I. E. Shparlinski. Computational and algorithmic problems in finite fields . Kluwer Academic Publishers, Dordrecht, The Netherlands, (1992).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and Weiss [36] and the references therein. Finally, since distinct irreducible factors over finite fields are important in most algorithms for factorizing polynomials, it is also of interest to investigate the measures of distinctness (in order to determine the complexity of the algorithms) cf. [34, 46, 18]. Such a metric notion is useful and widely used in different fields. In number theory, the number of prime factors (with or without multiplicity) has long been used as measures of compositeness of an integer; cf. 25, 40] In algorithmic theory, the introduction of the measures of presortedness, ....

I. E. Shparlinski. Computational and algorithmic problems in finite fields . Kluwer Academic Publishers, Dordrecht, The Netherlands, (1992).

....approximation theorems, irreducible polynomials, parabolic cylinder function, uniform asymptotics. 1 Introduction and main results Because of many applications in diverse areas (which may be exemplified by the huge number of items referenced in Lidl and Niederreiter s and Shparlinski s books [32, 36]) finite fields have received increasing interest in the literature. Finite fields are known to be very useful in finite geometries, combinatorics, algebraic coding theory, cryptology, combinatorial design theory, symbolic computations, pseudorandom number generation, and shift register ....

....fields have received increasing interest in the literature. Finite fields are known to be very useful in finite geometries, combinatorics, algebraic coding theory, cryptology, combinatorial design theory, symbolic computations, pseudorandom number generation, and shift register sequences; cf. [32, 36, 28]. Let F q be a finite field with q elements and n the set of monic polynomials over F q of degree n. Assuming that each of the q polynomials in n is equally likely, we are interested in the random variable# n , counting the total number (i.e. counted with multiplicities) of irreducible ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, Kluwer Academic Publishers Group, Dordrecht, 1992. Mathematics and its Applications (Soviet Series), 88. 23

....approximation theorems, irreducible polynomials, parabolic cylinder function, uniform asymptotics. 1 Introduction and main results Because of many applications in diverse areas (which may be exemplified by the huge number of items referenced in Lidl and Niederreiter s and Shparlinski s books [32, 36]) finite fields have received increasing interest in the literature. Finite fields are known to be very useful in finite geometries, combinatorics, algebraic coding theory, cryptology, combinatorial design theory, symbolic computations, pseudorandom number generation, and shift register ....

....fields have received increasing interest in the literature. Finite fields are known to be very useful in finite geometries, combinatorics, algebraic coding theory, cryptology, combinatorial design theory, symbolic computations, pseudorandom number generation, and shift register sequences; cf. [32, 36, 28]. Let F q be a finite field with q elements and P n the set of monic polynomials over F q of degree n. Assuming that each of the q polynomials in P n is equally likely, we are interested in the random This work was supported by National Science Council under the Grant NSC 85 2121 M 001 007. ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, Kluwer Academic Publishers Group, Dordrecht, 1992. Mathematics and its Applications (Soviet Series), 88.

....and Weiss [36] and the references therein. Finally, since distinct irreducible factors over finite fields are important in most algorithms for factorizing polynomials, it is also of interest to investigate the measures of distinctness (in order to determine the complexity of the algorithms) cf. [34, 46, 18]. Such a metric notion is useful and widely used in di#erent fields. In number theory, the number of prime factors (with or without multiplicity) has long been used as measures of compositeness of an integer; cf. 25, 40] In algorithmic theory, the introduction of the measures of presortedness, ....

I. E. Shparlinski. Computational and algorithmic problems in finite fields. Kluwer Academic Publishers, Dordrecht, The Netherlands, (1992).

.... number p, positive integer n and # 0, one can find an irreducible polynomial of degree n over F p using O 3 # 2 # 4 # log arithmetical operations in F p [Sho90] Moreover, if F is a field and f and g are two polynomials in F[x] of degree at most n, then f g [CK91] and f mod g [Shp92] can be computed using O(n log n log log n) arithmetic operations in F. Thus, F 2 k (the Galois field of order 2 ) can be represented as F 2 [x] # f for some irreducible polynomial f of degree k, and both the representation and the operations of F 2 k can be computed in polynomial time. ....

Igor E. Shparlinski. Computational and Algorithmic Problems in Finite Fields. Kluwer Academic, 1992.

....(x) divides a given polynomial f(x) 2 F 2 [x] Perhaps the most interesting case is the one when f(x) is a trinomial (i.e. a polynomial with exactly three nonzero terms) There is an extensive literature on the factorization of polynomials over finite fields; for good surveys, see ( 1] Ch. V) [2], Ch. 3 and 4) and ( 3] Ch. 2 ) In this paper we give several simple and natural statements which we could not find in the references above. Definition 1 Let g be fixed. For f(x) P 2 g Gamma2 i=0 a i x i 2 F 2 [x] and r 2 f1; 2; 2 g Gamma 2g, where gcd(r; 2 g Gamma 1) 1, ....

