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= Fusion Exponentiation = In group theory fusion exponentiation is a generalization of the usual exponential function $x\mapsto g^x$ in cyclic groups to direct products of cyclic groups, or vector-valued exponents.

Construction
Let $(G,\circ)$ be a cyclic group generated by two, not necessarily distinct, elements $g,h$. For a pair of integers $a,b$, the fusion exponential function to the base $$(a,b)\in G^2$$ is defined as

$$(g,h)^{(a,b)} := (g^ah^{-b}, g^bh^a),$$

where the symbolic notation alluding to the exponential function is justified by the same properties to hold as for powers:


 * $[(g,h)^{(a,b)}]^{(c,d)}=(g,h)^{(a,b)\circ (c,d)}$, where $\circ$ is the multiplication of the complex numbers, i.e., $(a,b)\circ (c,d)=(ac-bd, ad+bc)$
 * $(g,h)^{(a,b)}\cdot (g,h)^{(c,d)}=(g,h)^{(a,b)+(c,d)}=(g,h)^{(a+c,b+d)}$, where $\cdot$ is the component-wise product as for direct products of groups.

These two properties establish a ring homomorphism between $(\mathbb{N}\times\mathbb{N},+)$ to the Caley-Dickson extension $(GF^2,+,\circ)$  of a finite field $GF$, extended in the same way as the complex numbers are defined over the real numbers. Using the canonic embedding of $(GF,+,\circ)$ inside the extension $GF^2,+,\circ)$  via the mapping $x\mapsto (x,0)$  reveals the usual exponential function to appear as a special case of fusion exponentiation, as we have $(g,1)^{(a,0)}=(g^a1^0,g^01^1)=(g^a,1)$.

Carrying the construction further, it can be generalized to exponents of hihger but finite dimension $(a,b,c,\ldots)\in\mathbb{N}^k$ for any finite $k$.

Cryptography
Fusion exponentiation naturally induces discrete logarithm problems, which can be shown to be computationally equivalent to their standard counterparts. This means that fusion exponentiation does lend itself as a fundament of discrete-logarithm based cryptography, but does not add to intractability assumptions (it is equally well solvable by Shor's algorithm, although relations between the fusion Diffie-Hellman problem and the discrete logarithm problem are less well understood as for modulo arithmetic). Since the algebraic properties of fusion exponentiation are analogous to that of standard exponentiation, classical primitives like Diffie-Hellman key exchange, or ElGamal encryption have canonic counterparts in terms of fusion exponentiation.

Algebra
Fusion exponentiation lets us define generating elements for groups that are otherwise not cyclic: for example, the direct product of two groups $G, H$ is cyclic if and only if the respective group orders are coprime. This implies that the direct product of $G$ with itself cannot be cyclic. Fusion exponentiation allows to represent every such a structure with elements that allow a representing any group element as a fusion-power of a fixed vector of generators (not necessarily all pairwise distinct) $\mathbf g=(g,h)$ such that all $x\in G$  can be written as $x = \mathbf g^{(a,b)}$  for a pair $a,b$  of integers (its fusion discrete logarithm). In higher-order powers of direct products $G^k$ with $k>2$, a likewise representation by fusion-power of a fixed element is also possible.