Number theoretic methods in cryptography

Includes index

Includes bibliography (p. 165-177) and references

1 Introduction 2 Basic Notation and Definitions 3 Auxiliary Results II Approximation and Complexity of the Discrete Logarithm 4 Approximation of the Discrete Logarithm Modulo p 5 Approximation of the Discrete Logarithm Modulo p — 1 6 Approximation of the Discrete Logarithm by Boolean Functions 7 Approximation of the Discrete Logarithm by Real and Complex Polynomials III Complexity of Breaking the Diffie-Hellman Cryptosystem 8 Polynomial Approximation and Arithmetic Complexity of the Diffie-Hellman Key 9 Boolean Complexity of the Diffie-Hellman Key IV Other Applications 10 Trade-off between the Boolean and Arithmetic Depths of Modulo p Functions 11 Special Polynomials and Boolean Functions 12 RSA and Blum-Blum-Shub Generators of Pseudo-Random Numbers V Concluding Remarks 13 Generalizations and Open Questions 14 Further Directions.

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.