Exploration and reduction of data using principal component analysis

Digitally signing data includes collecting a group of signers, each having a public key and a corresponding secret key, a subgroup of signers each producing a partial digital signature of the data, and obtaining a combined signature of the data by combining the partial digital signatures of the data, where the combined digital signature keeps the subgroup of signers accountable for the data for which the subgroup of signers each produce a partial digital signature. Verifying a digital signature of data includes ascertaining members of a subgroup of signers that contributed to provide the digital signature of the data, determining a combined public key corresponding to individual secret keys of a the subgroup of signers, and using the combined public key to verify that the subgroup of signers have each contributed to provide the digital signature of the data.