A lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity is proved.
Abstract
We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $\Omega(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity. This leads to an optimal refutation of the Alon-Saks-Seymour conjecture, as well as an optimal communication lower bound for the Clique versus Independent Set problem, improving the previous results of Balodis, Ben-David, G\"{o}\"{o}s, Jain and Kothari (FOCS 2021, SICOMP 2023) by several doubly logarithmic factors. As further applications of our construction to query complexity and learning theory, we exhibit: (a) a family of Boolean functions that has an optimal quartic separation between certificate complexity and approximate degree, and (b) a sample compression lower bound of $\Omega(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.
We study the fundamental problem of implementing $m$ linearizable LL/SC objects with constant expected step complexity in a system of $n$ processes, using bounded base objects commonly available in hardware. Assuming that each process may have at most $\tau$ outstanding LL() operations, the best known deterministic algorithm requires $\Omega(n^2\tau + m)$ base objects (CAS and registers) [Blelloch and Wei, DISC 2020]. Previously, no comparable randomized algorithm was known. By employing randomization and FADD in addition to CAS and registers, we obtain a space bound of $O(n\tau+m)$ against the weak adaptive adversary. For $m=O(1)$ this matches a lower bound for algorithms using CAS and registers [Aghazadeh and Woelfel, PODC 2015]. In addition, our object can be employed by quiescently history-independent (QHI) algorithms: Whenever no operation on the object is pending and no process has an outstanding LL() operation, its internal memory state is uniquely determined by the values of the $m$ LL/SC objects. An important application is a recent QHI dynamic hashing algorithm, which uses $\Theta(m)$ hardware LL/SC objects to maintain a hash table of size $m$ [Attiya, Bender, Farach-Colton, Oshman, and Schiller, STOC 2025]. But LL/SC is not available in hardware, and prior to our work no wait-free or efficient lock-free software implementation of LL/SC with similar properties was known. Our work demonstrates that one can actually implement the hashing algorithm on available hardware, without an asymptotic increase in step and space complexity, under the reasonable assumption that $m=\Omega(n)$.
D. Bencivenga, Homa Habashi, Philipp Woelfel· 0 citations
We define and study the Structured Totient Preimage (STP) problem as a restricted reconstruction relation with a direct cryptographic motivation. Let $p_1,\ldots,p_k$ be distinct primes of the same bit length and reveal only $x=\prod_{i=1}^k(p_i-1)$. Given $(x,\lambda,k)$, STP asks for any set of $k$ distinct $\lambda$-bit primes satisfying this product. The relation is efficiently verifiable, but its reconstruction complexity is not known. We establish three concrete results. First, for factored $x$ we derive the exact number of ordered exponent allocations and a bound showing that direct reconstruction is polynomial for fixed $k$ when $\Omega(x)=O(\log\lambda)$; this rules out that regime as a basis for a strong hardness claim. Second, we give exhaustive algorithms for reconstruction and collision analysis. Third, we exhaustively evaluate 28 parameter pairs, with $2\leq k\leq5$, up to $\lambda=16$ for pairs and 4,588,935 prime sets in the largest census. The data quantify non-injectivity through collision participation, maximum multiplicity, and conditional ambiguity in bits. These results isolate STP from general inverse-totient computation and motivate a Structured Totient Preimage Assumption for explicitly growing parameter families. Under such an assumption, STP becomes a candidate preimage-resistant relation whose implications for commitments, proofs of knowledge of multiplicative witnesses, and authentication can be stated precisely. The paper establishes the computational foundation and parameter constraints for those constructions; it does not claim a security reduction or post-quantum hardness.
The usefulness of one-sided-error randomized reductions is demonstrated by showing that they can be conditionally derandomized when the target problem has an OR function, and a general theorem formalizing this derandomization is proved.
We consider the following problems: Given two $n \times n$ tables defining binary operations $+$ and $\cdot$ on a set $S$ of $n$ elements, decide whether $(S,+,\cdot)$ forms a ring or, respectively, a field. Recently, Dudek, Fischer, Gokaj, Jin, K\"unnemann, Mao, and Redzic (STOC 2026) obtained the following two (near-)optimal results: (1) A randomized $O(n^2\log(1/\delta))$-time algorithm for verifying rings. (2) A deterministic $O(n^2)$-time algorithm for verifying fields. Their algorithms build on machinery of Evra, Gadot, Klein, and Komargodski (FOCS 2024), which relies on Classification of Finite Simple Groups (CFSG). In this work, we give a deterministic $O(n^2)$-time algorithm for ring verification, resolving the deterministic complexity of this problem. As a corollary, we also obtain a deterministic $O(n^2)$-time algorithm for field verification. Our algorithms are elementary and avoid CFSG machinery entirely.
We show that, on trees, any locally checkable labeling problem (LCL) $\Pi$ that can be solved by an $n^{o(1)}$-dependent distribution can also be solved by an $O(\log n)$-round deterministic LOCAL algorithm. The result is obtained through a rake-and-compress-style decomposition of the input tree, and local simulations of the bounded dependent distribution on the components of the decomposition. As a corollary to our result, any LCL problem on trees can either be solved by an $O(\log n)$ deterministic LOCAL algorithm, or requires $n^{\Omega(1)}$ rounds to solve by a quantum-LOCAL algorithm.
We prove that certificate complexity can be quartically larger than approximate degree. More precisely, we construct a family of total Boolean functions $G$ with $$ C(G) = \tilde{\Omega}(\tilde{deg}(G)^4), $$ where $C$ denotes certificate complexity and $\tilde{deg}$ denotes $1/3$-approximate degree. This is optimal up to polylogarithmic factors, since every total Boolean function $f$ satisfies $C(f)\le O(\tilde{deg}(f)^4)$ by the classical block-sensitivity bounds of Nisan and Nisan--Szegedy. Thus the result closes the gap between these two measures and improves the previously best known separation $C(f)=\tilde{\Omega}(\tilde{deg}(f)^3)$ by Balodis, Ben-David, G\"o\"os, Jain, and Kothari. The construction starts from the partial function they used to quadratically separate $0$-certificate complexity from unambiguous $1$-certificate complexity. It already has the required certificate hardness, but its $0$-certificates are unstructured, which blocks the derivation of a low-degree verifier. We keep its $1$-condition and restrict the $0$-inputs to those certified by a structured family whose validity admits a low-degree approximant, while preserving the quadratic hardness. The partial function with its low-degree verifier is then fed through the cheat-sheet framework to yield the total function $G$ with the claimed separation. The main technical ingredient is an approximate polynomial that verifies the certificate in degree $\tilde{O}(\sqrt n)$. The verifier forms a low-degree count $W$ of the candidate $1$-certificates that remain compatible with the asserted $0$-certificate, and tests whether this count is zero. Crucially, the construction ensures that $W$ never exceeds $\tilde{O}(n)$, instead of the $\Theta(n^2)$ candidate pairs it counts bringing the verification down to degree $\tilde{O}(\sqrt n)$.
K. Balodis· 0 citations
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