Covered Topics

• Algebra
  • Complexity in all algebraic constructions
  • Algebraic complexity theory
  • Gröbner bases: structure, and solution of integer programs
  • bases of fields : finite fields, cyclotomic fields, special bases, optimal bases
  • explicit constructions : irreducible polynomials, factorizing polynomials
  • finite structures : Galois rings, codes over rings (Z_4, local and semi-local rings, Gray maps)
  • effective algebra : Nullstellensatz, linear algebra
  • multivariate polynomial problems, factorization/decomposition of multiv. polynomials, multivariate polynomial applications
  
   
• Codes and Algebra
  • algebraic codes : cyclic, double circulant, abelian codes
  • multiv. polynomial codes, concatenated codes, product codes
  • decoding methods and their complexity : algorithms for AG codes, Gröbner bases and decoding, trellis decoding, Turbo decoding, iterative algorithms for concatenated codes
  • AG-codes, algebraic geometry over finite fields
  • automorphisms of codes
  
   
• Codes and Combinatorics
  • Hadamard and weighing matrices, highly regular graphs from cosets, designs in codewords, r-partition designs, orthogonal arrays, (t,m,s)-nets
  • euclidean lattices (spherical codes and designs, decoding, vector quantizing, higher dimensional modulation)
  
   
• Modulation and Codes, Sequences
  • unequal error protection, coset codes, multidimensional trellis coded modulation, multilevel coding
  • lattices : Gaussian channel modulation
  • sequences (low correlation for spread spectrum and CDMA, Hamming correlation for FHMA, pseudo-random sequences for cryptography, complexity profile)
  
   
• Cryptography
  • public-key Cryptography, elliptic curve cryptosystems, computational number theory, authentication and digital signatures, Hash Functions, zero-knowledge proofs, pseudorandom numbers and sequences, stream ciphers and block ciphers