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Cryptology
"... Cryptology has advanced tremendously since 1976; this chapter provides a brief overview of the current stateoftheart in the field. Several major themes predominate in the development. One such theme is the careful elaboration of the definition of security for a cryptosystem. A second theme has be ..."
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Cryptology has advanced tremendously since 1976; this chapter provides a brief overview of the current stateoftheart in the field. Several major themes predominate in the development. One such theme is the careful elaboration of the definition of security for a cryptosystem. A second theme has been the search for provably secure cryptosystems, based on plausible assumptions about the difficulty of specific numbertheoretic problems or on the existence of certain kinds of functions (such as oneway functions). A third theme is the invention of many novel and surprising cryptographic capabilities, such as publickey cryptography, digital signatures, secretsharing, oblivious transfers, and zeroknowledge proofs. These themes have been developed and interwoven so that today theorems of breathtaking generality and power assert the existence of cryptographic techniques capable of solving almost any imaginable cryptographic problem.
The Computational Complexity of Randomness
, 2013
"... This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to wh ..."
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This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem’s output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling throughthelens ofcomputational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that “a little more time gives a lot more power to sampling algorithms.” Part II concerns settings where the algorithm’s output is random. Even when the specificationofacomputational problem involves no randomness, onecanstill consider randomized
Mathematical Foundations of Modern Cryptography: Computational Complexity Perspective
, 2002
"... Theoretical computer science has found fertile ground in many areas of mathematics. The approach has been to consider classical problems through the prism of computational complexity, where the number of basic computational steps taken to solve a problem is the crucial qualitative parameter. This ne ..."
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Theoretical computer science has found fertile ground in many areas of mathematics. The approach has been to consider classical problems through the prism of computational complexity, where the number of basic computational steps taken to solve a problem is the crucial qualitative parameter. This new approach has led to a sequence of advances, in setting and solving new mathematical challenges as well as in harnessing discrete mathematics to the task of solving realworld problems. In this talk, I will survey the development of modern cryptography — the mathematics behind secret communications and protocols — in this light. I will describe the complexity theoretic foundations underlying the cryptographic tasks of encryption, pseudorandomness number generators and functions, zero knowledge interactive proofs, and multiparty secure protocols. I will attempt to highlight the paradigms and proof techniques which unify these foundations, and which have made their way into the mainstream of complexity theory.
Open Problems on Exponential and Character Sums
, 2010
"... This is a collection of mostly unrelated open questions, at various levels of difficulty, related to exponential and multiplicative character sums. One may certainly notice a large proportion of selfreferences in the bibliography. By no means should this be considered as an indication of anything e ..."
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This is a collection of mostly unrelated open questions, at various levels of difficulty, related to exponential and multiplicative character sums. One may certainly notice a large proportion of selfreferences in the bibliography. By no means should this be considered as an indication of anything else than
Generating Random Factored Gaussian Integers, Easily
, 2013
"... We introduce an algorithm to generate a random Gaussian integer with the uniform distribution among those with norm at most N, along with its prime factorization. Then, we show that the algorithm runs in polynomial time. The hard part of this algorithm is determining a norm at random with a specific ..."
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We introduce an algorithm to generate a random Gaussian integer with the uniform distribution among those with norm at most N, along with its prime factorization. Then, we show that the algorithm runs in polynomial time. The hard part of this algorithm is determining a norm at random with a specific distribution. After that, finding the actual Gaussian integer is easy. We also consider the analogous problem for Eisenstein integers and quadratic integer rings. 1 1 Generating Random Factored Numbers, Easily Consider the following problem: Given a positive integer N, generatearandomintegerlessthanorequaltoN with uniform distribution, along with its factorization in polynomial time. (In this context, polynomial time refers to a polynomial in the number of digits of N, not the size of N. So,the running time of our algorithm should be O(log k N), for some real k.) At first glance, this seems very simple. Simply choose a random integer in the range [1,N] and factor it. However, there are no known polynomial time factorization algorithms. But,
Abstract The (True) Complexity of Statistical Zero Knowledge (Extended Abstract)
"... Statistical zeroknowledge is a very strong privacy constraint which is not dependent on computational limitations. In this paper we showthatgiven a complexity assumption a much weaker condition su ces to attain statistical zeroknowledge. As a result we are able to simplify statistical zeroknowled ..."
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Statistical zeroknowledge is a very strong privacy constraint which is not dependent on computational limitations. In this paper we showthatgiven a complexity assumption a much weaker condition su ces to attain statistical zeroknowledge. As a result we are able to simplify statistical zeroknowledge and to better characterize, on many counts, the class of languages that possess statistical zeroknowledge proofs. 1
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"... Abstract — The number field sieve of factoring 1024bit RSA keys has many steps involved, one of them being the ‘relation collection ’ step. This step consists of two parts, the first part is called Sieving and is used to generate smooth random numbers (numbers whose factors are less than a given bo ..."
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Abstract — The number field sieve of factoring 1024bit RSA keys has many steps involved, one of them being the ‘relation collection ’ step. This step consists of two parts, the first part is called Sieving and is used to generate smooth random numbers (numbers whose factors are less than a given boundary), and these smooth numbers are factored in the second part by special factoring methods like ECM, p1 and rho. The Sieving processes involve high complexity in generating smooth numbers, so there is a requirement for a simple mechanism which would generate large smooth numbers so that these numbers can be used as test vectors for the factorization methods like ECM, P1 and rho. In the project our aim is to generate the smooth numbers so that they can be as test vectors. We used Eric Bach and Adam Kalai’s algorithms which are developed for generating factored random numbers, and made modifications to these algorithms in order to generate smooth numbers. In this paper we describe these algorithms, modifications we made to them and how we implemented them. We also compare these algorithms and come to a conclusion as to which method is faster in generating smooth factored random numbers.
Asymptotic Semismoothness Probabilities
"... Abstract We call an integer semismooth with respect to y and z if each of its prime factors is ^ y, and all but one are ^ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(ff; fi) be the asymptotic probability that a random integer n is semismooth with ..."
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Abstract We call an integer semismooth with respect to y and z if each of its prime factors is ^ y, and all but one are ^ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(ff; fi) be the asymptotic probability that a random integer n is semismooth with respect to nfi and nff. We present new recurrence relations for G and related functions. We then give numerical methods for computing G, tables of G, and estimates for the error incurred by this asymptotic approximation.