We could perhaps then remove the

totient pair (1, N - 1) as it does not form a Goldbach sum, but this is unimportant.

Thus the order of K is [phi](d), where [phi] denotes Eulers

totient function.

[N.sub.X]: a large number, where [N.sub.X] = [p.sub.X] x [q.sub.X], [phi]([N.sub.X]) = ([p.sub.X] - 1)([q.sub.X] - 1), and [phi](*) is Euler's

totient function;

where [phi](n) = #{m [member of] N: 1 [less than or equal to] m [less than or equal to] n, (m, n) = 1} is Euler's

totient function and [[omega].sub.n] is a primitive nth root of unity for each n,

In this section, we study the performance of our proposed algorithms and compare them with the Euler's

totient based Algorithm, MDS algorithm, secure group communication, one way function tree, Binary tree based, Efficient large group key distribution, and Logical key hierarchy Algorithm described in [1][5][9][10][12][23][24] which are labeled as ETF, MDS, SKDC, OFT, Binary, ELK and LKH respectively.

[4] Chengliang Tian and Xiaoyan Li, On the Smarandache power function and Euler

totient function, Scientia Magna, 4(2008), No.

For a positive integer n, the Euler phi function (or Euler

totient function) [phi](n) is defined to be the number of positive integers less then n which are relatively prime to n.

For n [member of] [N.sub.+] the Jordan

totient function [J.sub.k], see [1], is given by [J.sub.k](n) = [absolute value of {x [member of] [Z.sup.k.sub.n]|(x,n) = 1}].

In RSA the cipher text C is obtained for the plaintext message M [member of] [z.sup.*.sub.N] as C = [M.sup.e] mod N, where N is the product of two large prime numbers of same length, e is the public key chosen such that it is relatively prime with the Euler

totient function [phi](N) and 1 < e < [phi](N).

"Euler's

Totient Function," "Does --(n) Properly Divide n-1," "Solutions of --(m)=--(n)," "Carmichael's Conjecture," "Gaps Between Totatives," "Iterations of --and--," "Behavior of --(--(n)) and --(--(n))." [section]B36-B42 in Unsolved Problems in Number Theory, [2.sup.n]d ed.

For the axial groups a n-fold symmetry axis first becomes possible with translational symmetry if the dimensionality equals the

totient of n, which is the number of positive integers less than or equal to n which are relatively prime (no common factors) to n (21).

Compute [phi](n) = [phi](p)[phi](q) = (p - 1)(q - 1) = n - (p + q - 1), where [phi] is Euler's

totient function.