Constrained PRFs for Bit-fixing (and More) from OWFs with Adaptive Security and Constant Collusion Resistance

Constrained pseudorandom functions (CPRFs) allow learning constrained PRF keys that can evaluate the PRF on a subset of the input space, or based on some sort of predicate. First introduced by Boneh and Waters [AC\u2713], Kiayias et al. [CCS\u2713] and Boyle et al. [PKC\u2714], they have been shown to be a useful cryptographic primitive with many applications. The full security definition of CPRFs requires the adversary to learn multiple constrained keys in an arbitrary order, a requirement for many of these applications. Unfortunately, existing constructions of CPRFs satisfying this security notion are only known from exceptionally strong cryptographic assumptions, such as indistinguishability obfuscation (IO) and the existence of multilinear maps, even for very weak constraints. CPRFs from more standard assumptions only satisfy selective security for a single constrained key query. In this work, we give the first construction of a CPRF that can adaptively issue a constant number of constrained keys for bit-fixing predicates (or more generally $t$-conjunctive normal form predicates), only requiring the existence of one-way functions (OWFs). This is a much weaker assumption compared with all previous constructions. In addition, we prove that the new scheme satisfies 1-key privacy (otherwise known as constraint-hiding). This is the only construction for any non-trivial predicates to achieve adaptive security and collusion-resistance outside of the random oracle model or relying on strong cryptographic assumptions. Our technique represents a noted departure from existing CPRF constructions

Tighter Security Proofs for GPV-IBE in the Quantum Random Oracle Model

In (STOC, 2008), Gentry, Peikert, and Vaikuntanathan proposed the first identity-based encryption (GPV-IBE) scheme based on a post-quantum assumption, namely, the learning with errors (LWE) assumption. Since their proof was only made in the random oracle model (ROM) instead of the quantum random oracle model (QROM), it remained unclear whether the scheme was truly post-quantum or not. In (CRYPTO, 2012), Zhandry developed new techniques to be used in the QROM and proved the security of GPV-IBE in the QROM, hence answering in the affirmative that GPV-IBE is indeed post-quantum. However, since the general technique developed by Zhandry incurred a large reduction loss, there was a wide gap between the concrete efficiency and security level provided by GPV-IBE in the ROM and QROM. Furthermore, regardless of being in the ROM or QROM, GPV-IBE is not known to have a tight reduction in the multi-challenge setting. Considering that in the real-world an adversary can obtain many ciphertexts, it is desirable to have a security proof that does not degrade with the number of challenge ciphertext. In this paper, we provide a much tighter proof for the GPV-IBE in the QROM in the single-challenge setting. In addition, we also show that a slight variant of the GPV-IBE has an almost tight reduction in the multi-challenge setting both in the ROM and QROM, where the reduction loss is independent of the number of challenge ciphertext. Our proof departs from the traditional partitioning technique and resembles the approach used in the public key encryption scheme of Cramer and Shoup (CRYPTO, 1998). Our proof strategy allows the reduction algorithm to program the random oracle the same way for all identities and naturally fits the QROM setting where an adversary may query a superposition of all identities in one random oracle query. Notably, our proofs are much simpler than the one by Zhandry and conceptually much easier to follow for cryptographers not familiar with quantum computation. Although at a high level, the techniques used for the single and multi-challenge setting are similar, the technical details are quite different. For the multi-challenge setting, we rely on the Katz-Wang technique (CCS, 2003) to overcome some obstacles regarding the leftover hash lemma

Private Puncturable PRFs From Standard Lattice Assumptions

A puncturable pseudorandom function (PRF) has a master key $k$ that enables one to evaluate the PRF at all points of the domain, and has a punctured key $k_x$ that enables one to evaluate the PRF at all points but one. The punctured key $k_x$ reveals no information about the value of the PRF at the punctured point $x$. Punctured PRFs play an important role in cryptography, especially in applications of indistinguishability obfuscation. However, in previous constructions, the punctured key $k_x$ completely reveals the punctured point $x$: given $k_x$ it is easy to determine $x$. A {\em private} puncturable PRF is one where $k_x$ reveals nothing about~$x$. This concept was defined by Boneh, Lewi, and Wu, who showed the usefulness of private puncturing, and gave constructions based on multilinear maps. The question is whether private puncturing can be built from a standard (weaker) cryptographic assumption. We construct the first privately puncturable PRF from standard lattice assumptions, namely from the hardness of learning with errors (LWE) and 1 dimensional short integer solutions (1D-SIS), which have connections to worst-case hardness of general lattice problems. Our starting point is the (non-private) PRF of Brakerski and Vaikuntanathan. We introduce a number of new techniques to enhance this PRF, from which we obtain a privately puncturable PRF. In addition, we also study the simulation based definition of private constrained PRFs for general circuits, and show that the definition is not satisfiable

