By Igor Shparlinski

The ebook introduces new methods of utilizing analytic quantity thought in cryptography and similar parts, corresponding to complexity concept and pseudorandom quantity generation.

Key subject matters and features:

- numerous reduce bounds at the complexity of a few quantity theoretic and cryptographic difficulties, linked to classical schemes resembling RSA, Diffie-Hellman, DSA in addition to with particularly new schemes like XTR and NTRU

- a chain of very fresh effects approximately convinced vital features (period, distribution, linear complexity) of a number of popular pseudorandom quantity turbines, resembling the RSA generator, Blum-Blum-Shub generator, Naor-Reingold generator, inversive generator, and others

- one of many crucial instruments is bounds of exponential sums, that are mixed with different quantity theoretic equipment akin to lattice relief and sieving

- a couple of open difficulties of other point of trouble and recommendations for additional research

- an in depth and updated bibliography

Cryptographers and quantity theorists will locate this booklet necessary. the previous can find out about new quantity theoretic options that have proved to be necessary cryptographic instruments, the latter approximately new demanding parts of functions in their skills.

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

Example text

JK there exist Cl, ... ,CK E IF, not all equal to zero, such that K L CiU(X + ji) = 0, x = 1,2, .... i=l Proof. It is easy to see that the set of all solutions of any linear recurrence relation over any field IF is a linear space of dimension n over IF, for example, see Chapter 8 of [343J. Therefore, any T 2 n + 1 solutions are linearly dependent. In particular, the sequences (u(x + jl)) , ... , (u(x + jK)) are linearly dependent. 1 which can be proved using similar arguments. 2. Let an N -element sequence W n , n = 0, ...

Ik ~ r there are at least s distin~t functions obtainable by making all 2k possible assignments to Ui1 , •.. , Uik . In particular, functions of the class P1,2 depend on all their variables. A restriction p of the set {I, ... ,r} is a mapping of this set to the set {O,l,*}, where o p(i) = 0 means that we substitute the value 0 for Xi i o p( i) Xi i o p(i) = * means that Xi remains a variable. = 1 means that we substitute the value 1 for Given a Boolean function B(X1 , ... ,Xr ) of r variables, we will denote by Bp the function obtained from B by applying the restriction Pi Bp will be a function of the variables Xi for which p(Xi ) = *, 1 ~ i ~ r.

If for some ao, ... , ak E IF q we have an identity k Ho(X) = ao + l: aiHi(X), i=l then k 2 T. Proof. The result is immediate for b = 0, so we can assume that b =I- O. We see that Hi(X) = fi(X)/gi(X) are non-constant rational functions, where fi,gi E IFp[X] with max{degJ;, deggi} = 1. Now we write gi(X) = CiX +di with ci,di E lF q . Clearing denominators in the identity of the lemma, we have: Ho(X) k k k i=l i=l i=l k IT (Ci X + di ) = ao IT (Ci X + di ) + l: adi(X) IT (cjX + dj ). j=1 j#i Comparing the coefficient of Xk+1 of the left-hand side and the right-hand side, we see that C1 ...

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