Divisible On-line/Off-line Signatures
On-line/Off-line signatures are used in a particular scenario where the signer must respond quickly once the message to be signed is presented. The idea is to split the signing procedure into two phases: the off-line and on-line phases. The signer can do some pre-computations in off-line phase before he sees the message to be signed. In most of these schemes, when signing a message $m$, a partial signature of $m$ is computed in the off-line phase. We call this part of signature the off-line signature token of message $m$. In some special applications, the off-line signature tokens might be exposed in the off-line phase. For example, some signers might want to transmit off-line signature tokens in the off-line phase in order to save the on-line transmission bandwidth. Another example is in the case of on-line/off-line threshold signature schemes, where off-line signature tokens are unavoidably exposed to all the players in the off-line phase. This paper discusses this exposure problem and introduces a new notion: divisible on-line/off-line signatures, in which exposure of off-line signature tokens in off-line phase is allowed. An efficient construction of this type of signatures is also proposed. Furthermore, we show an important application of divisible on-line/off-line signatures in the area of on-line/off-line threshold signatures.