129 research outputs found
Publicly-Verifiable Deletion via Target-Collapsing Functions
We build quantum cryptosystems that support publicly-verifiable deletion from
standard cryptographic assumptions. We introduce target-collapsing as a
weakening of collapsing for hash functions, analogous to how second preimage
resistance weakens collision resistance; that is, target-collapsing requires
indistinguishability between superpositions and mixtures of preimages of an
honestly sampled image.
We show that target-collapsing hashes enable publicly-verifiable deletion
(PVD), proving conjectures from [Poremba, ITCS'23] and demonstrating that the
Dual-Regev encryption (and corresponding fully homomorphic encryption) schemes
support PVD under the LWE assumption. We further build on this framework to
obtain a variety of primitives supporting publicly-verifiable deletion from
weak cryptographic assumptions, including:
- Commitments with PVD assuming the existence of injective one-way functions,
or more generally, almost-regular one-way functions. Along the way, we
demonstrate that (variants of) target-collapsing hashes can be built from
almost-regular one-way functions.
- Public-key encryption with PVD assuming trapdoored variants of injective
(or almost-regular) one-way functions. We also demonstrate that the encryption
scheme of [Hhan, Morimae, and Yamakawa, Eurocrypt'23] based on pseudorandom
group actions has PVD.
- with PVD for attribute-based encryption, quantum
fully-homomorphic encryption, witness encryption, time-revocable
encryption, assuming and trapdoored variants of injective (or
almost-regular) one-way functions.Comment: 52 page
Worms and Spiders: Reflection calculi and ordinal notation systems
We give a general overview of ordinal notation systems arising from
reflection calculi, and extend the to represent impredicative ordinals up to
those representable using Buchholz-style collapsing functions
Publicly-Verifiable Deletion via Target-Collapsing Functions
We build quantum cryptosystems that support publicly-verifiable deletion from standard cryptographic assumptions. We introduce target-collapsing as a weakening of collapsing for hash functions, analogous to how second preimage resistance weakens collision resistance; that is, target-collapsing requires indistinguishability between superpositions and mixtures of preimages of an honestly sampled image.
We show that target-collapsing hashes enable publicly-verifiable deletion (PVD), proving conjectures from [Poremba, ITCS\u2723] and demonstrating that the Dual-Regev encryption (and corresponding fully homomorphic encryption) schemes support PVD under the LWE assumption. We further build on this framework to obtain a variety of primitives supporting publicly-verifiable deletion from weak cryptographic assumptions, including:
- Commitments with PVD assuming the existence of injective one-way functions, or more generally, almost-regular one-way functions. Along the way, we demonstrate that (variants of) target-collapsing hashes can be built from almost-regular one-way functions.
- Public-key encryption with PVD assuming trapdoored variants of injective (or almost-regular) one-way functions. We also demonstrate that the encryption scheme of [Hhan, Morimae, and Yamakawa, Eurocrypt\u2723] based on pseudorandom group actions has PVD.
- with PVD for attribute-based encryption, quantum fully-homomorphic encryption, witness encryption, time-revocable encryption, assuming and trapdoored variants of injective (or almost-regular) one-way functions
Connecting the two worlds: well-partial-orders and ordinal notation systems
Kruskal claims in his now-classical 1972 paper [47] that well-partial-orders are among the most frequently rediscovered mathematical objects. Well partial-orders have applications in many fields outside the theory of orders: computer science, proof theory, reverse mathematics, algebra, combinatorics, etc.
The maximal order type of a well-partial-order characterizes that order’s strength. Moreover, in many natural cases, a well-partial-order’s maximal order type can be represented by an ordinal notation system. However, there are a number of natural well-partial-orders whose maximal order types and corresponding ordinal notation systems remain unknown. Prominent examples are Friedman’s well-partial-orders of trees with the gap-embeddability relation [76].
The main goal of this dissertation is to investigate a conjecture of Weiermann [86], thereby addressing the problem of the unknown maximal order types and corresponding ordinal notation systems for Friedman’s well-partial orders [76]. Weiermann’s conjecture concerns a class of structures, a typical member of which is denoted by T (W ), each are ordered by a certain gapembeddability relation. The conjecture indicates a possible approach towards determining the maximal order types of the structures T (W ). Specifically, Weiermann conjectures that the collapsing functions #i correspond to maximal linear extensions of these well-partial-orders T (W ), hence also that these collapsing functions correspond to maximal linear extensions of Friedman’s famous well-partial-orders
A Categorical Construction of Bachmann-Howard Fixed Points
Peter Aczel has given a categorical construction for fixed points of normal
functors, i.e. dilators which preserve initial segments. For a general dilator
we cannot expect to obtain a well-founded fixed point, as the
order type of may always exceed the order type of . In the present
paper we show how to construct a Bachmann-Howard fixed point of , i.e. an
order with an "almost" order preserving collapse
. Building
on previous work, we show that -comprehension is equivalent to the
assertion that is well-founded for any dilator .Comment: This version has been accepted for publication in the Bulletin of the
London Mathematical Societ
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