A shuffle with proof having a method for proof generating with small computational resources proportionate to the number of input encrypted messages and a corresponding method for verification. Shuffle is represented by a generalized transformation. Combining a proof that the transformation information is retained and a proof of a condition under which the transformation is met constitute the proof for shuffle. The two proofs are short proportional to the number of input encrypted messages. Transformation information retention is proved in such a manner that, since the response is generated from challenge value in dependency upon transformation, the condition under which the transformation is met is reflected in the response-challenge value relation. If the condition under which the transformation corresponding to the shuffle is selected as the condition for proof, the two proofs may constitute the proof for shuffle.