Number Theoretic Methods in Cryptography

1. Front Matter

   Pages i-ix

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2. Preliminaries

   1. Front Matter

      Pages 1-1

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   2. Introduction

      • Igor Shparlinski

      Pages 3-12

   3. Basic Notation and Definitions

      • Igor Shparlinski

      Pages 13-18

   4. Auxiliary Results

      • Igor Shparlinski

      Pages 19-36

3. Approximation and Complexity of the Discrete Logarithm

   1. Front Matter

      Pages 37-37

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   2. Approximation of the Discrete Logarithm Modulo p

      • Igor Shparlinski

      Pages 39-47

   3. Approximation of the Discrete Logarithm Modulo p - 1

      • Igor Shparlinski

      Pages 49-52

   4. Approximation of the Discrete Logarithm by Boolean Functions

      • Igor Shparlinski

      Pages 53-65

   5. Approximation of the Discrete Logarithm by Real and Complex Polynomials

      • Igor Shparlinski

      Pages 67-80

4. Complexity of Breaking the Diffie—Hellman Cryptosystem

   1. Front Matter

      Pages 81-81

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   2. Polynomial Approximation and Arithmetic Complexity of the Diffie—Hellman Key

      • Igor Shparlinski

      Pages 83-96

   3. Boolean Complexity of the Diffie—Hellman Key

      • Igor Shparlinski

      Pages 97-106

5. Other Applications

   1. Front Matter

      Pages 107-107

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   2. Trade-off between the Boolean and Arithmetic Depths of Modulo p Functions

      • Igor Shparlinski

      Pages 109-123

   3. Special Polynomials and Boolean Functions

      • Igor Shparlinski

      Pages 125-130

   4. RSA and Blum—Blum—Shub Generators of Pseudo-Random Numbers

      • Igor Shparlinski

      Pages 131-141

6. Concluding Remarks

   1. Front Matter

      Pages 143-143

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   2. Generalizations and Open Questions

      • Igor Shparlinski

      Pages 145-157

   3. Further Directions

      • Igor Shparlinski

      Pages 159-164

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.

• complexity
• complexity theory
• computer science
• cryptography
• finite field
• number theory

"This volume gives a thorough treatment of the complexity of the discrete logarithm problem in a prime field, as well as related problems. The final chapter on further directions gives an interesting selection of problems."

--Zentralblatt Math

• School of Mathematics, Physics, Computing and Electronics, Macquarie University, Australia

  Igor Shparlinski

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