# Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness

Springer Science & Business Media, Feb 12, 2003 - Computers - 411 pages
The book introduces new techniques that imply rigorous lower bounds on the com plexity of some number-theoretic and cryptographic problems. It also establishes certain attractive pseudorandom properties of various cryptographic primitives. These methods and techniques are based on bounds of character sums and num bers of solutions of some polynomial equations over finite fields and residue rings. Other number theoretic techniques such as sieve methods and lattice reduction algorithms are used as well. The book also contains a number of open problems and proposals for further research. The emphasis is on obtaining unconditional rigorously proved statements. The bright side of this approach is that the results do not depend on any assumptions or conjectures. On the downside, the results are much weaker than those which are widely believed to be true. We obtain several lower bounds, exponential in terms of logp, on the degrees and orders of o polynomials; o algebraic functions; o Boolean functions; o linear recurrence sequences; coinciding with values of the discrete logarithm modulo a prime p at sufficiently many points (the number of points can be as small as pI/2+O:). 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 rightmost bit of the discrete logarithm and defines whether the argument is a quadratic residue.

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### Contents

 Basic Notation and Definitions 17 Polynomials and Recurrence Sequences 27 Exponential Sums 37 Distribution and Discrepancy 61 Arithmetic Functions 67 Lattices and the Hidden Number Problem 83 Complexity Theory 103 Approximation and Complexity of the Discrete Logarithm 107
 Bit Security of the RSA Encryption and the Shamir Message Passing Scheme 211 Bit Security of the XTR and LUC Secret Keys 217 Bit Security of NTRU 223 Distribution of the RSA and Exponential Pairs 231 Exponentiation and Inversion with Precomputation 239 Pseudorandom Number Generators 247 RSA and BlumBlumShub Generators 249 NaorReingold Function 271

 Approximation of the Discrete Logarithm Modulo p 109 Approximation of the Discrete Logarithm Modulo p 1 123 Approximation of the Discrete Logarithm by Boolean Functions 129 Approximation of the Discrete Logarithm by Real Polynomials 143 Approximation and Complexity of the DiffieHellman Secret Key 157 Polynomial Approximation and Arithmetic Complexity of the DiffieHellman Secret Key 159 Boolean Complexity of the DiffieHellman Secret Key 179 Bit Security of the DiffieHellman Secret Key 189 Other Cryptographic Constructions 195 Security Against the Cycling Attack on the RSA and Timedrelease Crypto 197 The Insecurity of the Digital Signature Algorithm with Partially Known Nonces 201 Distribution of the ElGamal Signature 207
 1M Generator 279 Inversive Polynomial and Quadratic Exponential Generators 283 Subset Sum Generators 295 Other Applications 301 SquareFreeness Testing and Other NumberTheoretic Problems 303 Tradeoff Between the Boolean and Arithmetic Depths of Modulo p Functions 309 Polynomial Approximation Permanents and Noisy Exponentiation in Finite Fields 325 Special Polynomials and Boolean Functions 333 Concluding Remarks and Open Questions 341 Bibliography 367 Index 409 Copyright