By Gilbert Baumslag

ISBN-10: 3110372762

ISBN-13: 9783110372762

Cryptography has develop into crucial as financial institution transactions, bank card infor-mation, contracts, and delicate scientific info are despatched via inse-cure channels. This e-book is worried with the mathematical, specifically algebraic, features of cryptography. It grew out of many classes offered through the authors during the last 20 years at a number of universities and covers quite a lot of subject matters in mathematical cryptography. it really is essentially geared in the direction of graduate scholars and complicated undergraduates in arithmetic and computing device technology, yet can also be of curiosity to researchers within the area.

Besides the classical equipment of symmetric and personal key encryption, the ebook treats the maths of cryptographic protocols and several other certain issues such as

- Group-Based Cryptography

- Gröbner foundation equipment in Cryptography

- Lattice-Based Cryptography

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**Additional resources for A Course in Mathematical Cryptography**

**Sample text**

In many actual implementations, the encryption and decryption maps are bijective, and input and output block lengths are equal. The usual alphabet is A = {0, 1}, of course. Thus the encryption maps fk : {0, 1}n → {0, 1}n are permutations. Since there exist (2n )! such permutations, and since the size of the key space is generally much smaller than this huge number, only a small part of the set of possible encryption maps is realized in practice. For practical purposes it is essential that block ciphers are very efficient to encrypt and decrypt once one has the key.

The set of all such methods and rules to perform a cryptographical task is called a cryptographic protocol. 16 | 1 Basic Ideas of Cryptography We now list several cryptographic tasks. These will be discussed in more detail later in the book together with cryptographic protocols to implement them. (1) Authentication: This is the process of determining that a message, supposedly from a given person, both does come from that person and has not been tampered with. Included in authentication are the concepts of hash functions and digital signatures.

Let (an , . . , an ) be a sequence of length n of letters from the alphabet. Let n1 be the number of A???? s that occur, n2 the number of B???? s and so on. Then: n (n −1) (a) There are (n21 ) = 1 21 pairs of A’s in the sequence, (n22 ) pairs of B’s and so on. (b) There are altogether (n21 ) + ⋅ ⋅ ⋅ + (n226 ) pairs of equal letters. (c) The probability that two randomly chosen letters are equal is therefore given by pf = (n21 ) + ⋅ ⋅ ⋅ + (n226 ) (2n) . This number is called the Friedmann Coincidence Index.

### A Course in Mathematical Cryptography by Gilbert Baumslag

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