A Bit-fixing PRF with O(1) Collusion-Resistance from LWE

Constrained pseudorandom functions (CPRFs) allow learning modified PRF keys that can evaluate the PRF on a subset of the input space, or based on some sort of predicate. First introduced by Boneh and Waters [Asiacrypt 2013], they have been shown to be a useful cryptographic primitive with many applications. The full security definition of CPRFs requires the adversary to learn multiple constrained keys, a requirement for all of these applications. Unfortunately, existing constructions of CPRFs satisfying this security notion are only known from exceptionally strong cryptographic assumptions, such as indistinguishability obfuscation and the existence of multilinear maps, even for very weak predicates. CPRFs from more standard assumptions only satisfy security when one key is learnt. In this work, we give the first construction of a CPRF that can issue a constant number of constrained keys for bit-fixing predicates, from learning with errors (LWE). It also satisfies $1$-key privacy (otherwise known as constraint-hiding). Finally, our construction achieves fully adaptive security with polynomial security loss; the only construction to achieve such security under a standard assumption. Our technique represents a noted departure existing for CPRF constructions. We hope that it may lead to future constructions that can expose a greater number of keys, or consider more expressive predicates (such as circuit-based constraints)

Post-Quantum Insecurity from LWE

We show that for many fundamental cryptographic primitives, proving classical security under the learning-with-errors (LWE) assumption, does not imply post-quantum security. This is despite the fact that LWE is widely believed to be post-quantum secure, and our work does not give any evidence otherwise. Instead, it shows that post-quantum insecurity can arise inside cryptographic constructions, even if the assumptions are post-quantum secure. Concretely, our work provides (contrived) constructions of pseudorandom functions, CPA-secure symmetric-key encryption, message-authentication codes, signatures, and CCA-secure public-key encryption schemes, all of which are proven to be classically secure under LWE via black-box reductions, but demonstrably fail to be post-quantum secure. All of these cryptosystems are stateless and non-interactive, but their security is defined via an interactive game that allows the attacker to make oracle queries to the cryptosystem. The polynomial-time quantum attacker can break these schemes by only making a few classical queries to the cryptosystem, and in some cases, a single query suffices. Previously, we only had examples of post-quantum insecurity under post-quantum assumptions for stateful/interactive protocols. Moreover, there appears to be a folklore belief that for stateless/non-interactive cryptosystems with black-box proofs of security, a quantum attack against the scheme should translate into a quantum attack on the assumption. This work shows otherwise. Our main technique is to carefully embed interactive protocols inside the interactive security games of the above primitives. As a result of independent interest, we also show a 3-round quantum disclosure of secrets (QDS) protocol between a classical sender and a receiver, where a quantum receiver learns a secret message in the third round but, assuming LWE, a classical receiver does not

Privately Puncturing PRFs from Lattices: Adaptive Security and Collusion Resistant Pseudorandomness

A private puncturable pseudorandom function (PRF) enables one to create a constrained version of a PRF key, which can be used to evaluate the PRF at all but some punctured points. In addition, the constrained key reveals no information about the punctured points and the PRF values on them. Existing constructions of private puncturable PRFs are only proven to be secure against a restricted adversary that must commit to the punctured points before viewing any information. It is an open problem to achieve the more natural adaptive security, where the adversary can make all its choices on-the-fly. In this work, we solve the problem by constructing an adaptively secure private puncturable PRF from standard lattice assumptions. To achieve this goal, we present a new primitive called explainable hash, which allows one to reprogram the hash function on a given input. The new primitive may find further applications in constructing more cryptographic schemes with adaptive security. Besides, our construction has collusion resistant pseudorandomness, which requires that even given multiple constrained keys, no one could learn the values of the PRF at the punctured points. Private puncturable PRFs with collusion resistant pseudorandomness were only known from multilinear maps or indistinguishability obfuscations in previous works, and we provide the first solution from standard lattice assumptions