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Primality testing
, 1992
"... Abstract For many years mathematicians have searched for a fast and reliable primality test. This is especially relevant nowadays, because the RSA publickey cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful ..."
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Abstract For many years mathematicians have searched for a fast and reliable primality test. This is especially relevant nowadays, because the RSA publickey cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful in practice, but have the defect of occasionally giving the wrong answer, or taking a very long time to give an answer. Recently Agrawal, Kayal and Saxena found a deterministic polynomialtime primality test. I will describe their algorithm, mention some improvements by Bernstein and Lenstra, and explain why this is not the end of the story.
Two Observations on Probabilistic Primality Testing
, 1987
"... In this note, we make two loosely related observations on Rabin's probabilistic primality test. The first remark gives a rather strange and provocative reason as to why is Rabin's test so good. It turns out that a single iteration fails with a nonnegligible probability on a composite number of the ..."
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In this note, we make two loosely related observations on Rabin's probabilistic primality test. The first remark gives a rather strange and provocative reason as to why is Rabin's test so good. It turns out that a single iteration fails with a nonnegligible probability on a composite number of the form 4j +3 only if this number happens to be easy to split. The second observation is much more fundamental because is it not restricted to primality testing: it has profound consequences for the entire field of probabilistic algorithms. There we ask the question: how good is Rabin's algorithm? Whenever one wishes to produce a uniformly distributed random probabilistic prime with a given bound on the error probability, it turns out that the size of the desired prime must be taken into account. 1. Introduction In this note, we make two loosely related observations on Rabin's probabilistic primality test. The first remark gives a rather strange and provocative reason as to why is Rabin's te...
Complexity: A LanguageTheoretic Point of View
, 1995
"... this paper (see the discussion in [51, 58, 70, 126, 127, 120, 121, 130]); in what follows we shall superficially review this topic in connection with the related question: can computers think? ..."
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this paper (see the discussion in [51, 58, 70, 126, 127, 120, 121, 130]); in what follows we shall superficially review this topic in connection with the related question: can computers think?
A First Study of the Neural Network Approach in the RSA Cryptosystem
 IASTED 2002 Conference on Artificial Intelligence
, 2002
"... The RSA cryptosystem is supposed to be the first realization of a public key cryptosystem in 1977. Its (computational) security is relied upon the difficulty of factorization. In order to break the RSA cryptosystem it is enough to factorize is the product of two large prime numbers, . This i ..."
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The RSA cryptosystem is supposed to be the first realization of a public key cryptosystem in 1977. Its (computational) security is relied upon the difficulty of factorization. In order to break the RSA cryptosystem it is enough to factorize is the product of two large prime numbers, . This is equivalent to calculate ### ###### ######## is the Euler function. In this paper Neural Networks are trained in order the function to be computed.
Efficient Algorithms for Computing the Jacobi Symbol (Extended Abstract)
 JOURNAL OF SYMBOLIC COMPUTATION
, 1998
"... We present two new algorithms for computing the Jacobi Symbol: the rightshift and leftshift kary algorithms. For inputs of at most n bits in length, both algorithms take O(n 2 = log n) time and O(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid' ..."
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We present two new algorithms for computing the Jacobi Symbol: the rightshift and leftshift kary algorithms. For inputs of at most n bits in length, both algorithms take O(n 2 = log n) time and O(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid's algorithm for computing greatest common divisors. In practice, we found our new algorithms to be about two to three times faster for inputs of 100 to 1000 decimal digits in length. We also present parallel versions of both algorithms for the CRCW PRAM. One version takes O ffl (n= log log n) time using O(n 1+ffl ) processors, giving the first sublinear parallel algorithms for this problem, and the other version takes polylog time using a subexponential number of processors.
The Pseudosquares Prime Sieve
"... Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses t ..."
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Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p ≤ 2(log n) 2, but it is conjectured that p ∼ 1 log nlog log n. Thus, log2 the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n) 2) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, spaceefficient implementation of the wheel datastructure. 1
Uncertainty can be Better than Certainty: Some Algorithms for Primality Testing ∗
, 2006
"... First, some notation As usual, we say that f(n) = O(n k) if, for some c and n0, for all n ≥ n0, We say that if, for all ε> 0, f(n) ≤ cn k. f(n) = �O(n k) f(n) = O(n k+ε). The “ � O ” notation is useful to avoid terms like log n and log log n. For example, when referring to the SchönhageStra ..."
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First, some notation As usual, we say that f(n) = O(n k) if, for some c and n0, for all n ≥ n0, We say that if, for all ε> 0, f(n) ≤ cn k. f(n) = �O(n k) f(n) = O(n k+ε). The “ � O ” notation is useful to avoid terms like log n and log log n. For example, when referring to the SchönhageStrassen algorithm for nbit integer multiplication, it is easier to write than the (more precise) �O(n) O(nlog nlog log n).
Primality testing
, 2003
"... We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality ..."
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We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality testing, i.e. whether "PRIMES is in P". Recently Agrawal, Kayal and Saxena answered this question in the affirmative. They gave a surprisingly simple deterministic algorithm. We describe their algorithm, mention some improvements by Bernstein and Lenstra, and consider whether the algorithm is useful in practice. Finally, as a topic for future research, we mention a conjecture that, if proved, would give a fast and practical deterministic primality test.