With the advent of the big data era, the amount of data computation is getting larger and larger, and the computational load of clients is also increasing day by day. The advent of clouds allows clients to outsource their data to the cloud for computing services. Outsourced computation has greatly reduced the computational burden of clients, but also brings the issue of trust. Because the cloud is not trustworthy, clients need to verify the correctness of the remote computation results. In this paper, we mainly study the common norm operations, and propose two publicly verifiable schemes for 1-norm and 2-norm operations respectively, any client can publicly verify these two common norm operations under single-key setting by using our schemes. © Springer International Publishing AG 2018.