Based on the previously described content, in the technical solutions in the present specification, when public and private keys are determined, random number space Z*N, the subgroups of random number space Z*N, namely, quadratic residue group QRN and group ![custom character]()
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, and the subgroups of quadratic residue group QRN, namely, group QRNα and group QRNβ, are actually involved. In these groups, group QRNβ and group ![custom character]()
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are selected to constitute predetermined cyclic group QRNβ?![custom character]()
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in the present specification, and the size (2α) of predetermined cyclic group QRNβ?![custom character]()
?4![custom character]()
is related to the length of random number r. In other words, predetermined cyclic group QRNβ?![custom character]()
?1![custom character]()
of an appropriate size is selected, so that random number r with an appropriate length can be selected in the encryption calculation process, to strike a balance between encryption efficiency and security (a longer length of random number r indicates lower encryption efficiency and higher security; and on the contrary, a shorter length of random number r indicates higher encryption efficiency and lower security).
