# multiplicative inverse

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## multiplicative inverse

[‚məl·tə′plik·əd·iv ′in‚vərs]
(mathematics)
In a mathematical system with an operation of multiplication, denoted ×, the multiplicative inverse of an element e is an element ē such that e × ē = ē × e = 1, where 1 is the multiplicative identity.
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(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 , the GKD executes two modular multiplicative inverse operations by using Extended Euclidean Algorithm.
In the SAKDA , 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  t enc 1 dec AGKTP  1 Gen_[f.sub.t](x) + 1 hash 1 Gen_[f.sub.t](0) + 1 hash SAKDA  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  Gen_[f.sub.d](0): restore the constant term of [f.sub.d](x)  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).

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