Abstract
The purpose of the book which you have just completed was to highlight the most basic of the mathematical, and in particular the algebraic, aspects of modern cryptography. In the process we have covered quite a lot of ground, but even so we have barely scratched its surface. So in these concluding remarks, we shall indicate where you may dig deeper and also refer you to matters which have been left out of discussion altogether, some because the algebraic content is negligible or uninteresting, others simply because we wanted to keep the size of the book within reasonable bounds.
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There are also attribute based encryption schemes, in which case the public key is a function of the attributes of the relevant party, e.g. his/her rights and privileges: only parties who satisfy the relevant policy can decrypt.
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Merkle, R.C. and Hellman, M.E.: Hiding information in trapdoor knapsacks, IEEE Trans. Info. Th. IT-24, pp. 525–530. See also Odlyzko, A.M.: The rise and fall of knapsack cryptosystems in Cryptology and Computational Number Theory, edited by Carl Pomerance, Am. Math. Soc., 1990.
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Joux, A. and Stern, J.: Lattice reduction: a toolbox for the cryptanalyst, In his book, Algorithmic Cryptanalysis (CRC Press, Boca Raton, 2009) Joux devotes an entire chapter to lattice reduction techniques.
A good overview, but now rather dated, on lattice based techniques is that of Ngyuen, P.Q. and Stern,J.: The two faces of lattices in cryptology, Proc. CaLC ’01, Cryptography and Lattice Conference, LNCS 2148, Springer 2001, pp. 148–180.
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Gentry’s technique depends on the use of ideal lattices, which are not to be confused with lattices of ideals, which might be defined as we did for subgroups in Sect. 3.3. Instead an ideal lattice is a lattice (of the kind referred to above) which is itself isomorphic to an ideal in a ring of the form \(\mathbb{Z}[x]/ <f(x)>\) where f(x) is an irreducible polynomial.
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Read Philip Rogaway’s article The Moral Character of Cryptographic Work, IACR ePrint Archive, 2015/1162 for a view on this, and on many other important matters which are in danger of being ignored by the cryptological community.
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Bernstein, D.J., Buchmann, J. and Dahmen, E. (eds.): Post-Quantum Cryptography, Springer, 2009.
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And if you want to see in how many ways things can go wrong in protocol design, read Boyd and Mathurias’s Protocols for Key Authentication and Key Establishment, Springer 2003.
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Meijer, A.R. (2016). Where Do We Go from Here?. In: Algebra for Cryptologists. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-30396-3_11
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