Abstract
By making use of the theory of quadratic residues under the condition of strong prime, a method for studying the algebraic structure of Z*φ( n)of RSA( Rivest-Shamir-Adleman) algorithm was established in this work. A formula to determine the order of element in Z*φ( n)and an expression of maximal order were proposed; in addition, the numbers of quadratic residues and non-residues in the group Z*φ( n)were calculated. This work gave an estimate that the upper bound of maximal order was φ( φ( n)) /4 and obtained a necessary and sufficient condition on maximal order being equal to φ( φ( n)) /4.Furthermore, a sufficient condition for A1being cyclic group was presented, where A1was a subgroup composed of all quadratic residues in Z*φ( n), and a method of the decomposition of Z*φ( n)was also established. Finally, it was proved that the group Z*φ( n)could be generated by seven elements of quadratic non-residues and the quotient group Z*φ( n)/ A1was a Klein group of order 8.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
