Abstract
In this chapter we extend beyond integers modulo primes to consider finite fields of characteristic 2. For a more extensive presentation of finite fields, the reader should consult Lidl and Niederreiter [1]. For a different presentation of finite fields of characteristic 2, the reader could consult Golomb [2]. Finite field arithmetic in characteristic 2 is used in the Advanced Encryption Standard (AES). It can be preferable in other cryptosystems, because computer hardware works in binary, and thus the underlying arithmetic operations needed to encrypt and decrypt can be very fast.
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Notes
- 1.
The first several Mersenne primes are for \(r = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107\) and 127; certainly from the last four of these we would get extremely long sequences of deterministically-produced bits that do happen to satisfy standard tests for being a random sequence of bits.
References
R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, 2nd edn. (Cambridge University Press, Cambridge, 1997)
S. Golomb, Shift Register Sequences (Aegean Park Press, 1982)
N. Zierler, J. Brillhart, On primitive trinomials (mod 2). Inform. Control 13, 541–554 (1968)
N. Zierler, J. Brillhart, On primitive trinomials (mod 2) (part 2). Inform. Control 14, 566–569 (1969)
R.C. Mullin, I.M. Onyszchuk, S.A. Vanstone, R.M. Wilson, Optimal normal bases in GF(pn). Discrete Appl. Math. 22, 149–161 (1989)
D.E. Knuth, The Art of Computer Programming, Volume 2, Seminumerical Algorithms, 2nd edn. (Addison-Wesley, 1981)
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Buell, D. (2021). Finite Fields of Characteristic 2. In: Fundamentals of Cryptography. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-73492-3_6
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