By Igor Shparlinski

The ebook introduces new methods of utilizing analytic quantity idea in cryptography and comparable parts, akin to complexity conception and pseudorandom quantity generation.

Key issues and features:

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

- a sequence of very contemporary effects approximately convinced vital features (period, distribution, linear complexity) of a number of well-known pseudorandom quantity turbines, reminiscent of the RSA generator, Blum-Blum-Shub generator, Naor-Reingold generator, inversive generator, and others

- one of many valuable instruments is bounds of exponential sums, that are mixed with different quantity theoretic equipment corresponding to lattice aid and sieving

- a few open difficulties of alternative point of hassle and recommendations for additional research

- an in depth and updated bibliography

Cryptographers and quantity theorists will locate this e-book valuable. the previous can know about new quantity theoretic strategies that have proved to be worthwhile cryptographic instruments, the latter approximately new tough components of functions in their skills.

**Read Online or Download Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness PDF**

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**Additional info for Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness**

**Sample text**

7, we obtain the desired estimate. 16 has been used in [526] to obtain new bounds of Gauss sums in finite fields. 3 of [407]. 10. For any E > 0 there exists "I > 0 such that for any non-trivial multiplicative character X modulo p IX~I X(X)I :::; Hp-' for H :::: pI/HE and sufficiently large p. Part 1: Preliminaries 42 It is known that the Extended Riemann Hypothesis implies non-trivial upper bounds for much shorter sums. In particular, we use a bound which follows from one of the results of [230].

For the second configuration we merely ignore the congruence condition on Xi4 and remark that it is uniquely defined for each choice of XiI' Xi 2, Xi3 for which we have at most (p - 1)3/d l d 2 possibilities. 123 .... 11) we obtain ~ I~ e 4 p (ag X + cgxy)1 :::; 14((5/3)p14/3 + O(p13/:l). lO) we derive the desired result. D In fact, a more general estimate has been obtained in [94] which applies to elements of A E IF p of multiplicative order T :2': p3/ He . 14) We fix an element 9 E IF; of multiplicative order t and consider double exponential sums Sa(X, Y) = LL e p (ag XY ) , xEXyEY where a E lFp and X,Y ~ 7l.

6 with t = 4, we see that for each quadruple (XI,X2,X3,X4), there are at most values of y = 1, ... ,p - 1 which satisfy that system, where d = min maxgcd(xi l:Si:S4 Hi xJ,p -1). 12) 52 Part I: Preliminaries We claim that the number of quadruples (Xl, X2. X;l. 12) and which satisfy the first congruence of the above system of congruences does not exceed p3d-2 + p2. 12) holds belongs to one of the following two types of configuration. In the first case they can be separated into two pairs {i l ,i 2 ,i 3 ,i4 } = {1,2,3,4} and such that There are three choices of a partner for Xl (XiI' Xi2) and (Xi3' Xi4) with and this determines the pairings.