Stream Ciphers and Number Theory

ISBN: 9780444516312 出版年：2004 页码：493 Cusick, Thomas W Ding, Cunsheng Renvall, Ari R Elsevier Science_RM

Preface. Introduction. Applications of number theory. An outline of this book. Stream Ciphers. Stream cipher systems. Additive synchronous stream ciphers. Nonadditive synchronous stream ciphers. Stream ciphering with block ciphers. Cooperative distributed ciphering. Some keystream generators. Generators based on counters. Some number-theoretic generators. Cryptographic aspects of sequences. Minimal polynominal and linear complexity. Pattern distribution of key streams. Correlation functions. Sphere complexity and linear cryptanalysis. Higher order complexities. Harmony on binary NSGs. Security attacks. Primes, Primitive Roots and Sequences. Cyclotomic polynominals. Two basic problems from stream ciphers. A basic theorem and main bridge. Primes, primitive roots and binary sequences. Primes, primitive roots and ternary sequences. Primes, negord and sequences. Prime powers, primitive roots and sequences. Prime products and sequences. Binary sequences and primes. Ternary sequences and primes. On cryptographic primitive roots. Linear complexity of sequences over Z m . Period and its cryptographic importance. Cyclotomy and Cryptographic Functions. Cyclotomic numbers. Cyclotomy and cryptography. Cyclotomy and difference parameters. Cyclotomy and the differential cryptanalysis. Cryptographic cyclotomic numbers. Cryptographic functions from Z p to Z d . The case d = 2. The case d = 3. The case d = 4. The case d = 5. The case d = 6. The case d = 8. The case d = 10. The case d = 12. Cryptographic functions from Z pq to Z d . Whiteman's generalized cyclotomy and cryptography. Cryptographic functions from Z pq to Z 2 . Cryptographic functions from Z pq to Z 4 . Cryptographic functions from Z p2 to Z 2 . Cryptographic functions defined on GF(p m ) . The origin of cyclotomic numbers. Special Primes and Sequences. Sophie Germain primes and sequences. Their importance in stream ciphers. Their relations with other number-theoretic problems. The existence problem. A search for cryptographic Sophie Germain primes. Tchebychef primes and sequences. Their cryptographic significance. Existence and search problem. Other primes of form k x 2 n + 1 and sequences. Primes of form ( a n - 1)/( a - 1) and sequences. Mersenne primes and sequences. Cryptographic primes of form ((4 u ) n - 1)/(4 u - 1). Prime repunits and their cryptographic values. n ! +/- 1 and p# +/- 1 Primes and sequences. Twin primes and sequences over GF (2). The significance of twins and their sexes. Cryptographic twins and the sex distribution. Twin primes and sequences over GF (3). Other special primes and sequences. Prime distribution and their significance. Primes for stream ciphers and for RSA. Difference Sets and Cryptographic Functions. Rudiments of difference sets. Difference sets and autocorrelation functions. Differece sets and nonlinearity. Difference sets and information stability. Difference sets and linear approximation. Almost difference sets.