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PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
On derandomizing tests for certain polynomial identities
 In Proceedings of the Conference on Computational Complexity
, 2003
"... We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests. 1 ..."
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We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests. 1
A DETERMINISTIC VERSION OF POLLARD’S p − 1 ALGORITHM
"... Abstract. In this article we present applications of smooth numbers to the unconditional derandomization of some wellknown integer factoring algorithms. We begin with Pollard’s p − 1 algorithm, which finds in random polynomial time the prime divisors p of an integer n such that p − 1issmooth.Weshow ..."
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Abstract. In this article we present applications of smooth numbers to the unconditional derandomization of some wellknown integer factoring algorithms. We begin with Pollard’s p − 1 algorithm, which finds in random polynomial time the prime divisors p of an integer n such that p − 1issmooth.Weshow that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the kth cyclotomic method of factoring (k ≥ 2) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler’s totient function ϕ. We point out some explicit sets of integers n that are completely factorable in deterministic polynomial time given ϕ(n). These sets consist, roughly speaking, of products of primes p satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of p − 1. Finally, we prove that O(ln n) oracle queries for values of ϕ are sufficient to completely factor any integer n in less than exp (1 + o(1))(ln n) 1 3 (ln ln n) 2) 3 deterministic time. 1.
A VARIANT OF THE BOMBIERIVINOGRADOV THEOREM WITH EXPLICIT CONSTANTS AND APPLICATIONS
"... ABSTRACT. We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λ and a Dirichlet character χ. As a consequence of this result we prove an effective variant of t ..."
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ABSTRACT. We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λ and a Dirichlet character χ. As a consequence of this result we prove an effective variant of the BombieriVinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes. For integers a and q ≥ 1, let 1.
Authenticated DiffieHellman Key Exchange Algorithm
"... AbstractThe ability to distribute cryptographic keys has been a challenge for centuries. The DiffieHellman was the first practical solution to the problem.However, if the key exchange takes place in certain mathematical environments, the key exchange become vulnerable to a specific ManinMiddle a ..."
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AbstractThe ability to distribute cryptographic keys has been a challenge for centuries. The DiffieHellman was the first practical solution to the problem.However, if the key exchange takes place in certain mathematical environments, the key exchange become vulnerable to a specific ManinMiddle attack, first observed by Vanstone. This paper is an effort to solve a serious problem in DiffieHellman key exchange, that is, ManinMiddle attack. In this paper we have used RSA algorithm along with DiffieHellman to solve the problem. We explore the ManinMiddle attack, analyse the countermeasures against the attack. Index TermsCryptography, DiffieHellman, ManinMiddle attack, primality testing.