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arXiv:1901.08007 · IEEE International Symposium on Information Theory (ISIT)

Unique Information and Secret Key Decompositions

Johannes Rauh, Pradeep Kr. Banerjee, Eckehard Olbrich, Jürgen Jost

Published Jul 1, 2019• 8 citations
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Abstract

The unique information (UI) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the UI, which implies that the UI is never greater than one of the best known upper bounds on the two-way secret key rate. We conjecture that the UI lower bounds the two-way rate and discuss implications of the conjecture.

DOI
10.1109/isit.2019.8849550

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The unique information (UI) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the UI, which implies that the UI is never greater than one of the best known upper bounds on the two-way secret key rate. We conjecture that the UI lower bounds the two-way rate and discuss implications of the conjecture.
Given two channels that convey information about the same random variable, we introduce two measures of the unique information of one channel with respect to the other. The two quantities are based on the notion of generalized weighted Le Cam deficiencies and differ on whether one channel can approximate the other by a randomization at either its input or output. We relate the proposed quantities to an existing measure of unique information which we call the minimum-synergy unique information. We give an operational interpretation of the latter in terms of an upper bound on the one-way secret key rate and discuss the role of the unique informations in the context of nonnegative mutual information decompositions into unique, redundant and synergistic components.
The problem of constructing information measures with a well-defined interpretation is of fundamental significance in information theory. A good definition of an information measure entails certain desirable properties while also providing answers to operational problems. In this work, we investigate the properties of the unique information, an information measure that quantifies a deviation from the Blackwell order. Beyond providing an accessible introduction to the topic from a channel ordering perspective, we present a novel resource-theoretic characterization of unique information in a cryptographic task related to secret key agreement. Our operational view of unique information entails rich physical intuition that leads to new insights into secret key agreement in the context of non-negative decompositions of the mutual information into redundant and synergistic contributions. Through this lens, we illuminate new directions for research in partial information decompositions and information-theoretic cryptography.
The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable X i has on a target variable Y, relative to the other sources. For two sources, influence breaks down into the information that both X 0 and X 1 redundantly share with Y, what X 0 uniquely shares with Y, what X 1 uniquely shares with Y, and finally what X 0 and X 1 synergistically share with Y. Unfortunately, considerable disagreement has arisen as to how these four components should be quantified. Drawing from cryptography, we consider the secret key agreement rate as an operational method of quantifying unique information. Secret key agreement rate comes in several forms, depending upon which parties are permitted to communicate. We demonstrate that three of these four forms are inconsistent with the PID. The remaining form implies certain interpretations as to the PID’s meaning—interpretations not present in PID’s definition but that, we argue, need to be explicit. Specifically, the use of a consistent PID quantified using a secret key agreement rate naturally induces a directional interpretation of the PID. We further reveal a surprising connection between third-order connected information, two-way secret key agreement rate, and synergy. We also consider difficulties which arise with a popular PID measure in light of the results here as well as from a maximum entropy viewpoint. We close by reviewing the challenges facing the PID.
The secret-key rate measures the rate at which Alice and Bob can extract secret bits from sampling a joint probability distribution, unknown to an eavesdropper Eve. The secret-key rate has been bounded above by the intrinsic information and reduced intrinsic information. However, we prove that the reduced intrinsic information is 0 if and only if the intrinsic information is 0. This result implies that at least one of the following two conjectures is false: bound secrecy exists, or the reduced intrinsic information equals the secret-key rate. We give an explicit construction of an information-erasing binarization for a candidate for bound secrecy. We then introduce some approaches for proving the existence of bound secrecy, such as reducing the channel space, linearly transforming Bob's map, and perturbing a channel for Eve.
We present a simple method to obtain an upper bound on the achievable secret key rate in quantum key distribution (QKD) protocols that use only unidirectional classical communication during the public-discussion phase. This method is based on a necessary precondition for one-way secret key distillation; the legitimate users need to prove that there exists no quantum state having a symmetric extension that is compatible with the available measurements results. The main advantage of the obtained upper bound is that it can be formulated as a semidefinite program, which can be efficiently solved. We illustrate our results by analyzing two well-known qubit-based QKD protocols: the four-state protocol and the six-state protocol.
We study the problem of information-theoretically secure secret key agreement under the well-known source model and channel model. In both of these models the parties wish to create a shared secret key that is secure from an eavesdropper with unlimited computational resources. In the channel model, the first party can choose a sequence of inputs to a discrete memoryless channel, which has outputs at the other parties and at the eavesdropper. After each channel use, the parties can engage in arbitrarily many rounds of interactive authenticated communication over a public channel. At the end, each party should be able to generate the key. In the source model, the parties wishing to generate a secret key (as well as the eavesdropper) receive a certain number of independent identically distributed copies of jointly distributed random variables after which the parties are allowed interactive authenticated public communication, at the end of which each party should be able to generate the key. We derive new lower and upper bounds on the secret key rate under the source model and the channel model, and introduce a technique for proving that a given expression bounds the secrecy rate from above in the channel model. Our lower bounds strictly improve what is essentially the best known lower bound in both the source model and the channel model. Our upper bound in the channel model strictly improves the current state of art upper bound. We do not know whether our new upper bound in the source model represents an strict improvement but it includes the current best known bound as a special case.