The need to compute the multiplicative inverse of an element of a finite field (or Galois field) or of a finite ring occurs frequently in cryptography. The main application domains are asymmetric cryptosystems, for instance in the computation of the private-public key pair in RSA (see RSA public key encryption schems) or in the group operation of elliptic curve cryptosystems. The finite structures in asymmetric algorithms are typically and relatively large. A second application domains are inversions in small finite fields which occur in the context of block ciphers, e.g., within the S-box of the Advanced Encryption Standard (Rijndael/AES).
In the case of inversion in a finite integer ring or polynomial ring, the extended Euclidean algorithm can be used. Let u be the element whose inverse is to be computed and v the modulus. Note that u and v must be relatively prime in order for the inverse to exist. The extended Euclidean algorithm computes the coefficients s and t such that: \(us +...
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References
Guajardo, J. and C. Paar (2002). “Itoh–Tsujii Inversion in standard basis and its application in cryptography and codes.” Designs, Codes and Cryptography, 25, 207–216.
Itoh, T. and S. Tsujii (1988). “A fast algorithm for computing multiplicative inverses in GF2(m) using normal bases.” Information and Computation, 78, 171–177.
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Paar, C. (2005). Inversion in Finite Fields and Rings. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_207
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