arXiv subjects

Nadav Kohen

Publications and source records attributed to Nadav Kohen.

6 recordsLinked to original sources

arXiv2026

Enabling Threshold Custody for the Lightning Network with Nested Threshold Multi-Signatures

The Bitcoin Lightning Network secures hundreds of millions of dollars, yet channel endpoints rely on vulnerable single online keys. Although threshold signatures are routinely used to protect on-chain Bitcoin, no practical deployment has been possible for Lightning channels. This is because thresholdizing a Lightning party requires nesting a threshold signature scheme inside of an established two-party MuSig2 protocol without altering its nonce exchange or message flow. In this work, we resolve this limitation by formalizing nested threshold multi-signatures, a new cryptographic primitive for thresholdizing one participant inside a multi-signature protocol. As an instance of this primitive, we present Iceberg, the first construction for nested threshold MuSig2 signatures. Iceberg enables one side of a Lightning channel to operate as a $t$-of-$n$ threshold group while appearing to the counterparty as a standard MuSig2 participant. As a result, threshold custody can be deployed unilaterally on today's Lightning Network without requiring any modifications to Bitcoin, the Lightning protocol, or channel counterparties. We prove the security of Iceberg, integrate a prototype into a production Lightning node, and benchmark its performance. Our measurements show that thresholdizing a Lightning channel incurs only modest overhead, since a threshold group tolerating one corrupted member sustains over $93\%$ of the payment throughput of an unmodified endpoint.

cs.CR↗

arXiv2025

A Linear Representation for Constant Term Sequences mod $p^a$ with Applications to Uniform Recurrence

Many integer sequences including the Catalan numbers, Motzkin numbers, and the Apr{\'e}y numbers can be expressed in the form ConstantTermOf$\left[P^nQ\right]$ for Laurent polynomials $P$ and $Q$. These are often called ``constant term sequences''. In this paper, we characterize the prime powers, $p^a$, for which sequences of this form modulo $p^a$, and others built out of these sequences, are uniformly recurrent. For all other prime powers, we show that the frequency of $0$ is $1$. This is accomplished by introducing a novel linear representation of constant term sequences modulo $p^a$, which is of independent interest.

math.CO↗

arXiv2025

Density and Symmetry in the Generalized Motzkin Numbers mod $p$

We give a formula for the density of $0$ in the sequence of generalized Motzkin numbers, $M^{a, b}_n$, modulo a prime, $p$, in terms of the first $p$ generalized central trinomial coefficients $T^{a, b}_n\bmod p$ (with $n<p$). We apply our method to various other sequences to obtain similar formulas. We also prove that $T^{a, b}_{p-1-n}\equiv (b^2-4a^2)^{\frac{p-1}{2}-n}T^{a, b}_n\pmod p$ to obtain tight lower bounds for the density of $0$ in our sequences. This symmetry of the first $p$ central trinomial coefficients mod $p$ also appears in a couple of other applications, including the proof of a novel symmetry of the first $p-2$ Motzkin numbers that is of independent interest: $M^{a, b}_{p-3-n}\equiv (b^2-4a^2)^{\frac{p-3}{2}-n}M^{a, b}_n\pmod p$.

math.CO↗

arXiv2024

Uniform Recurrence in the Motzkin Numbers and Related Sequences mod $p$

Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we characterize the primes for which sequences of this form are uniformly recurrent modulo $p$. For all other primes, we show that $0$ has density $1$. This will be accomplished by showing that the study of these sequences mod $p$ can be reduced to the study of the generalized central trinomial coefficients, which are well-behaved mod $p$.

math.CO↗

arXiv2020

A Projective Representation of the Modular Group

Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest non-commutative torus which remains fixed by all elements of the mapping class group of the torus, $SL_2(\mathbb{Z})$. Also known as the modular group. We use this representation to associate a matrix to each element of $SL_2(\mathbb{Z})$; we then compute the trace and determinant of the associated matrix.

math.GT↗

arXiv2020

Consecutive Radio Labeling of Hamming Graphs

For a graph $G$, a $k$-radio labeling of $G$ is the assignment of positive integers to the vertices of $G$ such that the closer two vertices are on the graph, the greater the difference in labels is required to be. Specifically, $\vert f(u)-f(v)\vert\geq k + 1 - d(u,v)$ where $f(u)$ is the label on a vertex $u$ in $G$. Here, we consider the case when $G$ is the Cartesian products of complete graphs. Specifically we wish to find optimal labelings that use consecutive integers and determine when this is possible. We build off of a paper by Amanda Niedzialomski and construct a framework for discovering consecutive radio labelings for Hamming Graphs, starting with the smallest unknown graph, $K_3^4$, for which we provide an optimal labeling using our construction.

math.CO↗