Paper 2026/1946

Quantum Evolving Secret Sharing - revisited

Abstract

In evolving secret sharing, introduced by Komargodski et al., an infinite set of parties share a secret according to an infinite access structure (where minimal qualified sets are finite). In such schemes, par- ties arrive one by one, and and are given a share by the dealer. There is no additional communication incurred to a given party until the time of reconstruction, where share holders who show up combine their shares to reconstruct the secret if they form a qualified set. In this work, we revisit the notion of quantum evolving secret sharing schemes, we dub QESS, where a quantum secret is shared. Smoothly generalizing the above setting runs into difficulties due to the no-cloning theorem. In particular, values handed to previously arriving parties can not be copied by the dealer and used in the evaluation of future parties’ shares, as often done in evolving secret sharing schemes from the liter- ature. Indeed, attempts to formalize quantum evolving secret sharing, such as that on Chaudhury for the so called dynamic threshold setting and by Cohen et al. for general evolving access structures fell short from adhering to the minimally interactive communication pattern above. In this paper, we take a different approach, and consider a setting where several (possibly infinitely many) copies of a secret (known to the dealer) are generated by need. This approach has been used by Cakan et al. for (finite) quantum secret sharing (QSSS), allowing to circumvent the im- possibility of devising QSSS for monotone access structures which are not no-cloning - those that have disjoint pair of qualified sets. This setting is plausible in many applications, where the secret is known to the dealer at sharing time. Cakan et al. demonstrate that at most n copies of the secret prepared by the dealer, suffice to implement any monotone access structure. In the evolving setting, it is not clear how to devise QESS even for evolving no-cloning access structures. Introducing multiple copies of the secret allows to define QESS with a communication pattern similar to the classical setting, in which arbitrary (not necessarily no cloning) evolving access structures can be implemented. using c(t) ≤ t copies of the secret used to generate the shares of the first t parties - similarly to the finite setting. We study the copy number landscape for various evolving access struc- tures. For general access structures, c(t) ≤ t copies of the secret suffice to generate the shares of the first t parties - similarly to the finite setting. On the lower bound side, the no-cloning theorem implies c(t) = t is nec- essary for QESS. The main question we leave open is whether a single copy suffices for no-cloning evolving access structures, while 1 is the ex- act number of the finite setting. We prove that every no-cloning evolving access structure has a QESS with a finite copy number c(t) ≤ C, where the constant C depends on the particular access structure. Furthermore, for every T0, there exists a QESS as above with c(t) ≤ 1 for t ≤ T0. Our techniques generalize the hybrid technique of Cakan et al. abstract- ing their implicit approach of treating the access structure as a union of no-cloning access structures, and implementing each via exiting quantum erasure codes (QECC), which provide not privacy. Then, a copy of the secret, masked by a classical random key (via QOTP) is encoded by each of these codes. The classical keys are then classically shared via each of the access structure to provide privacy. While they rely on a union of no-cloning threshold access structures, we use arbitrary ones. This often reduces the resulting copy number already in the finite setting. Special care is required to share the QOTP keys for each access structure in the union, while keeping every party’s share finite.

Note: There was a little mistake in the keywords, QSSS instead of QESSS. I fixed it