Paper 2026/2436

Threshold BARGs with Statistical Soundness: Weighted Threshold Encryption without Obfuscation

Abstract

Non-interactive batch arguments (BARGs) for $\mathsf{NP}$ enable proving the validity of $k$ $\mathsf{NP}$ statements with a proof whose size is independent of $k$. Threshold BARGs are a natural generalization that enables proving that at least $t$ out of $k$ statements are valid. Unfortunately, existing threshold BARG constructions have the drawback that the size of the common reference string (CRS) grows polynomially with $k$. This is the case even when the statements have a short description (e.g., when considering index languages). Moreover, other than constructions based on indistinguishability obfuscation ($i\mathcal{O}$), prior constructions could only achieve computational soundness. In this work, we construct the first threshold BARG with statistical soundness for an expressive class of trapdoor threshold languages from either circular-secure fully homomorphic encryption or from witness encryption in combination with the learning with errors assumption. Moreover and importantly for our applications, the size of the CRS only scales polylogarithmically with $k$. Next, we combine our threshold BARG with a new technique for sampling pseudorandom permutations in order to build weighted threshold signatures with a statistical unforgeability guarantee. Then, in combination with witness encryption, we obtain a weighted threshold encryption scheme that supports $\lambda$-bit weights. Decryption is entirely non-interactive in our setting. Previously, such a scheme was only known from $i\mathcal{O}$. If we additionally work in the random oracle model, then we obtain a weighted threshold encryption scheme that supports arbitrary weights with silent setup. In the silent setup model, users choose their own public/secret keys and no longer need to rely on a central authority to generate and issue decryption keys.