## Cryptology ePrint Archive: Report 2021/698

Multi-Dimensional Sub/Super-Range Signatures

Masahito Ishizaka and Shinsaku Kiyomoto

Abstract: In time-specific signatures (TSS) [Paterson \& Quaglia, SCN'10] [Ishizaka \& Kiyomoto, ISC'20] with $T$ numerical values, each signer is given a secret-key associated with a numerical value $t\in[0,T-1]$ and each signature on a message is generated under a numerical range $[L,R]$ s.t. $0\leq L\leq R\leq T-1$. A signer with $t$ can correctly generate a signature under $[L,R]$ if $t$ is truly included in $[L,R]$, i.e., $t\in[L,R]$.

As a generalized primitive of TSS, we propose multi-dimensional \textit{sub}-range signatures (MDSBRS). As a related primitive, we also propose multi-dimensional \textit{super}-range signatures (MDSPRS). In MDSBRS (resp. MDSPRS) with $D\in\mathbb{N}$ dimensions, each secret-key is associated with a set of $D$ ranges $\{[l_i,r_i]\mid i\in[1,D]\}$ s.t. $0 \leq l_i\leq r_i\leq T_i-1$ and a threshold value $d\in[1,D]$, and it correctly produces a signature on any message under a set of $D$ ranges $\{[L_i,R_i]\mid i\in[1,D]\}$ s.t. $0 \leq L_i\leq R_i\leq T_i-1$, if and only if total number of key-ranges every one $[l_i,r_i]$ of which is a \textit{sub}-range (resp. \textit{super}-range) of the corresponded signature-range $[L_i,R_i]$, i.e., $L_i\leq l_i\leq r_i\leq R_i$ (resp. $l_i\leq L_i\leq R_i\leq r_i$), is more than $d-1$. We show that, by extending (or generalizing) an existing TSS scheme, we obtain MDSBRS and MDSPRS schemes each one of which is secure, i.e., existentially unforgeable and perfectly (signer-)private, under standard assumption and asymptotically efficient.

Category / Keywords: public-key cryptography / Multi-dimensional sub-range signatures, Multi-dimensional super-range signatures, Existential unforgeability, Perfect privacy