A finite field is a field with a finite number of elements. The number of elements or the order of a finite field is a power \(q = p^k\) of a prime number \(p \ge 2\), where \(k \ge 1\). The finite field with q elements is denoted F q or \(GF(q)\) (the latter meaning Galois Field). The prime p is the characteristic of the field, i.e., for all x ∈ F q , \(px = 0\).
(One generally refers to the finite field with q elements in the sense that all finite fields with a given order have the same structure, i.e., they are isomorphic to one another.)
Finite fields are commonly organized into three types in cryptography:
Characteristic-2 or binary fields, where \(p = 2\).
Prime-order fields, where \(p \ge 3\) and \(k = 1\).
Odd-characteristic extension fields, where \(p \ge 3\) and \(k > 1\).
Finite fields are widely employed in cryptography. The IDEA and Rijndael/AES algorithms, for instance, both involve operations over relatively small finite fields. Public-key cryptographygenerally...
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References
Lidl, R. and H. Niederreiter (1986). Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge.
Menezes, A.J., I.F. Blake, X. Gao, R.C. Mullin, S.A. Vanstone, and T. Yaghoobian (1992). Applications of Finite Fields. Kluwer Academic Publishers, Dordrecht.
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Kaliski, B. (2005). Finite Field. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_167
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