Message randomization and strong security in quantum stabilizer-based secret sharing for classical secrets

Abstract

We improve the flexibility in designing access structures of quantum stabilizer-based secret sharing schemes for classical secrets, by introducing message randomization in their encoding procedures. We generalize the Gilbert-Varshamov bound for deterministic encoding to randomized encoding of classical secrets. We also provide an explicit example of a ramp secret sharing scheme with which multiple symbols in its classical secret are revealed to an intermediate set, and justify the necessity of incorporating strong security criterion of conventional secret sharing. Finally, we propose an explicit construction of strongly secure ramp secret sharing scheme by quantum stabilizers, which can support twice as large classical secrets as the McEliece-Sarwate strongly secure ramp secret sharing scheme of the same share size and the access structure. Keywords Secret sharing • Quantum error-correcting code • Gilbert-Varshamove bound • Strong security Mathematics Subject Classification 94A62 • 81P70 • 94B65 This is one of several papers published in Designs, Codes and Cryptography comprising the "Special Issue on Coding and Cryptography 2019".

Advance sharing of quantum shares for classical secrets

“…As we can see from Gottesman's secret sharing [15], a dealer can distribute each qudit in the quantum state of shares corresponding to an identity operator I before a secret is given. Our proposed secret sharing scheme is a special case of Matsumoto's secret sharing scheme [25] and thus retains access structures of Matsumoto's secret sharing scheme [25]. In our paper, we clarify a necessary and sufficient condition on advance-shareable sets in our proposal.…”

“…In this section, we modify an encoding method of Matsumoto's secret sharing scheme [25] so that some shares can be distributed before a given secret.…”

“…In addition, the quantum state of shares has the form (I ⊗ U )|ψ , where U is a unitary operator for a given secret and thus a dealer can distribute the first qubit in the quantum state of shares to 1 participant before a given secret. Matsumoto generalized Gottesman's secret sharing [15] to an arbitrary number of participants and an arbitrary size of classical secrets [24,25]. Matsumoto's secret sharing scheme [25] is based on quantum stabilizer codes [2,8,9,14,19,22] and summarized as follows:…”

“…Matsumoto generalized Gottesman's secret sharing [15] to an arbitrary number of participants and an arbitrary size of classical secrets [24,25]. Matsumoto's secret sharing scheme [25] is based on quantum stabilizer codes [2,8,9,14,19,22] and summarized as follows:…”

“…We modify an encoding method of Matsumoto's secret sharing scheme [25] so that for any classical secrets, a dealer chooses a unitary operator of the form U = I ⊗ V , where I is the identity operator and V is a unitary operator. As we can see from Gottesman's secret sharing [15], a dealer can distribute each qudit in the quantum state of shares corresponding to an identity operator I before a secret is given.…”