hợp lý hoạt động OR (x, y là các biến Boolean), cũng bit hoạt động:-bit khôn ngoan hay (x, y là chuỗi bit) XOR hoạt động hợp lý (x, y là các biến Boolean), cũng bit hoạt động:-bit khôn ngoan xor (x , y là chuỗi bit) không thực thi phần bình luận trong các thuật toán nhận xét, bằng chứng hoặc ví dụ | x -jy logical operation OR x y are Boolean variables also bit operation bit-wise or x y are bit strings Vy logical operation XOR x y are Boolean variables also bit operation bit-wise xor x y are bit strings . non-executable comment parts in algorithms or protocols Ị 1 end of proof remark or example Chapter 3. Probability and Information Theory Section . Introduction Section . Basic Concept of Probability Section . Properties Section . Basic Calculation Section . Random Variables and their Probability Distributions Section . Birthday Paradox Section . Information Theory Section . Redundancy in Natural Languages Section . Chapter Summary Exercises Introduction Probability and information theory are essential tools for the development of modern cryptographic techniques. Probability is a basic tool for the analysis of security. We often need to estimate how probable it is that an insecure event may occur under certain conditions. For example considering Protocol Coin Flipping Over Telephone in Chapter 1 we need to estimate the probability for Alice to succeed in finding a collision for a given one-way function f which should desirably be bounded by a very small quantity and that for Bob to succeed in finding the parity of x when given f x 1 which should desirably be very close to - . Information theory is closely related to probability. An important aspect of security for an encryption algorithm can be referred to as uncertainty of ciphers an encryption algorithm should desirably output ciphertext which has a random distribution in the entire space of its ciphertext message space. Shannon quantifies the uncertainty of information by a notion which he names entropy. Historically the desire for achieving a high entropy in ciphers comes from the need for thwarting a cryptanalysis technique which makes use of the fact that natural languages contain redundancy which is related to frequent appearance of some known patterns in natural languages.
