Introduction to finite fields and their applications: List of Symbols

The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.

[1]  R. Tausworthe.  Random Numbers Generated by Linear Recurrence Modulo Two , 1965 .

[2]  A. Adrian Albert,et al.  Fundamental Concepts of Higher Algebra , 1958 .

[3]  Harald Niederreiter,et al.  A Public-Key Cryptosystem based on Shift Register Sequences , 1985, EUROCRYPT.

[4]  E. Berlekamp.  Factoring polynomials over large finite fields , 1970 .

[5]  D. Cantor,et al.  A new algorithm for factoring polynomials over finite fields , 1981 .

[6]  J. Hirschfeld.  Projective Geometries Over Finite Fields , 1980 .

[7]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[8]  R. Lidl,et al.  Applied abstract algebra , 1984 .

[9]  E. Watson.  Primitive Polynomials (Mod 2) , 1962 .

[10]  H. Zassenhaus.  On Hensel factorization, I , 1969 .

[11]  Michael A. Arbib,et al.  Topics in Mathematical System Theory , 1969 .

[12]  Jacobus H. van Lint,et al.  Introduction to Coding Theory , 1982 .

[13]  Larry L. Dornhoff,et al.  Applied Modern Algebra , 1978 .

[14]  Ian F. Blake,et al.  The mathematical theory of coding , 1975 .

[15]  A. Adrian Albert,et al.  An Introduction to Finite Projective Planes , 1970 .

[16]  Henry Beker,et al.  Cipher Systems: The Protection of Communications , 1982 .

[17]  Harald Niederreiter.  The Performance of k-Step Pseudorandom Number Generators under the Uniformity Test , 1984 .

[18]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[19]  James L. Massey,et al.  Shift-register synthesis and BCH decoding , 1969, IEEE Trans. Inf. Theory.

[20]  Robert J. McEliece,et al.  A public key cryptosystem based on algebraic coding theory , 1978 .

[21]  E. Polak,et al.  System Theory , 1963 .

[22]  S. M. Jennings,et al.  Multiplexed Sequences: Some Properties of the Minimum Polynomial , 1982, EUROCRYPT.

[23]  Martin E. Hellman,et al.  An improved algorithm for computing logarithms over GF(p) and its cryptographic significance (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[24]  H. Niederreiter.  On the cycle structure of linear recurring sequences. , 1976 .

[25]  D. Raghavarao.  Constructions and Combinatorial Problems in Design of Experiments , 1971 .

[26]  Abraham Bookstein,et al.  Cryptography: A new dimension in computer data security ? and ?. Wiley-Interscience, New York (1982). xxi + 775 pp., $43.95. ISBN 0471-04892-5. , 1985 .

[27]  Dorothy E. Denning,et al.  Cryptography and Data Security , 1982 .

[28]  S. Vanstone,et al.  Computing Logarithms in Finite Fields of Characteristic Two , 1984 .

[29]  Alan Ewert,et al.  A New Dimension , 1986 .

[30]  Kenneth P. Bogart,et al.  Introductory Combinatorics , 1977 .

[31]  Michael O. Rabin,et al.  Probabilistic Algorithms in Finite Fields , 1980, SIAM J. Comput..

[32]  V. Pless.  Introduction to the Theory of Error-Correcting Codes , 1991 .

[33]  W. W. Peterson,et al.  Error-Correcting Codes. , 1962 .

[34]  Solomon W. Golomb,et al.  Shift Register Sequences , 1981 .

[35]  John B. Fraleigh.  A first course in abstract algebra , 1967 .

[36]  N. Zierler.  Linear Recurring Sequences , 1959 .

[37]  Randolph Church.  Tables of Irreducible Polynomials for the First Four Prime Moduli , 1935 .

[38]  Jay P. Fillmore,et al.  Linear Recursive Sequences , 1968 .

[39]  Abraham Lempel,et al.  Cryptology in Transition , 1979, CSUR.

[40]  J. Dénes,et al.  Latin squares and their applications , 1974 .

[41]  Wayne Stahnke.  Primitive binary polynomials , 1973 .

[42]  Robert J. McEliece.  Factorization of polynomials over finite fields , 1969 .

[43]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[44]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[45]  T. Elgamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, CRYPTO 1984.

[46]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[47]  Ronald L. Rivest,et al.  A knapsack-type public key cryptosystem based on arithmetic in finite fields , 1988, IEEE Trans. Inf. Theory.

[48]  L. Dickson.  Linear Groups, with an Exposition of the Galois Field Theory , 1958 .