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1. A Modular Approach to Unclonable Cryptography

   Ananth, Prabhanjan; Behera, Amit (August 2024, Lecture notes in computer science)

   We explore a new pathway to designing unclonable cryptographic primitives. We propose a new notion called unclonable puncturable obfuscation (UPO) and study its implications for unclonable cryptography. Using UPO, we present modular (and in some cases, arguably, simple) constructions of many primitives in unclonable cryptography, including, public-key quantum money, quantum copy-protection for many classes of functionalities, unclonable encryption, and single-decryption encryption. Notably, we obtain the following new results assuming the existence of UPO: We show that any cryptographic functionality can be copy-protected as long as it satisfies a notion of security, which we term puncturable security. Prior feasibility results focused on copy-protecting specific cryptographic functionalities. We show that copy-protection exists for any class of evasive functions as long as the associated distribution satisfies a preimage-sampleability condition. Prior works demonstrated copy-protection for point functions, which follows as a special case of our result. We put forward two constructions of UPO. The first construction satisfies two notions of security based on the existence of (post-quantum) sub-exponentially secure indistinguishability obfuscation, injective one-way functions, the quantum hardness of learning with errors, and the two versions of a new conjecture called the simultaneous inner product conjecture. The security of the second construction is based on the existence of unclonable-indistinguishable bit encryption, injective one-way functions, and quantum-state indistinguishability obfuscation. 

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2. Succinct Arguments for QMA from Standard Assumptions via Compiled Nonlocal Games

   https://doi.org/10.1109/FOCS61266.2024.00078  

   Metger, Tony; Natarajan, Anand; Zhang, Tina (October 2024, IEEE)

   We construct a succinct classical argument system for QMA, the quantum analogue of NP, from generic and standard cryptographic assumptions. Previously, building on the prior work of Mahadev (FOCS '18), Bartusek et al. (CRYPTo ‘22) also constructed a succinct classical argument system for Q M A. However, their construction relied on post-quantumly secure indistinguishability obfuscation, a very strong primitive which is not known from standard cryptographic assumptions. In contrast, the primitives we use (namely, collapsing hash functions and a mild version of quantum homomorphic encryption) are much weaker and are implied by standard assumptions such as LWE. Our protocol is constructed using a general transformation which was designed by Kalai et al. (STOC '23) as a candidate method to compile any quantum nonlocal game into an argument system. Our main technical contribution is to analyze the soundness of this transformation when it is applied to a succinct self-test for Pauli measurements on maximally entangled states, the latter of which is a key component in the proof of MIP * = R E in Quantum complexity. 

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3. Towards Quantum One-Time Memories from Stateless Hardware

   https://doi.org/10.4230/lipics.tqc.2020.6  

   Broadbent, Anne; Gharibian, Sevag; Zhou, Hong-Sheng (January 2020, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Flammia, Steven T (Ed.)

   {"Abstract":["A central tenet of theoretical cryptography is the study of the minimal assumptions required to implement a given cryptographic primitive. One such primitive is the one-time memory (OTM), introduced by Goldwasser, Kalai, and Rothblum [CRYPTO 2008], which is a classical functionality modeled after a non-interactive 1-out-of-2 oblivious transfer, and which is complete for one-time classical and quantum programs. It is known that secure OTMs do not exist in the standard model in both the classical and quantum settings. Here, we propose a scheme for using quantum information, together with the assumption of stateless (i.e., reusable) hardware tokens, to build statistically secure OTMs. Via the semidefinite programming-based quantum games framework of Gutoski and Watrous [STOC 2007], we prove security for a malicious receiver, against a linear number of adaptive queries to the token, in the quantum universal composability framework, but leave open the question of security against a polynomial amount of queries. Compared to alternative schemes derived from the literature on quantum money, our scheme is technologically simple since it is of the "prepare-and-measure" type. We also show our scheme is "tight" according to two scenarios."]} 

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4. Public-Key Cryptography in the Fine-Grained Setting

   https://doi.org/10.1007/978-3-030-26954-8_20  

   LaVigne, R.; Lincoln, A.; Vassilevska Williams, V. (January 2019, Advances in Cryptology {\textendash} {CRYPTO} 2019)

   Cryptography is largely based on unproven assumptions, which, while believable, might fail. Notably if P=NP, or if we live in Pessiland, then all current cryptographic assumptions will be broken. A compelling question is if any interesting cryptography might exist in Pessiland. A natural approach to tackle this question is to base cryptography on an assumption from fine-grained complexity. Ball, Rosen, Sabin, and Vasudevan [BRSV’17] attempted this, starting from popular hardness assumptions, such as the Orthogonal Vectors (OV) Conjecture. They obtained problems that are hard on average, assuming that OV and other problems are hard in the worst case. They obtained proofs of work, and hoped to use their average-case hard problems to build a fine-grained one-way function. Unfortunately, they proved that constructing one using their approach would violate a popular hardness hypothesis. This motivates the search for other fine-grained average-case hard problems. The main goal of this paper is to identify sufficient properties for a fine-grained average-case assumption that imply cryptographic primitives such as fine-grained public key cryptography (PKC). Our main contribution is a novel construction of a cryptographic key exchange, together with the definition of a small number of relatively weak structural properties, such that if a computational problem satisfies them, our key exchange has provable fine-grained security guarantees, based on the hardness of this problem. We then show that a natural and plausible average-case assumption for the key problem Zero-k-Clique from fine-grained complexity satisfies our properties. We also develop fine-grained one-way functions and hardcore bits even under these weaker assumptions. Where previous works had to assume random oracles or the existence of strong one-way functions to get a key-exchange computable in O(n) time secure against O(n^2) time adversaries (see [Merkle’78] and [BGI’08]), our assumptions seem much weaker. Our key exchange has a similar gap between the computation of the honest party and the adversary as prior work, while being non-interactive, implying fine-grained PKC. 

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5. Pseudorandom Isometries

   Ananth, Prabhanjan; Gulati, Aditya; Kaleoglu, Fatih; Lin, Yao-Ting (April 2024, Springer Nature)

   We introduce a new notion called Q-secure pseudorandom isometries (PRI). A pseudorandom isometry is an efficient quantum circuit that maps an n-qubit state to an (n+m)-qubit state in an isometric manner. In terms of security, we require that the output of a q-fold PRI on \rho, for \rho \in Q, for any polynomial q, should be computationally indistinguishable from the output of a q-fold Haar isometry on \rho. By fine-tuning Q, we recover many existing notions of pseudorandomness. We present a construction of PRIs and assuming post-quantum one-way functions, we prove the security of Q-secure pseudorandom isometries (PRI) for different interesting settings of Q. We also demonstrate many cryptographic applications of PRIs, including, length extension theorems for quantum pseudorandomness notions, message authentication schemes for quantum states, multi-copy secure public and private encryption schemes, and succinct quantum commitments. 

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