Cohen H. A course in computational number theory. 3-rd edition, Springer-Verlag, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....complexity of our approach with the results of previously done work on genus 4 curves is illustrated in Table 2. Our improvements of the original algorithm proposed by Cantor are mainly based on following techniques: 1. Chinese remainder theorem 2. Montgomery s trick of simultaneous inversions [Coh93, Algorithm 10.3.4] 3. Reordering of normalization step [Tak02] 4. Karatsuba multiplication [KO63] 5. Using Karatsuba to reduce complexity of polynomial reduction 6. Smart choice of HEC An extensive description on how to apply the given techniques to reduce the complexity of the group ....

H. Cohen. A course in computational number theory. Graduate Texts in Math. 138. Springer-Verlag, Berlin, 1993. Third corrected printing 1996.

....arithmetic and thus, not applicable to the derivation of explicit formulae. Very recently further improvements were made by [MDM 02,Tak02] In [MDM 02] the authors were able to replace the two field inversions by only one, with the help of Montgomery s trick for simultaneous inversions [Coh93]. In [Tak02] one multiplication was saved through a displacement of one operation. All these improvements are for genus 2 curves and odd characteristic. The generalization to even characteristic was done in [Lan02a] where improved formulae for characteristic 2 curves are also given. There was also ....

H. Cohen. A course in computational number theory. Graduate Texts in Math. 138. Springer-Verlag, Berlin, 1993. Third corrected printing 1996.

....the reader may refer to [11, 12, 9, 10, 46, 43] and to the bibliographies therein. In symbolic computation, the problem of computing the exact value of the determinant is addressed for instance in relation with matrix normal forms problems [41, 28, 23, 51] or in computational number theory [16]. In this paper we survey the known major results for computing the determinant and its sign and give the corresponding references. Our discussion focuses on theoretical computational complexity This material is based on work supported in part by the National Science Foundation under grants ....

H. Cohen. A course in computational number theory. Springer-Verlag, 1996.

....of Z=dZ. UNCOMPUTABLY LARGE INTEGRAL POINTS 7 In particular, the above conditions can be checked within a number of bit operations polynomial in the degree of P , and singly exponential in the bit sizes of the coecients of P , via fast factorization of polynomials in 2 variables over Q and Z=dZ [Coh93]. Remark 4. The JST Theorem can be strengthened slightly in the following way: one can replace d in condition (3) with any positive integer d 0 such that d 0 P 1 ; d 0 P k 2 Z[x] 2. Geometric Background We rst point out that a complete account of computability, decidability, ....

Cohen, Henri, A Course in Computational Number Theory, Graduate Texts in Mathematics, 138, Springer-Verlag, Berlin, 1993.

....question; the reader may refer to [11,12,9,10,46,43] and to the bibliographies therein. In symbolic computation, the problem of computing the exact value of the determinant is addressed for instance in relation with matrix normal forms problems [41,28,23,51] or in computational number theory [16]. In this paper we survey the known major results for computing the determinant and its sign and give the corresponding references. Our discussion focuses on theoretical computational complexity aspects. For an input matrix A # Z nn with infinity matrix norm #A#, we report worst case bit ....

H. Cohen. A course in computational number theory. Springer-Verlag, 1996.

....is fast in most cases. In some cases however a branch of the algorithm with exponential complexity is needed. Chistov (1990) proved the existence of a polynomial time algorithm for factoring polynomials over local elds. The algorithm for factoring ideals of Buchmann and Lenstra described by Cohen (1993, section 6.2) can be used for factoring polynomials over a local eld in polynomial time. Its main disadvantage is that it needs an integral basis as an input. The algorithm by Montes (1999) is formulated as an algorithm for the decomposition of ideals over number elds and is based on ideas ....

H. Cohen, A Course in computational number theory, Springer Verlag, 1993.

....invocation of the algorithm there is a small probability of error. 1 Introduction One of the most fundamental invariants of a square matrix is the determinant. Applications for computing the determinant of a matrix are numerous. For integer matrices alone they include computational number theory [4], computational group theory [9] and computational geometry [2, 3] In this paper we present a new algorithm for the determinant which is faster than any previously known. For a matrix A n n this algorithm requires O n 3 logn log A 2 log detA log ....

....of A and s 1 ( s n , 0 the invariant factors of A. Once we have the Smith form, the determinant of A is s 1 s 2 523 s n (and this is how our algorithm for the determinant proceeds) The Smith normal form also has many applications in computational number theory and group theory [4] as well as computations in homology theory (e.g. Dumas Saunders [7] The best known deterministic algorithm to compute the Smith form of an integer matrix is by Storjo1 hann [18] and requires O n 4 log A . n 3 log 2 A bit operations (ignoring poly logarithmic ....

