Originally introduced in [5], the Itoh and Tsujii algorithm (ITA) is an exponentiation-based algorithm for inversion in finite fields which reduces the complexity of computing the inverse of a non-zero element in GF(2n), when using a normal basis representation, from \(n-2\) multiplications in GF(2n) and \(n-1\) cyclic shifts using the binary exponentiation method to at most \(2 \lfloor\log_2(n-1)\rfloor\) multiplications in GF(2n) and \(n-1\) cyclic shifts. As shown in [4], the method is also applicable to finite fields with a polynomial basis representation.
For the discussion that follows, it is important to point out that there are several possibilities to represent elements of a finite field. Thus, in general, given an irreducible polynomial \(P(x)\) of degree m over \(GF(q)\) and a root \(\alpha\) of \(P(x)\) (i.e., \(P(\alpha) = 0\)), one can represent an element \(A \in GF(q^m)\), \(q=p^n\) and p prime, as a polynomial in \(\alpha\), i.e., as \(A = a_{m-1} \alpha^{m-1} +...
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Guajardo, J. (2005). Itoh–Tsujii Inversion Algorithm. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_212
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