New algorithms for batch verification of standard ECDSA signatures

Abstract

In this paper, several algorithms for batch verification of ECDSA signatures are studied. The first of these algorithms is based upon the naive idea of taking square roots in the underlying field. In order to improve the efficiency beyond what can be achieved by the naive algorithm, two new algorithms are proposed which replace square-root computations by symbolic manipulations. Experiments carried out on NIST prime curves demonstrate a maximum speedup of above six over individual verification if all the signatures in the batch belong to the same signer, and a maximum speedup of about two if the signatures in the batch belong to different signers, both achieved by a fast variant of the second symbolic-manipulation algorithm. In terms of security, all the studied algorithms are equivalent to standard ECDSA* batch verification. These algorithms are practical only for small (≤8) batch sizes. The algorithms are also ported to the NIST Koblitz curves defined over fields of characteristic

A new batch verification scheme for ECDSA$$^*$$ signatures

“…The technique ECDSA Ã [7] is introduced to speedup the verification of individual ECDSA signature. The schemes S1 and S2 [8] are very efficient for smaller batch sizes. However, the efficiency of these techniques decreases as the batch size increases.…”

“…The other algorithms S1, S2 [8] concentrate on reducing the time required to calculate R.y. Usually, R.y is calculated by square root method in ECDSA.…”

“…There are various batch verification schemes introduced for verifying ECDSA signatures too. [7,8] have some of the popular schemes for ECDSA signatures. Even though the existing techniques try to reduce the computation time, most of them are not efficient for larger batch sizes [9].…”

An Improved ECDSA Scheme for Wireless Sensor Network

“…, p E a b and then Alice obtains a private key d and a public key Q d G   . Signer Alice generates the signature for the message m , as follows [10].…”

Fast batch modular exponentiation with common-multiplicand multiplication

“…Batch verification, which is firstly introduced by Naccache et al [40], is used to verify multiple signatures simultaneously consuming less time than verifying total individual signatures. Successive research results have achieved improvements in terms of efficiency and security [1,3,4,5,8,9,21,27,51]. There also have been researches for batch signature schemes [22,42,53] and batch generation of exponentiations [10,39].…”