Code-routing: a new attack on position verification

The cryptographic task of position verification attempts to verify one party's location in spacetime by exploiting constraints on quantum information and relativistic causality. A popular verification scheme known as $f$-routing involves requiring the prover to redirect a quantum system based on the value of a Boolean function $f$. Cheating strategies for the $f$-routing scheme require the prover use pre-shared entanglement, and security of the scheme rests on assumptions about how much entanglement a prover can manipulate. Here, we give a new cheating strategy in which the quantum system is encoded into a secret-sharing scheme, and the authorization structure of the secret-sharing scheme is exploited to direct the system appropriately. This strategy completes the $f$-routing task using $O(SP_p(f))$ EPR pairs, where $SP_p(f)$ is the minimal size of a span program over the field $\mathbb{Z}_p$ computing $f$. This shows we can efficiently attack $f$-routing schemes whenever $f$ is in the complexity class $\text{Mod}_p\text{L}$, after allowing for local pre-processing. The best earlier construction achieved the class L, which is believed to be strictly inside of $\text{Mod}_p\text{L}$. We also show that the size of a quantum secret sharing scheme with indicator function $f_I$ upper bounds entanglement cost of $f$-routing on the function $f_I$.Comment: 29 pages, v4 adds minor comment

不正検知可能な準最適 (2, 2, n) ランプ型秘密分散

In this research, we consider a strong ramp secret sharing scheme that can detect cheating. A cheating-detectable (k, L, n) ramp secret sharing scheme has been studied so far, and a strong ramp secret sharing scheme which achieves lower bounds on the size of shares and random number used in encoding (i. e., share generation), and the success probability of impersonation attack has been presented. Now a challenging task is to achieve the lower bound on the success probability of substitution attack.In this paper, we present a strong (2, 2, n) ramp secret sharing scheme that almost achieves the lower bound on the success probability of substitution attack. The proposed scheme is the first to almost achieve the lower bound. Moreover the proposed scheme also achieves other lower bounds such as those on the size of shares and random number used in encoding, and the success probability of impersonation attack. We take a unique strategy to construct the scheme. Most existing works present generic type verification functions which can detect cheating for any linear and strong (k, L, n) ramp scheme. On the other hand, our proposed verification function (one of those which we call limited type verification functions) can detect cheating when used with a linear and strong (2, 2, n) ramp scheme satisfying a certain property

Secret Sharing Based on a Hard-on-Average Problem

The main goal of this work is to propose the design of secret sharing schemes based on hard-on-average problems. It includes the description of a new multiparty protocol whose main application is key management in networks. Its unconditionally perfect security relies on a discrete mathematics problem classiffied as DistNP-Complete under the average-case analysis, the so-called Distributional Matrix Representability Problem. Thanks to the use of the search version of the mentioned decision problem, the security of the proposed scheme is guaranteed. Although several secret sharing schemes connected with combinatorial structures may be found in the bibliography, the main contribution of this work is the proposal of a new secret sharing scheme based on a hard-on-average problem, which allows to enlarge the set of tools for designing more secure cryptographic applications