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IEEE International Symposium on Information Theory (ISIT)

On the Source Model Key Agreement Problem

Hamidreza Abin, Amin Gohari

Published Jun 22, 2025• 2 citations
Publisher

Abstract

We consider the source model key agreement problem involving two legitimate parties and an eavesdropper who observe n i.i.d. samples of X and Y and Z respectively. In this paper, we focus on one of the simplest instances where the key capacity remains open, specifically when X and Y are binary random variables and Z is a function of the pair ($X, Y$). The best-known upper bound on the key capacity is characterized by an inf-max optimization problem that generally lacks a closedform solution. In this paper, we solve the optimization for some class of sources, thereby providing simple expressions for the upper bound. We provide general conditions under which the upper bound reduces to$I(X; Y)$. As an example, we consider the XOR setting in which X and Y are binary, and Z is the XOR of X and Y. The upper bound reduces to$I(X; Y)$for this source. Next, we conjecture that the rate$I(X; Y)$is not achievable for the XOR source, and provide some ideas that might be useful for developing a new upper bound.${ }^{1}$

DOI
10.1109/isit63088.2025.11195703

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We consider the source model key agreement problem involving two legitimate parties and an eavesdropper who observe n i.i.d. samples of X and Y and Z respectively. In this paper, we focus on one of the simplest instances where the key capacity remains open, specifically when X and Y are binary random variables and Z is a function of the pair (X,YX, YX,Y). The best-known upper bound on the key capacity is characterized by an inf-max optimization problem that generally lacks a closedform solution. In this paper, we solve the optimization for some class of sources, thereby providing simple expressions for the upper bound. We provide general conditions under which the upper bound reduces toI(X;Y)I(X; Y)I(X;Y). As an example, we consider the XOR setting in which X and Y are binary, and Z is the XOR of X and Y. The upper bound reduces toI(X;Y)I(X; Y)I(X;Y)for this source. Next, we conjecture that the rateI(X;Y)I(X; Y)I(X;Y)is not achievable for the XOR source, and provide some ideas that might be useful for developing a new upper bound.1{ }^{1}1
We consider the source model key agreement problem involving two legitimate parties and an eavesdropper who observe n i.i.d. samples of X, Y, and Z, respectively. In this paper, we focus on one of the simplest instances where the key capacity remains open, specifically when X and Y are binary random variables and Z is a function of the pair (X, Y). The best-known upper bound on the key capacity is characterized by an inf-max optimization problem that generally lacks a closed-form solution. We provide general conditions under which the upper bound reduces to I(X;Y). As an example, we consider the XOR setting in which X and Y are binary, and Z is the XOR of X and Y. The upper bound reduces to I(X;Y) for this source. Next, we conjecture that the rate I(X;Y) is not achievable for the XOR source and provide some ideas that might be useful for developing a new upper bound on the source model problem.
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.
A two-party key agreement problem with public discussion, known as the source model problem, is considered. By relating key agreement to hypothesis testing, a new coding scheme is developed that yields a sufficient condition to achieve a positive secret-key (SK) rate in terms of Rényi divergence. The merits of this coding scheme are illustrated by applying it to an erasure model for Eve's side information and by deriving an upper bound on Eve's erasure probabilities for which the SK capacity is zero. This bound strictly improves on the best known single-letter lower bound on the SK capacity. Moreover, the bound is tight when Alice's or Bob's source is binary, which extends a previous result for a doubly symmetric binary source. The results motivate a new measure for the correlation between two random variables which is of independent interest.
The two-party key agreement problem with public discussion, known as the source model problem, is considered for an erasure model for Eve's side information. By relating the key agreement problem to hypothesis testing, a new coding scheme is developed that yields an upper bound on the maximum erasure probability for which the secret-key (SK) capacity is zero. The bound is shown to be tight when Alice's or Bob's source is binary, and this shows that the new code achieves larger SK rates than the best known coding scheme. A full version of this paper with extensions to general models for Eve's side information is available in [1].
We consider the fundamental limits of the secret key generation problem when the sources are excited by the sender. In many practical communication settings, the channel may be influenced by the parties involved. Similar to recent works on probing capacity and channels with action-dependent states, our system model captures such a scenario. We derive single-letter expressions for the secret key capacity. Our coding strategy involves wiretap channel coding and a key generation scheme. By assuming that the eavesdropper receives a degraded version of the legitimate receiver's observation, we also obtain a capacity result that does not involve any auxiliary random variables. Finally, by studying two examples, we show that there is a fundamental tradeoff in between the amount of common randomness (i.e., the secret key rate) and the wiretap secrecy rate.
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, multiple terminals wish to create a shared secret key that is secure from a passive eavesdropper. The terminals have access to a noiseless public communication channel and an additional resource that depends on the model. In the source model, the resource is an external source that repeatedly beams correlated randomness to the terminals; whereas in the channel model, the resource is a secure but noisy discrete memoryless broadcast channel. We derive new lower and upper bounds on the secret key capacity under both the source model and the channel model. The technique used for deriving our bound for the source model is to find certain properties of functions of joint probability distributions which, applied to the joint distribution of the source, will imply that they dominate the secret key capacity, and then prove the bound by a verification argument. A similar technique is used for the channel model. Finally, we also define a problem of communication for omniscience by a neutral observer and establish the equivalence between this new problem and the problem of secret key agreement. This generalizes an earlier result of Csiszár and Narayan.