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Erschienen in:
2019 | OriginalPaper | Buchkapitel
14. Applications of Cosets
verfasst von : Lindsay N. Childs
Erschienen in: Cryptology and Error Correction
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Abstract
The idea of cosets is often thought of as a difficult concept in elementary abstract algebra. To help increase the comfort with cosets and the counting argument in the proof of Lagrange’s Theorem found in Chap. 10, this chapter gives have different applications of cosets to ideas from earlier chapters of the book. We give a proof of part of what is called the “Fundamental Theorem of Linear Algebra”, for vector spaces over the field of p elements \(\mathbb {Z}_{p}\). We look at Hamming codes and Euler’s Theorem from the point of view of cosets. The last two sections use cosets to look again at the strong pseudoprime test for finding large primes, and to obtain a result of Boneh that cracking an RSA cryptosystem by finding some decrypting exponent is essentially equivalent in difficulty to factoring the RSA modulus.