A Course in Number Theory and Cryptography (Graduate Texts by Neal Koblitz

By Neal Koblitz

This can be a considerably revised and up to date advent to mathematics themes, either old and sleek, which have been on the centre of curiosity in purposes of quantity thought, fairly in cryptography. As such, no heritage in algebra or quantity idea is believed, and the e-book starts off with a dialogue of the elemental quantity idea that's wanted. The process taken is algorithmic, emphasising estimates of the potency of the suggestions that come up from the idea, and one particular characteristic is the inclusion of modern functions of the idea of elliptic curves. broad workouts and cautious solutions are an essential component the entire chapters.

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Example text

Thus, since we are working in 21262, we are using "=" t o mean that the entries are congruent modulo 26. Just as in the real number case, a 2 x 2-matrix with real numbers a, b, c, d has a multiplicative inverse if and only if the determinant D = ad - bc is nonzero, and in that case the inverse matrix is 70 111. Cryptography 2 Enciphering hlatrices 71 with entries in a ring R can be multiplied by a column-vector z, y E (;) with + + 3y 72 + 9y x This gives a "linear map" from vectors to vectors, meaning that a linear combination (::~:\$f:), where kl and kz are in the ring R, is taken to (::;iI::ii).

I. , Springcr -Verlag, 1990. 8. 9. S. , Addison-Wesley, 1984. 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.

352 580 440 mod 676, which is "QY'I The digraph "ON" has numerical equivalent 377, and is taken to 359="NV': Notice that the digraphs change as a unit, anti t hero is no relation betwccn the encryption of one digraph and that of another one that has a letter in common with it or even consists of the same Iotters in the reversc ortler. To break a digraphic encryption system which uses an affinc transformation C aP+b mod N: we need to know the ciphertext correspondi~igto two different plaintext message units.