### MO419 – Probabilistic Algorithms – Flávio K. Miyazawa – IC/UNICAMP 2010 A survey on Probabilistic Algorithms to Primality Test

"... One of the longstanding problems in using encryption to encode messages is that the recipient of the message needs to know the key in order to decrypt the message. Clearly we somehow have to get the key to the participants so they can use it. We can’t send the key to them without encrypting *it*, or ..."

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One of the longstanding problems in using encryption to encode messages is that the recipient of the message needs to know the key in order to decrypt the message. Clearly we somehow have to get the key to the participants so they can use it. We can’t send the key to them without encrypting *it*, or someone might “eavesdrop ” and get it. But this puts us in an infinite loop: the

### NOTES ON SOME NEW KINDS OF PSEUDOPRIMES

"... Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprime to two bases only, where ..."

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Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprime to two bases only, where p = 2 or 3. In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2-pseudoprimes to the same bases. In Section 3, we give examples of Sylow p-pseudoprimes to the first several prime bases for the first several primes p. We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow p-pseudoprime to the same bases for all p ∈{2, 3, 5, 7, 11, 13}. In Section 4, we define n to be a k-fold Carmichael Sylow pseudoprime, ifitisaSylowp-pseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n − 1. We find and tabulate all three 3-fold Carmichael Sylow pseudoprimes < 1016. In Section 5, we define a positive odd composite n to be a Sylow uniform pseudoprime to bases b1,...,bk, or a Syl-upsp(b1,...,bk) for short, if it is a Sylp-psp(b1,...,bk) for all the first ω(n − 1) − 1 small prime factors p of n − 1, where ω(n − 1) is the number of distinct prime factors of n − 1. We find and tabulate all the 17 Syl-upsp(2, 3, 5)’s < 1016 and some Syl-upsp(2, 3, 5, 7, 11)’s < 1024. Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6. 1.

### Improved Bounds for Goldback Conjecture

"... : Goldach's conjecture states that every even integer greater or equal to 6 is the sum of two prime numbers. This result is still unproved. This conjecture has been numerically checked up to 4:10 11 on an IBM 3083 mainframe. We describe here an implementation on a less powerful machine which ..."

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: Goldach's conjecture states that every even integer greater or equal to 6 is the sum of two prime numbers. This result is still unproved. This conjecture has been numerically checked up to 4:10 11 on an IBM 3083 mainframe. We describe here an implementation on a less powerful machine which raises the bound to 10 12 . Key-words: Prime numbers; Goldbach's problem (R'esum'e : tsvp) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientifique Institut National de Recherche en Informatique (URA 227) Universit e de Rennes 1 -- Insa de Rennes et en Automatique -- unit e de recherche de Rennes Am'eliorations de bornes au sujet de la conjecture de Goldbach (premi`ere version) R'esum'e : La conjecture de Goldbach stipule que tout nombre pair sup'erieur ou 'egal `a 6 est somme de deux nombres premiers. Ce r'esultat est `a ce jour non d'emontr'e. Il a 'et'e v'erifi'e num'eriquement jusqu'`a 4:10 11 sur un IBM 3083. Nous d'ecrivons ici une impl'ementation...

### 1.2 Large Primes, Probabilistic Tests

"... Abstract. For large integers, the most efficient primality tests are pro-babilistic. However, for integers with a small fixed number of bits the best tests in practice are deterministic. Currently the best known tests of this type involve 3 rounds of the Miller-Rabin test for 32-bit integers and 7 r ..."

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Abstract. For large integers, the most efficient primality tests are pro-babilistic. However, for integers with a small fixed number of bits the best tests in practice are deterministic. Currently the best known tests of this type involve 3 rounds of the Miller-Rabin test for 32-bit integers and 7 rounds for 64-bit integers. Our main result in this paper: For 32-bit integers we reduce this to a single computation of a simple hash function and a single round of Miller-Rabin. Similarly, for 64-bit integers we can reduce the number of rounds to two (but with a significantly large precomputed table) or three. Up to our knowledge, our implementations are the fastest one-shot deterministic primality tests for 32-bit and 64-bit integers to date. We also provide empirical evidence that our algorithms are fast in prac-tice and that the data segment is roughly as small as possible for an algorithm of this type. 1

### Short effective intervals containing primes

, 2000

"... We prove that every interval xð1 D1Þ; x contains a prime number with D 28 314 000 and provided xX10 726 905 041: The proof combines analytical, sieve and algorithmical methods. ..."

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We prove that every interval xð1 D1Þ; x contains a prime number with D 28 314 000 and provided xX10 726 905 041: The proof combines analytical, sieve and algorithmical methods.

### Article electronically published on November 2, 2004 FINDING C3-STRONG PSEUDOPRIMES

"... Abstract. Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3)thesignature, andH = h1h2h3 the height of N, respectively. We call N a C3-number if it is a Carmichael numberwitheachprimefactorqi≡3 ..."

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Abstract. Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3)thesignature, andH = h1h2h3 the height of N, respectively. We call N a C3-number if it is a Carmichael numberwitheachprimefactorqi≡3 mod 4. If N is a C3-number and a strong pseudoprime to the t bases bi for 1 ≤ i ≤ t, wecallNaC3-spsp(b1,b2,...,bt). Since C3-numbers have probability of error 1/4 (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of ψm (the smallest strong pseudoprime to all the first m prime bases). If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. In this paper, we first describe an algorithm for finding C3-spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 C3-spsp(2)’s < 1024 with heights < 109. We then give an overview of the 21978 C3spsp(2)’s and tabulate 54 of them, which are C3-spsp’s to the first 8 prime bases up to 19; three numbers are spsp’s to the first 11 prime bases up to 31. No C3-spsp’s < 1024 to the first 12 prime bases with heights < 109 were found. We conjecture that there exist no C3-spsp’s < 1024 to the first 12 prime bases with heights ≥ 109 and so that

### Article electronically published on February 17, 2000 FINDING STRONG PSEUDOPRIMES TO SEVERAL BASES

"... Dedicated to the memory of P. Erdős (1913–1996) Abstract. Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic primality testing algorithm which is not only easier to implement but also f ..."

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Dedicated to the memory of P. Erdős (1913–1996) Abstract. Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic primality testing algorithm which is not only easier to implement but also faster than either the Jacobi sum test or the elliptic curve test. Thanks to Pomerance et al. and Jaeschke, ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ9,ψ10 and ψ11 were given by Jaeschke. In this paper we tabulate all strong pseudoprimes (spsp’s) n<1024 to the first ten prime bases 2, 3, ·· · , 29, which have the form n = pq with p, q odd primes and q −1 =k(p −1),k=2, 3, 4. There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp’s to both bases 31 and 37. As a result the upper bounds for ψ10 and ψ11 are lowered from 28- and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for ψ12 is obtained. The main tools used in our methods are the biquadratic residue characters and cubic residue characters. We propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases. Comparisons of effectiveness with both Jaeschke’s and Arnault’s methods are given. 1.