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Internet Security Cryptographic Principles, Algorithms and Protocols - Chapter 5

Asymmetric Public-key Cryptosystems Mật mã khóa công khai đã trở thành công ngay sau khi Whitefield Diffie và Martin Hellman (1976) đề xuất khái niệm sáng tạo của một chương trình trao đổi quan trọng theo cấp số nhân. Từ năm 1976, nhiều thuật toán khóa công khai đã được đề xuất, nhưng nhiều người trong số họ đã bị phá vỡ. Trong số rất nhiều các thuật toán là vẫn được coi là an toàn, nhất là không thực tế. Chỉ có một vài thuật toán khóa công khai an toàn và thực tế. . | 5 Asymmetric Public-key Cryptosystems Public-key cryptography became public soon after Whitefield Diffie and Martin Hellman 1976 proposed the innovative concept of an exponential key exchange scheme. Since 1976 numerous public-key algorithms have been proposed but many of them have since been broken. Of the many algorithms that are still considered to be secure most are impractical. Only a few public-key algorithms are both secure and practical. Of these only some are suitable for encryption. Others are only suitable for digital signatures. Among these numerous public-key cryptography algorithms only four algorithms RSA 1978 and ElGamal 1985 Schnorr 1990 and ECC 1985 are considered to be suitable for both encryption and digital signatures. Another public-key algorithm that is designed to only be suitable for secure digital signatures is DSA 1991 . The designer should bear in mind that the security of any encryption scheme depends on the length of the key and the computational work involved in breaking a cipher. 5.1 Diffie-Hellman Exponential Key Exchange In 1976 Diffie and Hellman proposed a scheme using the exponentiation modulo q a prime as a public key exchange algorithm. Exponential key exchange takes advantage of easy computation of exponentials in a finite field GF q with a prime q compared with the difficulty of computing logarithms over GF q with q elements 1 2 . q 1 . Let q be a prime number and a a primitive element of the prime number q. Then the powers of a generate all the distinct integers from 1 to q 1 in some order. For any integer Y and a primitive element a of prime number q a unique exponent X is found such that Y aX mod q 1 X q 1 Then X is referred to as the discrete logarithm of Y to the base a over GF q X log aYover GF q 1 Y q 1 Internet Security. Edited by M.Y. Rhee 2003 John Wiley Sons Ltd ISBN 0-470-85285-2 162 INTERNET SECURITY Calculation of Y from X is comparatively easy using repeated squaring but computation of X from Y is typically .