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arXiv:1812.10862 · IEEE International Symposium on Information Theory (ISIT)

Secure Modulo Sum via Multiple Access Channel

Masahito Hayashi

Published Jul 12, 2021• 6 citations
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Abstract

We discuss secure computation of modular sum when multiple access channel from distinct players$A_{1}, \ldots, A_{c}$to a third party (Receiver) is given. Then, we define the secure modulo sum capacity as the supremum of the transmission rate of modulo sum without information leakage of other information. We derive its useful lower bound, which is numerically calculated under a realistic model that can be realizable as a Gaussian multiple access channel (MAC).

DOI
10.1109/isit45174.2021.9518041

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We discuss secure computation of modular sum when multiple access channel from distinct playersA1,…,AcA_{1}, \ldots, A_{c}A1​,…,Ac​to a third party (Receiver) is given. Then, we define the secure modulo sum capacity as the supremum of the transmission rate of modulo sum without information leakage of other information. We derive its useful lower bound, which is numerically calculated under a realistic model that can be realizable as a Gaussian multiple access channel (MAC).
The problem of entanglement-assisted summation over a quantum multiple access channel (Σ\SigmaΣ-QMAC) is intro-duced, involvingSSSservers,KKKclassical(Fd)(\mathbb{F}_{d})(Fd​)data streams that are replicated arbitrarily across various subsets of servers, and a receiver who wishes to compute the sum of theKKKdata streams. Independent of the data, entangled quantum systemsQ1,Q2,⋯ ,QS\mathcal{Q}_{1}, \mathcal{Q}_{2}, \cdots, \mathcal{Q}_{S}Q1​,Q2​,⋯,QS​are prepared in advance and distributed to the corresponding servers. Each servers,s∈[S]s, s\in[S]s,s∈[S]locally manipulates its quantum systemQs\mathcal{Q}_{s}Qs​according to its classical data and sendsQs\mathcal{Q}_{s}Qs​to the receiver. The total communication cost islog⁡d∣Q1∣+log⁡d∣Q2∣+⋯+log⁡d∣QS∣\log_{d}\vert \mathcal{Q}_{1}\vert +\log_{d}\vert \mathcal{Q}_{2}\vert +\cdots+\log_{d}\vert \mathcal{Q}_{S}\vertlogd​∣Q1​∣+logd​∣Q2​∣+⋯+logd​∣QS​∣qudits, where∣Qs\vert \mathcal{Q}_{s}∣Qs​denotes the dimension ofQs\mathcal{Q}_{s}Qs​. Based on a measurement of the composite systemQ1Q2⋯QS\mathcal{Q}_{1}\mathcal{Q}_{2}\cdots \mathcal{Q}_{S}Q1​Q2​⋯QS​, the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits(Fd(\mathbf{F}_{d}(Fd​symbols) of the desired sum computed by the receiver per qudit (d-dimsional quantum system) of download. The capacityCCCis the supremum of the set of all achievable rates. As the main result of this work, the precise capacity ofΣ\SigmaΣ-QMAC is obtained, from which it follows that quantum entanglements allow a factor of 2 gain in capacity (superdense coding gain) relative to capacity with no entanglements, in all cases (anyS,K,FdS, K, \mathbf{F}_{d}S,K,Fd​and any data replication pattern) provided that the entanglement-assisted capacity does not exceed 1 dit/qudit (Holevo bound). Coding schemes based on a recentNNN-sum box abstraction are sufficient to achieve capacity.
In this paper, the problem of securely computing a function over the binary modulo-2 adder multiple-access wiretap channel is considered. The problem involves a legitimate receiver that wishes to reliably and efficiently compute a function of distributed binary sources while an eavesdropper has to be kept ignorant of them. In order to characterize the corresponding fundamental limit, the notion of secrecy computation-capacity is introduced. Although determining the secrecy computation-capacity is challenging for arbitrary functions, it surprisingly turns out that if the function perfectly matches the algebraic structure of the channel and the joint source distribution fulfills certain conditions, the secrecy computation-capacity equals the computation capacity, which is the supremum of all achievable computation rates without secrecy constraints. Unlike the case of securely transmitting messages, no additional randomness is needed at the encoders nor does the legitimate receiver need any advantage over the eavesdropper. The results therefore show that the problem of securely computing a function over a multiple-access wiretap channel may significantly differ from the one of securely communicating messages.
In this paper, we study the problem of secret communication over a Compound Multiple Access Channel (MAC). In this channel, we assume that one of the transmitted messages is confidential that is only decoded by its corresponding receiver and kept secret from the other receiver. For this proposed setting (compound MAC with confidential messages), we derive general inner and outer bounds on the secrecy capacity region. Also, as examples, we investigate `Less noisy' and `Gaussian' versions of this channel, and extend the results of the discrete memoryless version to these cases. Moreover, providing numerical examples for the Gaussian case, we illustrate the comparison between achievable rate regions of compound MAC and compound MAC with confidential messages.
Inspired by recent work by Christensen and Popovski on secure 2-user product computation for finite-fields of prime-order over a quantum multiple access channel, the generalization toKKKusers and arbitrary finite fields is explored. Asymptotically optimal (capacity-achieving for large alphabet) schemes are proposed. Additionally, the capacity of modulo-ddd(d≥2d\geq 2d≥2) secureKKK-sum computation is shown to be2/K2/K2/Kcomputations/qudit, generalizing a result of Nishimura and Kawachi beyond binary, and improving upon it for oddKKK.
We show that the sum secure degrees of freedom (d.o.f.) of the K-user Gaussian multiple access (MAC) wiretap channel is K(K-1)/K(K-1)+1. Our achievability is based on real interference alignment and structured cooperative jamming. Each user divides its message into K - 1 sub-messages, and sends a linear combination of signals carrying these sub-messages together with a structured cooperative jamming signal. All cooperative jamming signals are aligned in a single dimension at the legitimate receiver allowing for reliable decoding of the message carrying signals by the legitimate receiver. Each cooperative jamming signal is aligned with K-1 message signals at the eavesdropper limiting the information leakage rate to the eavesdropper. We provide a matching converse establishing the exact sum secure d.o.f. of the Gaussian MAC wiretap channel as K(K-1)/K(K-1)+1.
Several security models of multiple-access channel (MAC) are investigated. First, we study the degraded MAC with confidential messages, where two users transmit their confidential messages (no common message) to a destination, and each user obtains a degraded version of the output of the MAC. Each user views the other user as a eavesdropper, and wishes to keep its confidential message as secret as possible from the other user. Measuring each user's uncertainty about the other user's confidential message by equivocation, the inner and outer bounds on the capacity-equivocation region for this model have been provided. The result is further explained via the binary and Gaussian examples. Second, the discrete memoryless multiple-access wiretap channel (MAC-WT) is studied, where two users transmit their corresponding confidential messages (no common message) to a legitimate receiver, while an additional wiretapper wishes to obtain the messages via a wiretap channel. This new model is considered into two cases: the general MAC-WT with cooperative encoders, and the degraded MAC-WT with non-cooperative encoders. The capacity-equivocation region is totally determined for the cooperative case, and inner and outer bounds on the capacity-equivocation region are provided for the non-cooperative case. For both cases, the results are further explained via the binary examples.