Batch arguments (BARGs) are non-interactive arguments for conjunctions of NP statements, with proof size that is sublinear in the number of statements. Several previous works studied the communication complexity of BARGs, focusing both on the CRS size and on the additive overhead of the proof, defined as the difference between the proof size and the size $m$ of a single NP witness: - Devadas et al.~[FOCS 22] constructed BARGs with additive overhead that is independent of $m$, however, their CRS size is polynomial in $m$. - Paneth and Pass [FOCS 22] constructed BARGs where the CRS size is independent of $m$, but with higher additive overhead $m^{1-\epsilon}$. Under the hardness of LWE, we construct BARGs where both the CRS size the additive overhead of the proof are independent of $m$. Such BARGs can be recursively composed an unbounded polynomial number of times without losing succinctness. Along the way, we also considerably simplify the construction of fully local somewhere extractable hash functions used in the construction of Devadas et al.

We study the design of Oblivious RAMs (ORAMs) that allow a client to access memory outsourced to a remote, untrusted server without revealing the client’s data access pattern. We are interested in concretely efficient constructions and prior works have yielded different ORAM frameworks with various trade-offs. Tree-based constructions such as RingORAM [Ren et al., USENIX’15] obtain low communication overhead, but require client storage of linear position maps and two roundtrip queries. Hierarchical schemes such as FutORAMa [Asharov et al., CCS’23] further reduce communication at the cost of more roundtrips during queries. Finally, SQRT-ORAM [Goldreich, STOC ’87] enables fast queries of one roundtrip and one block of communication at the cost of larger amortized communication costs. We present two new constructions, LatORAM and Lat 2 ORAM, that simultaneously obtain the positive traits of all three types of ORAM constructions. Online queries are blazing fast with one roundtrip and a single block of communication like SQRT-ORAM. Fixing the client memory sizes for comparison, the online communication cost of our constructions are 5-8x smaller than RingORAM and 5-10x smaller than FutORAMa even though both RingORAM and FutORAM a require multiple roundtrips per online query. Furthermore, our total amortized communication is also up to 50% smaller. To obtain our constructions, we present a new lazy approach of lateral stash growth that delays large shuffles. Of independent interest, we present improved oblivious merging schemes for specific settings important for our ORAMs. Our constructions solely rely on symmetric cryptography.

We study a new pairing, beyond the Weil and Tate pairing. The Weil pairing is a non-degenerate pairing $E[m] \times E[m] \to \mu_{m}$, which operates on the kernel of $[m]$. Similarly, when $\mu_{m} \subseteq \mathbb{F}_q^*$, the Tate pairing is a non-degenerate pairing $E[m](\mathbb{F}_q) \times E(\mathbb{F}_q) / [m]E(\mathbb{F}_q) \to \mu_{m}$, which connects the kernel and the rational cokernel of $[m]$. We define a pairing \[ \langle{\quad}\rangle_m : E(\mathbb{F}_q) / [m]E(\mathbb{F}_q) \times E(\mathbb{F}_q) / [m]E(\mathbb{F}_q) \to \mu_{m}\] on the rational cokernels of $[m]$, filling the gap left by the Weil and Tate pairing. When $E[m] \subseteq E(\mathbb{F}_q)$, this pairing is non-degenerate, and can be computed using three Tate pairings, and two discrete logarithms in $\mu_{m}$, assuming a basis for $E[m]$. For $m = \ell$ prime, this pairing allows us to study $E(\mathbb{F}_q) / [\ell]E(\mathbb{F}_q)$ directly and to simplify the computation for a basis of $E[\ell^k]$, and more generally the Sylow $\ell$-torsion. This finds natural applications in isogeny-based cryptography when computing $\ell^k$-isogenies.