H. Cohen. A Course in Computational Number Theory. Springer, 1996.

....invocation of the algorithm there is a small probability of error. 1 Introduction One of the most fundamental invariants of a square matrix is the determinant. Applications for computing the determinant of a matrix are numerous. For integer matrices alone they include computational number theory [4], computational group theory [9] and computational geometry [2, 3] In this paper we present a new algorithm for the determinant which is faster than any previously known. For a matrix A 2 Z n n this algorithm requires O(n 3 (logn logkAk) 2 p log j detAj log 2 n) Research was ....

....form of A and s 1 ; s n 2 Z 0 the invariant factors of A. Once we have the Smith form, the determinant of A is s 1 s 2 s n (and this is how our algorithm for the determinant proceeds) The Smith normal form also has many applications in computational number theory and group theory [4] as well as computations in homology theory (e.g. Dumas Saunders [7] The best known deterministic algorithm to compute the Smith form of an integer matrix is by Storjo1 hann [18] and requires O(n 4 logkAk n 3 log 2 kAk) bit operations (ignoring poly logarithmic factors) When the matrix ....

H. Cohen. A Course in Computational Number Theory. Springer, 1996.

....in parallel. Solutions produced are small and space required is essentially linear in the output size. 1 Introduction Computing integer solutions to systems of linear Diophantine equations is a classical mathematical problem with many interesting applications in number theory (see, e.g. Cohen 1993), group theory (see, e.g. Newman 1972) and combinatorics (see, e.g. Gibbons 1996) Given an input matrix A # Z nn and vector w # Z n1 , the problem is to find integer vectors v # Z n1 such that Av = w. It appears to be considerably harder to compute integer solutions than solutions ....

H. Cohen. A Course in Computational Number Theory. Springer, 1993.

....algorithm as well as complexity analyses and experimental computing times. 2 Preliminaries In this section we introduce the notation that will be used and give some basic definitions and results that will be needed. We refer the reader interested in more details to the books by Cohen [2], Hecke [6] and Marcus [15] Lowercase Greek letters, with the exception of OE, will denote algebraic numbers. For each ff there is a unique M 2 Z[t] such that M is a primitive, irreducible polynomial with positive leading coefficient having ff as a root. M is called the minimal polynomial of ....

H. Cohen. A Course in Computational Number Theory. Graduate Texts in Mathematics. Springer-Verlag, 1993.

....all values x 0 2 F q r and then calculating the number of roots of 4 STEVEN D. GALBRAITH f(x 0 ; y) in F q r . From the values t r = q r 1 #C(F q r ) P 2g i=1 r i one can obtain the coefficients of P (X) using Newton s identities am = 1 m ( t m P m 1 i=1 a i t m i ) see Cohen [5] Proposition 4.3.3) This naive algorithm takes time O(q g (log q g ) c ) for some constant c, which can also be written as O(q g ) One method to speed this up is to compute #C(F q r ) for r = 1; g 1 and then to try all values of #C(F q g ) q g 1) i.e. all integers in ....

H. Cohen, A course in computational number theory, Springer GTM 138, 1993.

....via remark 8. Thus the number of necessary random bits, the number of bits of any integer in [x j ; x j 1 ) and the time needed to compute fx j ; x j 1 g are all polynomial in the size of F . Finally, via [Pra75] any prime p2 [x j ; x j 1 ) can be certified (as being a prime) within NP. Also, via [Coh93], a putative root of F mod p can indeed be verified in polynomial time. So we really need just one call to an NP oracle. Our algorithm is thus indeed an AM algorithm. Xi Remark 9 Results weaker than Main Theorems 4 and 5 would have sufficed as well and perhaps shortened the proof (e.g. Koi96, ....

....g 1 (x) j 0 mod ff . g k (x) j 0 mod ff is all of Z=ffZ. Xi Remark 12 The JST Theorem can be strengthened slightly in the following way: one can replace ff in condition (3) with any positive integer ff 0 such that ff 0 f 1 ; ff 0 f k 2 Z[x] Also, the techniques of [Coh93] can easily support the stated complexity bound. Proof of Main Theorem 3: It follows easily from the Hurwitz genus formula for curves [Sil95] that 8x 9y f(v 0 ; x; y) 0 = Z fv = v 0 g defines a curve with a singular component. The set of such v 0 2C is of course finite (by Bertini s theorem ....

Cohen, Henri, A Course in Computational Number Theory, Graduate Texts in Mathematics, 138, SpringerVerlag, Berlin, 1993.

.... UNCOMPUTABLY LARGE INTEGRAL POINTS 7 In particular, the above conditions can be checked within a number of bit operations polynomial in the degree of P , and singly exponential in the bit sizes of the coefficients of P , via fast factorization of polynomials in 2 variables over Q and Z=dZ [Coh93]. Xi Remark 4. The JST Theorem can be strengthened slightly in the following way: one can replace d in condition (3) with any positive integer d 0 such that d 0 P 1 ; d 0 P k 2 Z[x] 2. Geometric Background We first point out that a complete account of computability, ....

