Time Space Tradeoffs

Tradeoffs between offline precomputation and online query time for a cryptanalytic search task.

Status	Statement	Tags
	Balanced Root-n Preprocessing PIR The client-space-times-bandwidth product is settled up to n^{o(1)} by the source’s own matching bounds. What is open is whether client space, bandwidth and per-query server computation can all sit at O~(n^{1/2}) simultaneously; the best known point has client space and server computation O~(n^{2/3}). 4 open	Private Information Retrieval Time Space Tradeoffstight-bound
	Two-block sponge attack, quadratic advice Open: whether the multi-instance two-block sponge attack of advantage (S^2T4/C²⁾S lifts to an auxiliary-input attack of advantage S^2T4/C^2, matching the best proved security bound. Settled in the multi-instance model; no partial result in the auxiliary-input model. 7 open	Collision Finding Collision Resistant Hashing Time Space Tradeoffspromlower-boundadaptation (ai)
	Quantum two-block MD collisions Open, in both directions and with no partial result: whether the best known quantum preprocessing attack, of advantage ST^2/N + T^3/N, is optimal for two-block Merkle-Damgård collisions. The source reports that no matching quantum bounds are known at any block length. 7 open	Collision Finding Collision Resistant Hashing Quantum Query Complexity Quantum Random Oracle Model Time Space Tradeoffsqromtight-bound
	CSAT with Small Space and Preprocessing Conjecture 1.2 of the source, a parameterized family rather than a single claim. Corollary 1.3 turns each parameter setting into a limitation on succinct IOPs. The source attaches decreasing confidence as the class T grows, calling the largest setting only (arguably) unlikely to be false rather than confidently believed. 4 open	Interactive Oracle Proofs Proof Size Lower Bounds Space Bounded Computation Time Space Tradeoffslower-boundbarrier (ai)
	r-round DLOG tradeoff Open for every intermediate r: the r = 1 endpoint is the source’s own theorem and r = T is Corrigan-Gibbs–Kogan, with nothing proved in between. The source conjectures the interpolating formula and says a matching attack exists at every r, so the missing half is the lower bound. 6 open	Adaptivity Discrete Logarithm Generic Group Model Time Space Tradeoffsggmtight-bound
	3-way collision, oblivious sequential curve Open: closing the factor-root-S gap between the proved oblivious lower bound and a matching table-and-hunt algorithm for finding a 3-way collision with an oblivious sequential branching program. 2 open	Collision Finding Random Oracle Model Time Space Tradeoffsromtight-boundadaptation (ai)
	Non-adaptive DDH bound Open, and asserted rather than asked: the source proves 1/2 + O~(T^2/N + sqrt(ST/N)), states twice that it conjectures this is not tight for DDH, and names 1/2 + O~(T^2/N + ST/N) as the right answer. The same theorem’s square-DDH bound is sharp, with a matching attack, which is what makes the DDH case a question rather than a suspicion. 6 open	Adaptivity Discrete Logarithm Generic Group Model Time Space Tradeoffsggmtight-boundadaptation (ai)
	STB conjecture (Merkle-Damgård) Open for non-constant B in the regime ST^2 > 2^n, where the best published security bound and the best published attack are a factor of up to S apart; settled at B=1, B=2, every constant B, B~T, and every 2<B<T with ST^2 <= 2^n. 6 open	Collision Finding Collision Resistant Hashing Random Oracle Model Time Space Tradeoffsromtight-boundbarrier (ai)
	STB conjecture (sponge) Open for every B >= 2 except B ~ T: at B=2 in the regime ST^3 > C, and at B >= 3 with a gap of about T/B. The source asks for a proof or a refutation and takes no position. 7 open	Collision Finding Collision Resistant Hashing Time Space Tradeoffspromtight-boundbarrier (ai)
	Tight \(k\)-collision time-space tradeoff The exact tight time-space tradeoff for k>=3 collisions under preprocessing is open; only the k=2 case has a matching upper and lower bound. 2 open	Collision Finding Random Oracle Model Time Space Tradeoffsromtight-bound