Number Theoretic Methods in Cryptography

Complexity lower bounds

Igor Shparlinski (Autor)

Buch | Softcover

IX, 182 Seiten

2012 | 1. Softcover reprint of the original 1st ed. 1999
Springer Basel (Verlag)
978-3-0348-9723-5 (ISBN)

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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.

I Preliminaries.- 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.

"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

Erscheint lt. Verlag	8.10.2012
Reihe/Serie	Progress in Computer Science and Applied Logic
Zusatzinfo	IX, 182 p.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	311 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Complexity • Complexity theory • Computer Science • cryptography • finite field • Number Theory
ISBN-10	3-0348-9723-5 / 3034897235
ISBN-13	978-3-0348-9723-5 / 9783034897235
Zustand	Neuware