| Reference | |M| | |σ| | |pk| | Sec. Loss | Assumptions |
|---|---|---|---|---|---|
| HJ [37] | 1 | 10d + 6 | 13 | 8 | DLIN |
| ACDKNO [3] | (n1, 0) | (7, 4) | (5, n1 + 12) | O(qs) | SXDH, XDLIN1 |
| ACDKNO [3] | (n1, n2) | (8, 6) | (n2 + 6, n1 + 13) | O(qs) | SXDH, XDLIN1 |
| LPY [45] | (n1, 0) | (10, 1) | (16, 2n1 + 5) | O(qs) | SXDH, XDLIN2 |
| KPW [41] | (n1, 0) | (6, 1) | (0, n1 + 6) | O(qs2) | SXDH |
| KPW [41] | (n1, n2) | (7, 3) | (n2 + 1, n1 + 7) | O(qs2) | SXDH |
| JR [39] | (n1, 0) | (5, 1) | (0, n1 + 6) | O(qs log qs) | SXDH |
| Ours (Sect. 4.2) | (n1, 0) | (13, 12) | (18, n1 + 11) | O(λ) | SXDH |
| Ours (Sect. 4.3) | (n1, n2) | (14, 14) | (n2 + 19, n1 + 12) | O(λ) | SXDH |
Table 1: Object sizes and loss of security among structure-preserving signature schemes with assumptions in the standard model. Smallest possible parameters are set to parameterized assumptions. Notation (x,y) means x and y elements in $\mathbb{G}_1$ and $\mathbb{G}_2$, respectively. The $|M|$, $|\sigma|$, $|pk|$ columns mean the number of messages, the number of group elements in a signature, and the number of group elements in a public key, respectively. The “Sec. Loss” column means reduction costs. The “Assumptions” column means the underlying assumptions for proving security. For HJ, parameter d limits number of signing to $2^d$. Parameters $q_s$ and $\lambda$ represent number of signing queries and security parameter, respectively.
The only tightly secure SPS under compact assumptions is that by Hofheinz and Jager [37]. Their tree-based construction, however, yields unacceptably large signatures consisting of hundreds of group elements. For other SPS schemes under compact assumptions the security is proven using a hybrid argument that repeat reductions in $q_s$. Thus, their security loss is $O(q_s)$ [3, 45] or even $O(q_s^2)$ [41], as shown in Table 1.
| Reference | |M| | #(s.mult) in signing | #(PPEs) | #(Pairings) | |
|---|---|---|---|---|---|
| Plain | Batched | ||||
| KPW [41] | (6, 1) | 3 | $n_1 + 11$ | $n_1 + 10$ | |
| JR [39] | (n1, 0) | (6, 1) | 2 | $n_1 + 8$ | $n_1 + 6$ |
| Ours (Sect. 4.2) | (15, 15) | 15 | $n_1 + 57$ | $n_1 + 16$ | |
| KPW [41] | (8, 3.5) | 4 | $n_1 + n_2 + 15$ | $n_1 + n_2 + 14$ | |
| Ours (Sect. 4.3) | (n1, n2) | (17.5, 16) | 16 | $n_1 + n_2 + 61$ | $n_1 + n_2 + 18$ |
Table 2: Comparison of factors relevant to computational efficiency against SPS schemes having smallest signature sizes. Third column indicates number of scalar multiplications in $\mathbb{G}_1$ and $\mathbb{G}_2$ for signing. Multi-scalar multiplication is counted as 1.5. For JR, a constant pairing is included. Column “Batched” shows the number of pairings in a verification when pairing product equations are merged into one by using a batch verification technique [14].
The non-tightness of security reductions does not necessarily mean the existence of a forger with reduced complexity, but the security guarantees given by non-tight reductions are quantitatively weaker than those given by tight reductions. Recovering from the security loss by increasing the security parameter is not a trivial solution when bilinear groups are involved. The security in source and target groups should be balanced, and computational efficiency is influenced by the choice of curves, pairings, and parameters such as embedding degrees, and the presence of dedicated techniques. In practice, an optimal setting for a targeted security parameter is determined by actual benchmarks, e.g., [30, 6, 33, 25], and only standard security parameters such as 128, 192, and 256, have been investigated. One would thus have to hop to the next standard security level to offset the security loss in reality. Besides, we stress that increasing the security parameter for a building block in structure-preserving cryptography is more costly than usual as it results in losing efficiency in all other building blocks using the same bilinear groups. Thus, the demand for tight security is stronger in structure-preserving cryptography.
Even in ordinary (i.e. non-structure-preserving) signature schemes, most of the constructions satisfying tight security are either in the random oracle model, e.g. [9, 40, 23, 4], rely on q-type or strong RSA assumptions, e.g., [16, 46], or lead to large signatures and/or keys, e.g., [23, 43]. Hofheinz presented the first tightly secure construction with compact signatures and keys under a standard compact assumption