Therefore, an internal direct product of group QRNβ and group ?1
 constitutes the predetermined cyclic group. For example, the predetermined cyclic group can be represented as QRNβ?
?1
. By setting |QRNβ|=α, it can be determined that the size of the predetermined cyclic group is k=2α with reference to the size of group 
?1
 being 2. Therefore, when private key sk=α=a·k is satisfied, a=α/k=1/2 can be determined.