By Gilbert Baumslag
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 publication is anxious with the mathematical, specifically algebraic, elements of cryptography. It grew out of many classes awarded through the authors over the last 20 years at a variety of universities and covers quite a lot of issues in mathematical cryptography. it's basically geared in the direction of graduate scholars and complicated undergraduates in arithmetic and computing device technological know-how, yet can also be of curiosity to researchers within the area.
Besides the classical tools of symmetric and personal key encryption, the publication treats the math of cryptographic protocols and several other special themes such as
- Group-Based Cryptography
- Gröbner foundation equipment in Cryptography
- Lattice-Based Cryptography
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Extra info for A Course in Mathematical Cryptography
St − αt α2t .. Each column is encrypted with a single alphabet (or a Caesar cipher). We have always the same shift cipher. Hence the coincidence index is about pL (each column is a shifted plain text). The probability that two letters from different columns are equal 1 is about 26 . Each column has mt letters. The number of pairs of letters, which are in the same column, is m m(m − t) t( t ) = . 2t 2 The number of pairs of letters that are in different columns is t m 2 m2 (t − 1) ( )( ) = .
The key k is chosen and it generates the keystream s1 , s2 , . . , sn , . . For each si there is an encryption map fsi and a corresponding decryption map gei . Let F denote the overall encryption map. Then we would have F(a1 a2 ⋅ ⋅ ⋅ ak ) = fs1 (a1 )fs2 (a2 ) ⋅ ⋅ ⋅ fsk (ak ). In practice the plaintext characters are usually single bits 0, 1. Stream ciphers typically encrypt and decrypt at higher speeds than block ciphers and can usually be decrypted synchronously with the encryption. Stream ciphers are an attempt to approximate the security of so-called one-time pads.
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. 5. Given a sequence of letters of length m representing a Vigènere encrypted cipher text and let pF be its Friedman Coincidence Index.
A Course in Mathematical Cryptography by Gilbert Baumslag