By Neal Koblitz
The aim of this e-book is to introduce the reader to mathematics subject matters, either historic and glossy, which were on the heart of curiosity in functions of quantity thought, quite in cryptography. No heritage in algebra or quantity conception is believed, and the e-book starts with a dialogue of the fundamental quantity conception that's wanted. The strategy taken is algorithmic, emphasizing estimates of the potency of the ideas that come up from the idea. a different function is the inclusion of contemporary program of the speculation of elliptic curves. wide workouts and cautious solutions were integrated in the entire chapters. simply because quantity thought and cryptography are fast-moving fields, this re-creation includes tremendous revisions and up-to-date references.
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Extra info for A Course in Number Theory and Cryptography
R r ) = 1. Any A E A12(Z/NZ) Suppose tliat. N = 7 ~ n wllcre can be considered in M2(Z/nzZ) or A12(Z/nZ) hy simply rctlucing the entries rnod~~lo nz or n. ~. (a) Prove that the map that tnkrs A to t IIP pair (A. A) is a 1-to-1 corresporidcncc betwoe11 h12(Z/NZ) : u ~ tllir. scbtA12(Z/irrZ) x A12(Z/rlZ) of all pairs of matrices, one modulo rn m(1 m e mod1110n. -' 111. Cryptography 2 Enciphering Iblatrices 81 (b) Prove that the map in part (a) gives a 1-to-1 correspondence between the set M2(Z/NZ)* of invertible matrices mod N and the set M2(Z/mZ)* x M2(Z/nZ)* For 11 a prime, find tlic number of elements in M2(Z/pZ)* in two ways, and check that your answers agree: (a) Count the number of solutions in F, of the equation ad - bc = 0, and subtract this from the number of elements in M2(Z/pZ).
To an element of z / N 2 Z . , to a pair of integers with x and y each considered modulo N. For example, if we're using the 26-letter alphabet A-Z with numerical equivalents 0-25, respectively, then the digraph NO corresponds to the vector See the diagram at the top of the next page. (i) (t:). We picture each digraph P as a point on an N x N square array. That. is, we have an "xy-plane," except that each axis, rather than being a copy 66 III. Cryptography 2 Enciphering Matrices 67 For a fixed matrix, this function from one vector to another vector is called a linear transformation, meaning that it preserves sums and constant multiples of vectors.
R. Lid1 and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge Univ. Press, 1986. 10. V. Pless, Introduction to the Theory of Error-Correcting Codes, Wiley, 1982. 11. D. , Chelsea Publ. , 1985. 1 Some simple cryptosystems 55 can represent the situation schematically by the diagram Any such set-up is called a cryptosystem. The first step in inventing a cryptosystxm is to "label" all possible plaintext message units and all possible ciphertext message units by means of mat hematical objects from which functions can be easily constructed.
A Course in Number Theory and Cryptography by Neal Koblitz