Cohen, Henri, A Course in Computational Number Theory, Graduate Texts in Mathematics, 138, Springer-Verlag, Berlin, 1993.

....then there is a relation 1 P 1 Delta Delta Delta r P r j O (mod 2E(Q) for some 1 ; r 2 f0; 1g not all zero. This means that there is a point Q 2 E(Q) with 2Q = 1 P 1 Delta Delta Delta r P r : 14) Note that we can find Q using exact arithmetic, although if E(Q)[2] 6= 0, there will be more than one possible choice for Q. To ease our exposition, we will assume that E(Q) 2] 0, so Q is uniquely determined. We then pick an index j with j = 1, replace P j by Q, and repeat the procedure. There are two possibilities. First, if P 1 ; P r are actually ....

....r 2 f0; 1g not all zero. This means that there is a point Q 2 E(Q) with 2Q = 1 P 1 Delta Delta Delta r P r : 14) Note that we can find Q using exact arithmetic, although if E(Q) 2] 6= 0, there will be more than one possible choice for Q. To ease our exposition, we will assume that E(Q)[2] = 0, so Q is uniquely determined. We then pick an index j with j = 1, replace P j by Q, and repeat the procedure. There are two possibilities. First, if P 1 ; P r are actually independent, then after just a few iterations we end up with a set of points which are independent in ....

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H. Cohen, A Course in Computational Number Theory, GTM 138, Springer-Verlag, 1993.

....1) our new algorithm improves on the cost of the best known sequential algorithm by a factor of almost n 1 Gamma . 1 Introduction Computing integer solutions to systems of linear Diophantine equations is a classical mathematical problem with many interesting applications in number theory (see Cohen 1993), group theory (see Newman 1972) and combinatorics (see, e.g. Kramer Mesner 1976) Given an input matrix A 2 Z n Thetan and vector w 2 Z n Theta1 , the problem is to find integer vectors v 2 Z n Theta1 such that Av = w. It appears to be considerably harder to compute integer solutions ....

H. Cohen. A Course in Computational Number Theory. Springer, 1993.

....as ddh. We use Shoup s terminology. To disprove ddh one may first try to come up with a ddh algorithm that works in all groups. Indeed, such an algorithm would be devastating. However, the best known generic algorithm for ddh is a generic discrete log algorithm, namely the Baby Step Giant Step [9]. When applied in a group of prime order p this algorithm runs in time O ffl ( p p) Shoup shows that this is the best possible generic algorithm for ddh. We discuss the implications of this result at the end of the section. Definition 3.1 (Shoup) An encoding function on the additive group Z ....

H. Cohen, "A course in computational number theory", Springer-Verlag.

....places. Its Galois group is isomorphic to the class group Cl k ; hence the degree [H k : k] is the class number h k . There now exist very satisfactory algorithms to compute the discriminant, the ring of integers and the class group of a number field, and especially of a quadratic field (see [3] and [16] For the computation of the Hilbert class field, however, there exists an e#cient version only for complex quadratic fields, using complex multiplication (see [18] and a general method for all number fields, using Kummer theory, which is not really satisfactory except when the ground ....

....compute Fn # Ei(nA) directly (see below) 6. Return the values Fn for 1 # n # N and terminate the algorithm. For small values of n, we compute the exponential integral by standard means since the Taylor series converges slowly. One can find explicit formulas to compute the function Ei in [3], Proposition 5.6.12. Note that this type of method for computing Ei and more generally for confluent hypergeometric functions has already been studied in detail, in particular with respect to its numerical stability. See [19, 23, 24] Finally, we compute the Artin root number W (#) We will ....

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H. Cohen, A Course in Computational Number Theory, GTM 138, Springer-Verlag, 1993 MR 94i:11105

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Cohen H. A course in computational number theory. 3-rd edition, Springer-Verlag, 1996.

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H. Cohen, A Course in Computational Number Theory, Graduate Texts in Math. 138. Springer-Verlag, 1993.

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H.Cohen, A course in computational number theory, Graduate Texts in Math. 138, Springer-Verlag, 1993, Third corrected printing, 1996.

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H. Cohen. A course in computational number theory. Graduate Texts in Math. 138. Springer-Verlag, Berlin, 1993. Third corrected printing 1996.

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H. Cohen, A Course in Computational Number Theory, Graduate Text 138, Springer 1993.

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Cohen, H., A course in computational number theory, GTM 138, SpringerVerlag, Berlin, 1993.

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H. Cohen, A course in Computational Number Theory, Graduate Texts in Mathematics 138, Springer, 1993.

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H. Cohen, A Course in Computational Number Theory , Graduate Text 138, Springer 1993.

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Cohen, H.: A course in computational number theory, Graduate Texts in Math. 138, Springer-Verlag, Berlin Heidelberg New York 1993. 254 Ren'e Schoof

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