General Multi-Signatures and the Subset-f Problem Sample Clauses

General Multi-Signatures and the Subset-f Problem. ‌ Although the proof of Lemma 6.5 depends on the specific LOSSW multi-signature scheme, the overall approach only depends on certain properties of that scheme; in particular, there is no inherent reliance on the structure of multiplication of keys and Subset-Product. Motivated by this observation, in this section, we start by exploring hardness of Subset-ƒ problems for a more general class of functions ƒ , focusing on the class of elementary symmetric polynomials φl. We begin by demonstrating (worst-case) NP-hardness results for Subset-φl. We then abstract out the properties used in Lemma 6.5 (deemed “SNARG-compliance”), and show that existence of SRDS based on a SNARG-compliant multi-signature scheme implies existence of SNARGs for corresponding Subset- φl problems. The Subset-ƒ problem. We first define the following analogous variant of average-case Subset- Product problem for more general functions ƒ . We restrict our attention to the natural setting of symmetric functions ƒ ; one can extend to arbitrary ƒ , e.g., given a canonical ordering of inputs. | | L∈ ⊆ → ∈ Definition 6.6 (Average-Case Subset-ƒ ). Let s = s(n) be an integer, let R be a ring, and let ƒ : Rs R an efficiently computable symmetric function. Given an instance x = (a1, . . . , an, t) Rn+1, the (s, R)-Subset-ƒ problem is the problem of deciding if there exists a subset S [n] of size S = s, such that ƒ ((ai)i S) = t. Such instances are said to be in the (s, R)-Subset-ƒ language f . We consider the average-case version of this problem characterized by the following two distri- butions:
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