International Association for Cryptologic Research

International Association
for Cryptologic Research


Hart Montgomery

Affiliation: Fujitsu Laboratories of America


Minicrypt Primitives with Algebraic Structure and Applications 📺
Algebraic structure lies at the heart of Cryptomania as we know it. An interesting question is the following: instead of building (Cryptomania) primitives from concrete assumptions, can we build them from simple Minicrypt primitives endowed with some additional algebraic structure? In this work, we affirmatively answer this question by adding algebraic structure to the following Minicrypt primitives:One-Way Function (OWF)Weak Unpredictable Function (wUF)Weak Pseudorandom Function (wPRF) The algebraic structure that we consider is group homomorphism over the input/output spaces of these primitives. We also consider a “bounded” notion of homomorphism where the primitive only supports an a priori bounded number of homomorphic operations in order to capture lattice-based and other “noisy” assumptions. We show that these structured primitives can be used to construct many cryptographic protocols. In particular, we prove that: (Bounded) Homomorphic OWFs (HOWFs) imply collision-resistant hash functions, Schnorr-style signatures and chameleon hash functions.(Bounded) Input-Homomorphic weak UFs (IHwUFs) imply CPA-secure PKE, non-interactive key exchange, trapdoor functions, blind batch encryption (which implies anonymous IBE, KDM-secure and leakage-resilient PKE), CCA2 deterministic PKE, and hinting PRGs (which in turn imply transformation of CPA to CCA security for ABE/1-sided PE).(Bounded) Input-Homomorphic weak PRFs (IHwPRFs) imply PIR, lossy trapdoor functions, OT and MPC (in the plain model). In addition, we show how to realize any CDH/DDH-based protocol with certain properties in a generic manner using IHwUFs/IHwPRFs, and how to instantiate such a protocol from many concrete assumptions.We also consider primitives with substantially richer structure, namely Ring IHwPRFs and L-composable IHwPRFs. In particular, we show the following: Ring IHwPRFs with certain properties imply FHE.2-composable IHwPRFs imply (black-box) IBE, and L-composable IHwPRFs imply non-interactive $$(L+1)$$ (L+1)-party key exchange. Our framework allows us to categorize many cryptographic protocols based on which structured Minicrypt primitive implies them. In addition, it potentially makes showing the existence of many cryptosystems from novel assumptions substantially easier in the future.
Symmetric Primitives with Structured Secrets 📺
Securely managing encrypted data on an untrusted party is a challenging problem that has motivated the study of a wide variety of cryptographic primitives. A special class of such primitives allows an untrusted party to transform a ciphertext encrypted under one key to a ciphertext under another key, using some auxiliary information that does not leak the underlying data. Prominent examples of such primitives in the symmetric setting are key-homomorphic (weak) PRFs, updatable encryption, and proxy re-encryption. Although these primitives differ significantly in terms of their constructions and security requirements, they share two important properties: (a) they have secrets with structure or extra functionality, and (b) all known constructions of these primitives satisfying reasonably strong definitions of security are based on concrete public-key assumptions, e.g., DDH and LWE. This raises the question of whether these objects inherently belong to the world of public-key primitives, or they can potentially be built from simple symmetric-key objects such as pseudorandom functions. In this work, we show that the latter possibility is unlikely. More specifically, we show that:Any (bounded) key-homomorphic weak PRF with an abelian output group implies a (bounded) input-homomorphic weak PRF, which has recently been shown to imply not only public-key encryption  but also a variety of primitives such as PIR, lossy TDFs, and even IBE.Any ciphertext-independent updatable encryption scheme that is forward and post-compromise secure implies PKE. Moreover, any symmetric-key proxy re-encryption scheme with reasonably strong security guarantees implies a forward and post-compromise secure ciphertext-independent updatable encryption, and hence PKE. In addition, we show that unbounded (or exact) key-homomorphic weak PRFs over abelian groups are impossible in the quantum world. In other words, over abelian groups, bounded key-homomorphism is the best that we can hope for in terms of post-quantum security. Our attack also works over other structured primitives with abelian groups and exact homomorphisms, including homomorphic one-way functions and input-homomorphic weak PRFs.
Algebraic Pseudorandom Functions with Improved Efficiency from the Augmented Cascade
Dan Boneh Hart Montgomery Ananth Raghunathan
We construct an algebraic pseudorandom function (PRF) that is more efficient than the classic Naor- Reingold algebraic PRF. Our PRF is the result of adapting the cascade construction, which is the basis of HMAC, to the algebraic settings. To do so we define an augmented cascade and prove it secure when the underlying PRF satisfies a property called parallel security. We then use the augmented cascade to build new algebraic PRFs. The algebraic structure of our PRF leads to an efficient large-domain Verifiable Random Function (VRF) and a large-domain simulatable VRF.