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Generating a Product of Three Primes with an Unknown Factorization
 Proc. 3rd Algorithmic Number Theory Symposium (ANTSIII
, 1998
"... We describe protocols for three or more parties to jointly generate a composite N = pqr which is the product of three primes. After our protocols terminate N is publicly known, but neither party knows the factorization of N . Our protocols require the design of a new type of distributed primality te ..."
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We describe protocols for three or more parties to jointly generate a composite N = pqr which is the product of three primes. After our protocols terminate N is publicly known, but neither party knows the factorization of N . Our protocols require the design of a new type of distributed primality test for testing that a given number is a product of three primes. We explain the cryptographic motivation and origin of this problem.
A oneparameter quadraticbase version of the Baillie–PSW probable prime test
 Math. Comp
"... Abstract. The wellknown BailliePSW probable prime test is a combination of a RabinMiller test and a “true ” (i.e., with (D/n) =−1) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT ..."
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Abstract. The wellknown BailliePSW probable prime test is a combination of a RabinMiller test and a “true ” (i.e., with (D/n) =−1) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT) with probability of error less than 1/7710, and pointed out that the lack of counterexamples to the BailliePSW test indicates that the true probability of error may be much lower. In this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter: Tu = T mod (T 2 − uT + 1), and define the basecounting functions: B(n) =#{u:0 ≤ u<n, nis a psp(Tu)} and SB(n) =#{u:0 ≤ u<n, nis an spsp(Tu)}. Then we give explicit formulas to compute B(n) and SB(n), and prove that, for odd composites n, B(n) <n/2 and SB(n) <n/8, and point out that these are best possible. Finally, based on oneparameter quadraticbase pseudoprimes, we provide a probable prime test, called the OneParameter QuadraticBase Test (OPQBT), which passed by all primes ≥ 5 andpassedbyanoddcompositen = p r1 1 pr2 2 ···prs s (p1 <p2 < ·· · <ps odd primes) with probability of error τ(n). We give explicit formulas to compute τ(n), and prove that
ON THE EXISTENCE AND NONEXISTENCE OF ELLIPTIC PSEUDOPRIMES
"... Abstract. In a series of papers, D. Gordon and C. Pomerance demonstrated that pseudoprimes on elliptic curves behave in many ways very similar to pseudoprimes related to Lucas sequences. In this paper we give an answer to a challenge that was posted by D. Gordon in 1989. The challenge was to either ..."
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Abstract. In a series of papers, D. Gordon and C. Pomerance demonstrated that pseudoprimes on elliptic curves behave in many ways very similar to pseudoprimes related to Lucas sequences. In this paper we give an answer to a challenge that was posted by D. Gordon in 1989. The challenge was to either prove that a certain composite N ≡ 1mod4didnotexist, orto explicitly calculate such a number. In this paper, we both present such a specific composite (for Gordon’s curve with CM by Q ( √ −7)), as well as a proof of the nonexistence (for curves with CM by Q ( √ −3)). We derive some criteria for the group structure of CM curves that allow testing for all composites, including N ≡ 3 mod 4 which had been excluded by Gordon. This gives rise to another type of examples of composites where strong elliptic pseudoprimes are not Euler elliptic pseudoprimes. 1.
Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.7 On a Compositeness Test for (2 p + 1)/3
"... In this note, we give a necessary condition for the primality of (2 p + 1)/3. 1 ..."
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In this note, we give a necessary condition for the primality of (2 p + 1)/3. 1
ABSOLUTE QUADRATIC PSEUDOPRIMES
"... Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1. ..."
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Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.
A Simple Derivation for the Frobenius Pseudoprime Test
, 2008
"... Probabilistic compositeness tests are of great practical importance in cryptography. Besides prominent tests (like the wellknown MillerRabin test) there are tests that use Lucassequences for testing compositeness. One example is the socalled Frobenius test that has a very low error probability. ..."
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Probabilistic compositeness tests are of great practical importance in cryptography. Besides prominent tests (like the wellknown MillerRabin test) there are tests that use Lucassequences for testing compositeness. One example is the socalled Frobenius test that has a very low error probability. Using a slight modification of the above mentioned Lucas sequences we present a simple derivation for the Frobenius pseudoprime test in the version proposed by Crandall and Pommerance in [CrPo05]. 1 Lucas and Frobenius Pseudoprimes For f(x) = x 2 − ax + b ∈ Z[x] the Lucas sequences are given by
A GENERALIZATION OF MILLER’S PRIMALITY THEOREM PEDRO BERRIZBEITIA AND AURORA OLIVIERI
"... Abstract. For any integer r we show that the notion of ωprime to base a introduced by Berrizbeitia and Berry, 2000, leads to a primality test for numbers n congruent to 1 modulo r, which runs in polynomial time assuming the Extended Riemann Hypothesis (ERH). For r = 2 we obtain Miller’s classical r ..."
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Abstract. For any integer r we show that the notion of ωprime to base a introduced by Berrizbeitia and Berry, 2000, leads to a primality test for numbers n congruent to 1 modulo r, which runs in polynomial time assuming the Extended Riemann Hypothesis (ERH). For r = 2 we obtain Miller’s classical result. 1.