Signature verification involves time-consuming operations. There are two mechanisms supposed to accelerate signature verification. One is batch verification, and the other is server-aided verification. Batch verification aims to verify signatures in a batch to reduce re-operations. Server-aided verification aims to delegate part of computational load to a powerful but untrusted server to ease the on-line computational burden. Both of the two mechanisms should carefully avoid violating the security guarantee of the original signature schemes. In the thesis the cryptanalysis on both batch verification and server-aided verification is provided with the security notions.