* Identity element of

scalar multiplication: 1v = v, where 1 is the real number one.

In the NA[F.sub.k] algorithm, the converted NAF values are buffered in memory until

scalar multiplication or modular exponentiation is completed.

Finally, we prove that

scalar multiplication is continuous.

The proposed protocol requires more time in

scalar multiplication and XOR operation.

The implementation of the

scalar multiplication multiplyScalar() is straightforward as it corresponds to multiplying an arbitrary [C.sub.i]--we choose [C.sub.1] for simplicity--of the respective MPO by the scalar at hand.

The addition [[direct sum].sub.M] and

scalar multiplication [[cross product].sub.M] for the set [parallel][V.sub.s][parallel] = {[+ or -][parallel]a[parallel]; a [member of] [V.sub.s]} in the axiom (VV) of gyrovector space are defined by the equations

It has been found that the cost of the bilinear parings is approximately 20 times more than that of the

scalar multiplication over elliptic curve group [34].

The first

scalar multiplication ([pp.sub.i] x [PS2.sub.j]) in [V.sub.ijk] can be pre-computed whenever the pseudonym is generated for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the current autonomous network as it saves one

scalar multiplication during signature generation.

A linear topological space X over the real field R is said to be a paranormed space if there is a subadditive function g:X [member of] R such that g([theta]) = 0, g(x) = g(-x), and

scalar multiplication is continuous, i.e., [absolute value of [[alpha].sub.n] - [alpha]] [right arrow] 0 and g([x.sub.n] - x) [right arrow] 0 imply g([[alpha].sub.n][x.sub.n] - ax) [right arrow] 0 for all x's in X and a's in R, where [theta] is the zero vector in the linear space X.

This addition and

scalar multiplication are called Blaschke addition and

scalar multiplication.

Temporary registers store intermediate results during the

scalar multiplication. Add blockperforms finite field addition and subtraction, with simple XOR gates.

Therefore we can construct an IT2FNN-2 that computes all FBF expansion combinations with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the form of the proposed IT2FNN-2, and Y is closed under

scalar multiplication.