agreement in one-shot setting is inherently uneven: for some inputs, almost all communication complexity inevitably falls on only one party.
We study the connection between mixing properties for bipartite graphs and materialization of the mutual information in one-shot settings. We show that mixing properties of a graph imply impossibility to extract the mutual information shared by the ends of an edge randomly sampled in the graph. We apply these impossibility results to some questions motivated by information-theoretic cryptography. In particular, we show that communication complexity of a secret key agreement in one-shot setting is inherently uneven: for some inputs, almost all communication complexity inevitably falls on only one party.
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.
We propose a linear algebraic method, rooted in the spectral properties of graphs, that can be used to prove lower bounds in communication complexity. Our proof technique effectively marries spectral bounds with information-theoretic inequalities. The key insight is the observation that, in specific settings, even when data sets X and Y are closely correlated and have high mutual information, the owner of X cannot convey a reasonably short message that maintains substantial mutual information with Y. In essence, from the perspective of the owner of Y, any sufficiently brief message m=m(X) would appear nearly indistinguishable from a random bit sequence. We employ this argument in several problems of communication complexity. Our main result concerns cryptographic protocols. We establish a lower bound for communication complexity of multi-party secret key agreement with unconditional, i.e., information-theoretic security. Specifically, for one-round protocols (simultaneous messages model) of secret key agreement with three participants we obtain an asymptotically tight lower bound. This bound implies optimality of the previously known omniscience communication protocol (this result applies to a non-interactive secret key agreement with three parties and input data sets with an arbitrary symmetric information profile). We consider communication problems in one-shot scenarios when the parties' inputs are not produced by any i.i.d. sources, and there are no ergodicity assumptions on the input data. In this setting, we found it natural to present our results using the framework of Kolmogorov complexity.
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, the size of which is much smaller than the mutual information between x and y . However, we discuss a natural class 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 the spectral properties of appropriate graphs and the expander mixing lemma, as well as information-theoretic techniques, including constraint information inequalities and Muchnik’s conditional descriptions. A preliminary version of this article was published in the proceedings of MFCS 2020. In the present version, we give full proofs of all theorems and get rid of the assumption that the number of random bits used in the communication protocols is polynomial.
We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties—one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair—can establish via a probabilistic protocol with interaction on a public channel. For ℓ > 2, the longest shared secret that can be established from a tuple of strings ( x 1 , …, x ℓ ) by ℓ parties—each one having one component of the tuple and the complexity profile of the tuple—is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length for protocols with public randomness. We also show that if the communication complexity drops below the established threshold, then only very short secret keys can be obtained.
This paper studies an information-theoretic one-shot variable-length secret key agreement problem with public discussion. Let X and Y be jointly distributed random variables, each taking values in some measurable space. Alice and Bob observe X and Y respectively, can communicate interactively through a public noiseless channel, and want to agree on a key length and a key that is approximately uniformly distributed over all bit sequences with the agreed key length. The public discussion is observed by an eavesdropper, Eve. The key should be approximately independent of the public discussion, conditional on the key length. We show that the optimal expected key length is close to the mutual information I(X;Y) within a logarithmic gap. Moreover, an upper bound and a lower bound on the optimal expected key length can be written down in terms of I(X;Y) only. This means that the optimal one-shot performance is always within a small gap of the optimal asymptotic performance regardless of the distribution of the pair (X,Y). This one-shot result may find applications in situations where the components of an i.i.d. pair source (Xn,Yn) are observed sequentially and the key is output bit by bit, or in situations where the random source is not an i.i.d. or ergodic process.
We study the randomness required to share possibly correlated secret bits among parties connected by a graph. A dealer places shares on the edges so that each party can recover its own secret from its incident shares and learn nothing about the others beyond what its own secret reveals. Anilkumar et al. completely determined the minimum randomness required for three binary secrets. We extend this study to four secrets and obtain results for arbitrary numbers of secrets on general graphs. For four parties on a complete graph, we determine the minimum number of random states for every set of permitted secret combinations: the possible values are one, two, three and four. We also characterize when one random bit suffices on an arbitrary graph.