# Number Theoretic Methods in Cryptography

## Complexity lower bounds

• Igor Shparlinski
Book

Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 17)

1. Front Matter
Pages i-ix
2. ### Preliminaries

1. Front Matter
Pages 1-1
2. Igor Shparlinski
Pages 3-12
3. Igor Shparlinski
Pages 13-18
4. Igor Shparlinski
Pages 19-36
3. ### Approximation and Complexity of the Discrete Logarithm

1. Front Matter
Pages 37-37
2. Igor Shparlinski
Pages 39-47
3. Igor Shparlinski
Pages 49-52
4. Igor Shparlinski
Pages 53-65
5. Igor Shparlinski
Pages 67-80
4. ### Complexity of Breaking the Diffie—Hellman Cryptosystem

1. Front Matter
Pages 81-81
2. Igor Shparlinski
Pages 83-96
3. Igor Shparlinski
Pages 97-106
5. ### Other Applications

1. Front Matter
Pages 107-107
2. Igor Shparlinski
Pages 109-123
3. Igor Shparlinski
Pages 125-130
4. Igor Shparlinski
Pages 131-141
6. ### Concluding Remarks

1. Front Matter
Pages 143-143
2. Igor Shparlinski
Pages 145-157
3. Igor Shparlinski
Pages 159-164

### Introduction

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.

### Keywords

complexity complexity theory computer science cryptography finite field number theory

#### Authors and affiliations

• Igor Shparlinski
• 1
1. 1.School of Mathematics, Physics, Computing and ElectronicsMacquarie UniversityAustralia

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-0348-8664-2
• Copyright Information Birkhäuser Verlag 1999
• Publisher Name Birkhäuser, Basel
• eBook Packages
• Print ISBN 978-3-0348-9723-5
• Online ISBN 978-3-0348-8664-2