Number Theoretic Methods in Cryptography: Complexity lower bounds

Description

The book introduces new techniques which imply rigorous lower bounds on the complexity of some number theoretic and cryptographic problems. These methods and techniques are based on bounds of character sums and numbers of solutions of some polynomial equations over finite fields and residue rings. It also contains a number of open problems and proposals for further research. We obtain several lower bounds, exponential in terms of logp, on the de­ grees and orders of • polynomials; • algebraic functions; • Boolean functions; • linear recurring sequences; coinciding with values of the discrete logarithm modulo a prime p at suf­ ficiently many points (the number of points can be as small as pI/He). These functions are considered over the residue ring modulo p and over the residue ring modulo an arbitrary divisor d of p - 1. The case of d = 2 is of special interest since it corresponds to the representation of the right­ most bit of the discrete logarithm and defines whether the argument is a quadratic residue. We also obtain non-trivial upper bounds on the de­ gree, sensitivity and Fourier coefficients of Boolean functions on bits of x deciding whether x is a quadratic residue. These results are used to obtain lower bounds on the parallel arithmetic and Boolean complexity of computing the discrete logarithm. For example, we prove that any unbounded fan-in Boolean circuit. of sublogarithmic depth computing the discrete logarithm modulo p must be of superpolynomial size Front Matter....Pages i-ix Front Matter....Pages 1-1 Introduction....Pages 3-12 Basic Notation and Definitions....Pages 13-18 Auxiliary Results....Pages 19-36 Front Matter....Pages 37-37 Approximation of the Discrete Logarithm Modulo p ....Pages 39-47 Approximation of the Discrete Logarithm Modulo p - 1....Pages 49-52 Approximation of the Discrete Logarithm by Boolean Functions....Pages 53-65 Approximation of the Discrete Logarithm by Real and Complex Polynomials....Pages 67-80 Front Matter....Pages 81-81 Polynomial Approximation and Arithmetic Complexity of the Diffie—Hellman Key....Pages 83-96 Boolean Complexity of the Diffie—Hellman Key....Pages 97-106 Front Matter....Pages 107-107 Trade-off between the Boolean and Arithmetic Depths of Modulo p Functions....Pages 109-123 Special Polynomials and Boolean Functions....Pages 125-130 RSA and Blum—Blum—Shub Generators of Pseudo-Random Numbers....Pages 131-141 Front Matter....Pages 143-143 Generalizations and Open Questions....Pages 145-157 Further Directions....Pages 159-164 Back Matter....Pages 165-182