(4) the calculation of the multiplicative inverse 1/(1 + [alpha]), which is carried out as described in Section 6.3.

Moreover, an exact determination of the remainder is accomplished for the relevant case of the multiplicative inverse function.

In the SAKDA [20], the GKD executes two modular multiplicative inverse operations by using Extended Euclidean Algorithm.

In the SAKDA [20], each member executes one modular multiplicative inverse operation and one modular exponentiation to get the group key.

The computation overhead Scheme GKD Each group member GKMP [19] t enc 1 dec AGKTP [4] 1 Gen_[f.sub.t](x) + 1 hash 1 Gen_[f.sub.t](0) + 1 hash SAKDA [20] 2 inv + 1 hash 1 inv + 1 exp +x + 1 hash Our Scheme t Gen_[f.sub.1](x) + 1 hash or 1 Gen_[f.sub.1](0) + 1 hash or 1 Gen_[f.sub.1](x) + 1 hash + 1 enc 1 dec + 1 hash t: the number of group members Gen_[f.sub.d](x): generate a polynomial of degree d [4] Gen_[f.sub.d](0): restore the constant term of [f.sub.d](x) [4] inv: modular multiplicative inverse exp: modular exponentiation enc/dec: symmetric encryption/decryption

In order to design a residue to binary converter, we first need to obtain the multiplicative inverse values and substitute these values with the modulus set in the conversion algorithm formulas.

The multiplicative inverse of ([2.sup.2n+1]) modulo ([2.sup.2n+1] - 1) is [k.sub.1] = 1.

a) The proof for 1 as the multiplicative inverse of itself is trivial.

b) The proof for ([2.sup.n] 1) as the multiplicative inverse of itself can be given as follows for some integer z that is less than [2.sup.n]: ([2.sup.n] - 1) * ([2.sup.n] - 1) = [2.sup.2n] - [2.sup.n+1] + 1 = [2.sup.n] ([2.sup.n] - 2) + 1 = [2.sup.n] (z) + 1 = 1 mod ([2.sup.n])

As we saw above, the multiplicative inverse of 2 mod 7 is 4, and 4 is the span of the generating interval of the perfect fifth.

In an MP (therefore W[F.sup.*]) set of N elements, the spans of the generating intervals are the multiplicative inverses rood N of the multiplicities of the step intervals.

Therefore 5, the

multiplicative inverse of 3 (mod 7), is also a primitive root (mod 7) and these are the only distinct residues which are primitive roots (mod 7).