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

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Springer Science & Business Media, Feb 12, 2003 - Computers - 411 pages
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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
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