Abstract
This chapter is about polynomials and their use. After learning the basics we discuss Lagrange’s interpolation needed for Shamir’s secret sharing scheme that we discuss in Chap. 6. Then, after proving some further results on polynomials, we give a construction of a finite field whose cardinality is a power of a prime. The field of cardinality \(p^n\) is constructed as polynomials over \(\mathbb {Z}_p\) modulo an irreducible polynomial of degree n. This field is an extension of \(\mathbb {Z}_p\) and in this context we discuss minimal annihilating polynomials which we will need in Chap. 7 for the construction of good error-correcting codes. Finally, we discuss permutation polynomials and a cryptosystem based on them.
A polynomial walks into a bar and asks for a drink. The barman declines: “We don’t cater for functions.”
An old math joke.
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Notes
- 1.
A purist would talk about a zero of the polynomial f(x) but a root of the equation \(f(x)=0\). We are not making this distinction.
- 2.
Those familiar with basics of abstract algebra will recognise the quotient-ring of F[x] by the principal ideal generated by m(x).
- 3.
See Sect. 3.1.3 for a brief historic note about this mathematician.
- 4.
In Rivest, Ronald L., et al. “The RC6 block cipher”. in First Advanced Encryption Standard (AES) Conference. 1998 it was used that polynomial \(f(x)=x(2x+1)\) is a permutation polynomial in \(\mathbb {Z}_w[x]\), where w is the word size of the machine.
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Slinko, A. (2020). Polynomials. In: Algebra for Applications. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-44074-9_5
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DOI: https://doi.org/10.1007/978-3-030-44074-9_5
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