The book introduces new techniques that imply rigorous lower bounds on the com­ plexity of some number-theoretic and cryptographic problems. These functions are considered over the residue ring modulo p and over the residue ring modulo an arbitrary divisor d of p - 1.

I Preliminaries.- 1 Basic Notation and Definitions.- 2 Polynomials and Recurrence Sequences.- 3 Exponential Sums.- 4 Distribution and Discrepancy.- 5 Arithmetic Functions.- 6 Lattices and the Hidden Number Problem.- 7 Complexity Theory.- II Approximation and Complexity of the Discrete Logarithm.- 8 Approximation of the Discrete Logarithm Modulop.- 9 Approximation of the Discrete Logarithm Modulop -1.- 10 Approximation of the Discrete Logarithm by Boolean Functions.- 11 Approximation of the Discrete Logarithm by Real Polynomials.- III Approximation and Complexity of the Diffie-Hellman Secret Key.- 12 Polynomial Approximation and Arithmetic Complexity of the.- Diffie-Hellman Secret Key.- 13 Boolean Complexity of the Diffie-Hellman Secret Key.- 14 Bit Security of the Diffie-Hellman Secret Key.- IV Other Cryptographic Constructions.- 15 Security Against the Cycling Attack on the RSA and Timed-release Crypto.- 16 The Insecurity of the Digital Signature Algorithm with Partially Known Nonces.- 17 Distribution of the ElGamal Signature.- 18 Bit Security of the RSA Encryption and the Shamir Message Passing Scheme.- 19 Bit Security of the XTR and LUC Secret Keys.- 20 Bit Security of NTRU.- 21 Distribution of the RSA and Exponential Pairs.- 22 Exponentiation and Inversion with Precomputation.- V Pseudorandom Number Generators.- 23 RSA and Blum-Blum-Shub Generators.- 24 Naor-Reingold Function.- 25 1/M Generator.- 26 Inversive, Polynomial and Quadratic Exponential Generators.- 27 Subset Sum Generators.- VI Other Applications.- 28 Square-Freeness Testing and Other Number-Theoretic Problems.- 29 Trade-off Between the Boolean and Arithmetic Depths of ModulopFunctions.- 30 Polynomial Approximation, Permanents and Noisy Exponentiation in Finite Fields.- 31 Special Polynomials and BooleanFunctions.- VII Concluding Remarks and Open Questions.

The book introduces new ways of using analytic number theory in cryptography and related areas, such as complexity theory and pseudorandom number generation.

Key topics and features:

- various lower bounds on the complexity of some number theoretic and cryptographic problems, associated with classical schemes such as RSA, Diffie-Hellman, DSA as well as with relatively new schemes like XTR and NTRU

- a series of very recent results about certain important characteristics (period, distribution, linear complexity) of several commonly used pseudorandom number generators, such as the RSA generator, Blum-Blum-Shub generator, Naor-Reingold generator, inversive generator, and others

- one of the principal tools is bounds of exponential sums, which are combined with other number theoretic methods such as lattice reduction and sieving

- a number of open problems of different level of difficulty and proposals for further research

- an extensive and up-to-date bibliography

Cryptographers and number theorists will find this book useful. The former can learn about new number theoretic techniques which have proved to be invaluable cryptographic tools, the latter about new challenging areas of applications of their skills.

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