# Unclonable Secret Keys

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## Description

Unclonable Secret Keys, Is A Well-Researched Topic, It Is To Be Used As A Guide Or Framework For Your Research

## Abstract

We propose a novel concept of securing cryptographic keys which we call “Unclonable Secret Keys,” where any cryptographic object is modified so that its secret key is an unclonable quantum bit-string whereas all other parameters such as messages, public keys, ciphertexts, signatures, etc., remain classical. We study this model in the authentication and encryption setting giving a plethora of definitions and positive results as well as several applications that are impossible in a purely classical setting.

In the authentication setting, we define the notion of one-shot signatures, a fundamental element in building unclonable keys, where the signing key not only is unclonable, but also is restricted to
signing only one message even in the paradoxical scenario where it is generated dishonestly. We propose a construction relative to a classical oracle and prove its unconditional security. Moreover, we provide numerous applications including a signature scheme where an adversary can sign as many messages as it wants and yet it cannot generate two signing keys for the same public key. We show
that one-shot signatures are sufficient to build a proof-of-work-based decentralized cryptocurrency with several ideal properties: it does not make use of a blockchain, it allows sending money over insecure classical channels and it admits several smart contracts. Moreover, we demonstrate that a weaker version of one-shot signatures, namely privately verifiable tokens for signatures, are sufficient to reduce any classically queried stateful oracle to a stateless one. This effectively eliminates, in a provable manner, resetting attacks to hardware devices (modeled as oracles).

In the encryption setting, we study different forms of unclonable decryption keys. We give constructions that vary on their security guarantees and their flexibility. We start with the simplest setting
of secret key encryption with honestly generated keys and show that it exists in the quantum random oracle model. We provide a range of extensions, such as public key encryption with dishonestly generated keys, predicate encryption, broadcast encryption and more.

1 Introduction 1
1.1 Hybrid Quantum/Classical Cryptography . . . . . . . . . . . . . . . . . . . . . 1
1.2 Unclonable Keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 A Philosophical Note: Travelling over the Phone . . . . . . . . . . . . . . . . . . 17
2 Preliminaries 18
2.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Deterministic Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Common Reference String Model . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Quantum Tokens for Digital Signatures . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Privately Verifiable One-shot Signatures . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Quantum Retrieval Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 Witness Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 One-Shot Signatures and Applications 27
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Equivocal Collision Resistant Hash Functions . . . . . . . . . . . . . . . . . . . 34
3.3 One-shot Chameleon Hash Functions . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 One-shot Signatures and Budget Signatures . . . . . . . . . . . . . . . . . . . . 53
3.5 Quantum Lightning and Quantum Money . . . . . . . . . . . . . . . . . . . . 58
3.6 Ordered Signatures and Applications . . . . . . . . . . . . . . . . . . . . . . . 64

3.7 Proofs of Quantumness and Min-Entropy . . . . . . . . . . . . . . . . . . . . . 72
4 Stateful Oracle to Stateless Oracle Transformation 76
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 The transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Security Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Unclonable Decryption 86
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 New Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Secret-key Encryption with Honestly Generated Keys . . . . . . . . . . . . . . . 97
5.4 Public-key Encryption with Dishonestly Generated Keys . . . . . . . . . . . . . 101
5.5 Broadcast Encryption with Unclonable Decryption . . . . . . . . . . . . . . . . 104
5.6 Splittable Attribute-based Encryption . . . . . . . . . . . . . . . . . . . . . . . 109
5.7 Revocable Time-Released Encryption . . . . . . . . . . . . . . . . . . . . . . . 112
Appendix A Summary of Security Definitions 120
A.1 One-Shot Authentication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Single Decryptor Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
References 128