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

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.

---

---

Content:
Front Matter....Pages i-ix
Introduction....Pages 1-14
Front Matter....Pages 15-15
Basic Notation and Definitions....Pages 17-26
Polynomials and Recurrence Sequences....Pages 27-36
Exponential Sums....Pages 37-60
Distribution and Discrepancy....Pages 61-65
Arithmetic Functions....Pages 67-81
Lattices and the Hidden Number Problem....Pages 83-102
Complexity Theory....Pages 103-106
Front Matter....Pages 107-107
Approximation of the Discrete Logarithm by Boolean Functions....Pages 109-122
Approximation of the Discrete Logarithm by Real Polynomials....Pages 123-128
Front Matter....Pages 129-141
Polynomial Approximation and Arithmetic Complexity of the Diffie-Hellman Secret Key....Pages 143-156
Boolean Complexity of the Diffie-Hellman Secret Key....Pages 157-157
Bit Security of the Diffie—Hellman Secret Key....Pages 159-177
Front Matter....Pages 179-188
Security Against the Cycling Attack on the RSA and Timed-release Crypto....Pages 189-194
The Insecurity of the Digital Signature Algorithm with Partially Known Nonces....Pages 195-195
Distribution of the ElGamal Signature....Pages 197-200
Bit Security of the RSA Encryption and the Shamir Message Passing Scheme....Pages 201-206
Bit Security of the XTR and LUC Secret Keys....Pages 207-210
Front Matter....Pages 211-215
Bit Security of NTRU....Pages 217-221
Distribution of the RSA and Exponential Pairs....Pages 195-195
Exponentiation and Inversion with Precomputation....Pages 223-229
Front Matter....Pages 231-237
RSA and Blum—Blum—Shub Generators....Pages 239-245
Naor—Reingold Function....Pages 247-247
1/M Generator....Pages 249-270
Inversive, Polynomial and Quadratic Exponential Generators....Pages 271-277
Subset Sum Generators....Pages 279-282
Front Matter....Pages 283-294
Square-Freeness Testing and Other Number-Theoretic Problems....Pages 295-299
Trade-off Between the Boolean and Arithmetic Depths of Modulo p Functions....Pages 301-301
Polynomial Approximation, Permanents and Noisy Exponentiation in Finite Fields....Pages 303-308
Special Polynomials and Boolean Functions....Pages 309-323
Front Matter....Pages 325-332
Concluding Remarks and Open Questions....Pages 333-339
Back Matter....Pages 341-341
....Pages 341-365