By Gilbert Baumslag

ISBN-10: 3110372762

ISBN-13: 9783110372762

Cryptography has turn into crucial as financial institution transactions, bank card infor-mation, contracts, and delicate scientific details are despatched via inse-cure channels. This ebook is worried with the mathematical, particularly algebraic, facets of cryptography. It grew out of many classes provided by way of the authors during the last two decades at a number of universities and covers a variety of subject matters in mathematical cryptography. it really is 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 equipment of symmetric and personal key encryption, the publication treats the maths of cryptographic protocols and several other distinctive subject matters such as

- Group-Based Cryptography

- Gröbner foundation equipment in Cryptography

- Lattice-Based Cryptography

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

**Example text**

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.

If P1 and P2 are two decision problems then we say that P1 reduces to P2 in polynomial time if there exists a polynomial time algorithm that constructs an answer for P2 given a solution to P1 . Hence the problem of breaking the ElGamal cryptosystem reduces to the discrete log problem (see next chapter). Finally a decision problem is NP-complete if any other problem in NP can be reduced to it in polynomial time. A decision problem is NP-hard if any NP problem reduces to it. A good cryptographic decision problem should be at least NP-hard.

If we use an affine cipher on N, N ≥ 2, single letters with n → an + b with b not congruent to 0 modulo N and (a − 1, N) = 1, show that there is always a unique fixed letter. (This can be used in cryptanalysis). Let N ∈ ℕ with N ≥ 2 and n ????→ an + b with (a, N) = 1 be an affine cipher on an N letter alphabet. Show that if any two letters are guessed n1 → m1 , n2 → m2 with (n1 − n2 , N) = 1 then the code can be broken. Let A be an N-letter alphabet. As before consider A as ℤN and treat them as the modular ring.

### A Course in Mathematical Cryptography by Gilbert Baumslag

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