1. A computer-implemented method for validating a batch of implicit certificates, the method comprising: accessing key-pair-validation data for a batch of implicit certificates, wherein the key-pair-validation data include key-pair-validation values for each of the implicit certificates in the batch, and the key-pair-validation values for each implicit certificate are either valid or invalid according to a key-pair-validation function, wherein the batch of implicit certificates includes l implicit certificates and the key-pair-validation data include l data sets, each of the data sets i includes the key-pair-validation values for one of the implicit certificates, and the key-pair-validation values for each implicit certificate include a private key value b i , a public key reconstruction value P i , a certificate authority public key value C i , and a hash value e i ; modifying at least one of the key-pair-validation values for each implicit certificate; and evaluating, by a computing system, a batch-validation function for the batch of implicit certificates based on the key-pair-validation data that include the modified key-pair validation values, wherein evaluating the batch-validation function identifies whether the key-pair-validation data for the batch of implicit certificates include key-pair-validation values that are invalid.