By Gilbert Baumslag

Cryptography has turn into crucial as financial institution transactions, bank card infor-mation, contracts, and delicate clinical details are despatched via inse-cure channels. This booklet is anxious with the mathematical, specially algebraic, points of cryptography. It grew out of many classes awarded through the authors during the last 20 years at a variety of universities and covers a variety of issues in mathematical cryptography. it's essentially geared in the direction of graduate scholars and complex undergraduates in arithmetic and laptop 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 maths of cryptographic protocols and a number of other precise themes such as

- Group-Based Cryptography

- Gröbner foundation equipment in Cryptography

- Lattice-Based Cryptography

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**Example text**

3) Transform the sequence of plaintext units into a sequence of residue classes m0 , m1 , . . in ℤn . (4) Encrypt the mi into ci = mi + xi ∈ ℤn . The secret key is s ∈ ℤn . Before moving onto the hardware procedure we make some comments on the cryptanalysis of this software generated stream cipher. (1) The integer n should be very large and the residue classes should occur with the same probability. Further the function f should not have a fixed point. To accomplish this we must choose f and s ∈ ℤn so that the period length x0 , x1 , .

For a given key k ∈ K the encryption function is given by Fk (p) = p ⊕ k for p ∈ P. We assume that the distribution on all three sets is the uniform distribution and a key k is only used once. The resulting encryption protocol is called a one-time pad or Vernam one-time pad. Shannon proved that the one-time pad, under the assumptions provided in the definition, is perfectly secure, as long as the keys are randomly chosen and used only once. 4 (Shannon). A one-time key pad has perfect secrecy if the keys are randomly chosen from the uniform distribution of keys and a key is used only once.

In symmetric key cryptography, both the encryption and decryption keys are supposedly known only to the sender and receiver, usually referred to as Bob and Alice. In public key cryptography, the encryption map is public knowledge but only the receiver knows how to decode. In this chapter we look at the standard methods for efficient symmetric key cryptography, while in Chapter 7 we discuss public key methods in detail. In general, symmetric key encryption methods are much more efficient in terms of time and storage requirements then public key methods.