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Let be an integer. Denote by the multiplicative order of modulo integer . We prove that there is a positive constant such that if , then where It was known for in [P. Kurlberg and C. Pomerance, On a problem of Arnold: The average multiplicative order of a given integer, Algebra Number Theory 7 (2013) 981-999] in which they refer to [F. Luca and I. E. Shparlinski, Average multiplicative orders of elements modulo , Acta Arith. 109(4) (2003) 387-411. [ABSTRACT FROM AUTHOR]

Abstract: In this work a new efficient modulo 2 n +1 modified Booth multiplication algorithm for both operands in the weighted representation is proposed. Furthermore, the same algorithm is extended to realize modulo 2 n +1 multiply-add units. The derived partial products are reduced by an inverted end around carry-save adder tree to two operands, which are finally added by a modulo 2 n +1 adder. The performance and efficiency of the proposed multipliers are evaluated and compared against the earlier modulo 2n+1 multipliers, based on a single gate level model. Comparisons based on experimental CMOS implementations for both the multiply and multiply-add units are also given. The proposed multipliers yield area and power savings by an average of 15% and 10% respectively, while the corresponding area and power savings of the proposed multiply-add units are 14% and 21% respectively. [Copyright &y& Elsevier]