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27
It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
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Cited by 18 (2 self)
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The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
Interpolation of ShiftedLacunary Polynomials (Extended Abstract)
"... Given a “black box” function to evaluate an unknown rational polynomial f ∈Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t∈Z>0, the shift α∈Q, the exponents 0≤e1< ..."
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Cited by 11 (1 self)
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Given a “black box” function to evaluate an unknown rational polynomial f ∈Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t∈Z>0, the shift α∈Q, the exponents 0≤e1< e2<···<et, and the coefficients c1,...,ct∈Q\{0} such that f (x)=c1(x−α) e1 + c2(x−α) e2 +···+ct(x−α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size and in particular is logarithmic in deg f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information. We give both an unconditional deterministic algorithm which is polynomialtime but has a rather high complexity, and a more practical probabilistic algorithm which relies on some unknown constants.
Efficient CMconstructions of elliptic curves over finite fields
 MATH. COMP.
, 2007
"... We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) hasorderN. Although it is unproved that this can be done for all N, a heuristic analysis shows that the algorithm has an expect ..."
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Cited by 6 (3 self)
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We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) hasorderN. Although it is unproved that this can be done for all N, a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 ω(N) log N, whereω(N) isthe number of distinct prime factors of N. In the cryptographically relevant case where N is prime, an expected run time O((log N) 4+ε) can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order N =10 2004 and N = nextprime(10 2004)=10 2004 +4863.
Cyclotomy primality proofs and their certificates. Mathematica Goettingensis
, 2006
"... Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein ..."
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Cited by 4 (2 self)
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Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein
THERE ARE INFINITELY MANY PERRIN PSEUDOPRIMES
"... Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zeroden ..."
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Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zerodensity estimates for Hecke Lfunctions. 1. Background In a 1982 paper [1], Adams and Shanks introduced a probable primality test based on third order recurrence sequences. The following is a version of that test. Consider sequences An = An(r, s) defined by the following relations: A−1 = s, A0 = 3, A1 = r, and An = rAn−1 − sAn−2 + An−3. Let f(x) = x 3 − rx 2 + sx − 1 be the associated polynomial and ∆ its discriminant. (Perrin’s sequence is An(0, −1).) Definition. The signature mod m of an integer n with respect to the sequence Ak(r, s) is the 6tuple (A−n−1, A−n, A−n+1, An−1, An, An+1) mod m. Definitions. An integer n is said to have an Ssignature if its signature mod n is congruent to (A−2, A−1, A0, A0, A1, A2). An integer n is said to have a Qsignature if its signature mod n is congruent to (A, s, B, B, r, C), where for some integer a with f(a) ≡ 0 mod n, A ≡ a −2 + 2a, B ≡ −ra 2 + (r 2 − s)a, and C ≡ a 2 + 2a −1. An integer n is said to have an Isignature if its signature mod n is congruent to (r, s, D ′ , D, r, s), where D ′ + D ≡ rs − 3 mod n and (D ′ − D) 2 ≡ ∆. Definition. A Perrin pseudoprime with parameters (r, s) is an odd composite n such that either
FINDING THE GROUP STRUCTURE OF ELLIPTIC CURVES OVER FINITE FIELDS
, 2005
"... We show that an algorithm of V. Miller to compute the group structure of an elliptic curve over a prime finite field runs in probabilistic polynomial time for almost all curves over the field. Important to our proof are estimates for some divisor sums. ..."
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Cited by 1 (1 self)
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We show that an algorithm of V. Miller to compute the group structure of an elliptic curve over a prime finite field runs in probabilistic polynomial time for almost all curves over the field. Important to our proof are estimates for some divisor sums.
Interpolation of ShiftedLacunary Polynomials
, 2010
"... Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 ..."
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Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 <e2 < ·· · <et, and the coefficients c1,...,ct ∈ Q \{0} such that f(x) =c1(x − α) e1 + c2(x − α) e2 + ···+ ct(x − α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, which may be logarithmic in the degree of f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.