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arXiv:2606.25080

Quantum Primitive for Output-Hiding Function Sharing

Olivia R. Hartzell

quant-phSubmitted Jun 23, 2026• Announced Jun 25, 2026• 0 citations
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Abstract

A quantum information-theoretic primitive is introduced for determining a discrete-valued function that depends on multiple parties' local private inputs. The primitive permits the parties to mutually learn each others' local inputs, and thereby determine function values, while their individual systems remain independent of these inputs. The resulting function values are shared among the parties, but may remain information-theoretically hidden from any external observer, as well as from adversarial state-preparation or measurement processes within the quantum system, in every iteration. In particular, while classically producing a shared function with these information-theoretic properties requires the use of private keys or hidden randomness, in the proposed setting it is achieved using quantum resources alone. I outline the primitive's general properties while applications across a broad range of secure quantum communication and computation settings including: quantum key distribution, multi-party coordination and decision schemes, function evaluation, and in some settings, protocols for fairly generated private coins, are relegated to further publications.

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A quantum information-theoretic primitive is introduced for determining a discrete-valued function that depends on multiple parties' local private inputs. The primitive permits the parties to mutually learn each others' local inputs, and thereby determine function values, while their individual systems remain independent of these inputs. The resulting function values are shared among the parties, but may remain information-theoretically hidden from any external observer, as well as from adversarial state-preparation or measurement processes within the quantum system, in every iteration. In particular, while classically producing a shared function with these information-theoretic properties requires the use of private keys or hidden randomness, in the proposed setting it is achieved using quantum resources alone. I outline the primitive's general properties while applications across a broad range of secure quantum communication and computation settings including: quantum key distribution, multi-party coordination and decision schemes, function evaluation, and in some settings, protocols for fairly generated private coins, are relegated to further publications.
Applications of the proposed primitive: Quantum Primitive for Output-Hiding Function Sharing, are discussed for secure quantum communications and computation protocols. In particular, QKD applications provide notable enhanced security and efficiency properties relative to status-quo protocols. Additionally, we provide examples when parties may wish to encode joint functions, or decisions, which remain information-theoretically hidden from external parties and those internal to the quantum system, without additional private keys, hidden randomness, or classical communication. In particular these applications may be useful in domains such as; financial transactions, joint signaling or coordination decisions, and navigation systems, among others.
Secure multi-party computation (MPC) is a cryptographic primitive that lets a num- ber N of mutually distrustful parties compute a function f (x1, . . . , xN ) on their private inputs xi such that at the end of the MPC protocol, each honest party obtains the func- tion’s correct output and no adversary controlling a certain subset of parties learns any- thing about the honest parties’ inputs beyond what can be inferred from the function’s output value. In 2002, [Smi01] and [CGS02] introduced the notion of multi-party quan- tum computation (MPQC) in which arbitrary quantum circuits can be evaluated in a distributed manner, secure against a quantum adversary. Few protocols for MPQC are known. We present a fundamental study of the building blocks of MPC and MPQC in the information-theoretic setting with a focus on protocols built on top of (quantum) se- cret sharing schemes. In particular, we point out structural similarities and differences and compile an extensive list of theoretical feasibility results for the maximum number of corrupted parties within classical and quantum secret sharing and MPC protocols.
We propose a quantum function secret sharing scheme in which the communication is exclusively classical. In this primitive, a classical dealer distributes a secret quantum circuit CCC by providing shares to ppp quantum parties. The parties on an input state ∣ψ⟩\ketψ∣ψ⟩ and a projection ΠΠΠ, compute values yiy_iyi​ that they then classically communicate back to the dealer, who can then compute ∥ΠC∣ψ⟩∥2\lVert ΠC|ψ\rangle\rVert^2∥ΠC∣ψ⟩∥2 using only classical resources. Moreover, the shares do not leak much information about the secret circuit CCC. Our protocol for quantum secret sharing uses the {\em Cayley path}, a tool that has been extensively used to support quantum primacy claims. More concretely, the shares of CCC correspond to randomized version of CCC which are delegated to the quantum parties, and the reconstruction can be done by extrapolation. Our scheme has two limitations, which we prove to be inherent to our techniques: First, our scheme is only secure against single adversaries, and we show that if two parties collude, then they can break its security. Second, the evaluation done by the parties requires exponential time in the number of gates.
In the secure two-party computation problem, two parties wish to compute a (possibly randomized) function of their inputs via an interactive protocol, while ensuring that neither party learns more than what can be inferred from only their own input and output. For semi-honest parties and information-theoretic security guarantees, it is well-known that, if only noise-less communication is available, only a limited set of functions can be securely computed; however, if interaction is also allowed over general communication primitives (multi-input/output channels), there are “complete” primitives that enable any function to be securely computed. The general set of complete primitives was characterized recently by Maji, Prabhakaran, and Rosulek leveraging an earlier specialized characterization by Kilian. Our contribution in this paper is a simple, self-contained, alternative derivation using elementary information-theoretic tools.
One-sided output secure function evaluation is a cryptographic primitive where the two mutually distrustful players, Alice and Bob, both have a private input to a bivariate function. Bob obtains the value of the function for the given inputs, while Alice receives no output. It is known that this primitive cannot be securely implemented if the two players only have access to noiseless classical and quantum communication. In this work, we first show that Bob can extract the function values for all his possible inputs from any implementation of a non-trivial function that is correct and preserves the privacy of Bob's input. Our result holds in the non-asymptotic setting where the players have finite resources and the error is a constant. Then we consider protocols for secure function evaluation in a setup where the two players have access to trusted distributed randomness as a resource. Building upon the first result, we prove a bound on the efficiency of such cryptographic reductions for any non-trivial function in terms of the conditional entropies of the trusted randomness. From this result, we can derive lower bounds on the number of instances of different variants of OT needed to securely implement a given function.