I. E. Shparlinski, "Computational and Algorithmic Problems in Finite Fields", Kluwer Academic Publishers, Dordrecht / Boston / London, 1992.

.... integer n and # 0, one can find an irreducible polynomial of degree n over F p using O # n 3 # p 1 2 # n 4 # log 2 p # arithmetical operations in F p [Sho90] Moreover, if F is a field and f and g are two polynomials in F[x] of degree at most n, then f g [CK91] and f mod g [Shp92] can be computed using O(n log n log log n) arithmetic operations in F. Thus, F 2 k (the Galois field of order 2 k ) 2 can be represented as F 2 [x] # f #, for some irreducible polynomial f of degree k, and both the representation and the operations of F 2 k can be computed in ....

Igor E. Shparlinski. Computational and Algorithmic Problems in Finite Fields. Kluwer Academic, 1992.

....C 1 and C 2 such that P k;u (ff) C 1 q uk (log mu 1) 2 Gamma C 2 2 mu mq uk=2 : 13 In fact, if g(ff) is primitive in F q um , then Norm(g(ff) is primitive in F q m . Thus, Corollary 4.3 follows from Corollary 4.2. Taking k = 1, Corollary 4. 3 reduces to Shparlinski s result [Shp]. Taking u = 1, Corollary 4.3 reduces to Lemma 4.2. Proof of Theorem 4.1. We can assume that m 1. The idea is to remove those elements g(ff) that are not normal and then apply Shoup s lemma. Let oe be the Frobenius automorphism oe(fi) fi q . The additive group F q m is a cyclic F q [T ....

I.E. Shparlinski, Computational and Algorithmic Problems in Finite Fields, Kluwer Academic Publishers, 1992.

.... integer n and # 0, one can find an irreducible polynomial of degree n over F p using O n 3 # p 1 2 # n 4 # log 2 p arithmetical operations in F p [Sho90] Moreover, if F is a field and f and g are two polynomials in F[x] of degree at most n, then f g [CK91] and f mod g [Shp92] can be computed using O(n log n log log n) arithmetic operations in F. Thus, F 2 k (the Galois field of order 2 k ) can be represented as F 2 [x] h f i, for some irreducible polynomial f of degree k, and both the representation and the operations of F 2 k can be computed in polynomial ....

Igor E. Shparlinski. Computational and Algorithmic Problems in Finite Fields. Kluwer Academic, 1992.

.... brute force approach to computing all points on C via finding, for each a 2 F q , all b 2 F q with f(a; b) 0, takes O (n 2 q 3=2 ) operations in F q , using the fastest known deterministic algorithms to factor the univariate polynomial f(a; y) for all a 2 F q (Shoup 1990; Section 1. 1 of Shparlinski 1992, von zur Gathen Shoup 1992) We present a deterministic method that uses O (n 5 q) operations, i.e. polynomial time per point; the method only works in the case of a prime field F q , with q = p prime, and does not work for exceptional curves. Shoup (1990) has exhibited a deterministic ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, vol. 88 of Mathematics and its applications. Kluwer Academic Publishers, 1992.

.... into account the equality X 0sq Gamma2 fi fi fi X 1th exp(2 ist= fi fi fi 2 = X 1t 1 ;t 2 h X 0sq Gamma2 exp(2 is(t 1 Gamma t 2 ) h(q Gamma 1) we obtain W ( n 2 Gamma 3n)q 1=2 n 2 ) 2 q Gamma 1 Delta (h(q Gamma 1) Gamma h 2 ) 26 von zur Gathen Shparlinski X 0aq Gamma2 (#C( a; h) Gamma h#C=q) 2 2n 2 h 2 = q Gamma 1) 2( n 2 Gamma 3n)q 1=2 n 2 ) 2 h 2n 2 h(1 ( n Gamma 3)q 1=2 n) 2 ) 2n 2 h(nq 1=2 n) 2 8n 4 hq: 2 We now show that hypothesis B is not a severe restriction, in that it is satisfied after a ....

....(von zur Gathen Shparlinski 1995b) certainly a primitive root is sufficient. Results about the construction, distribution and density of primitive roots can be found in Lidl Niederreiter (1983) see Shparlinski (1992b) for a survey and also von zur Gathen Giesbrecht (1990) Perel muter Shparlinski (1990) , Shoup (1992) Shparlinski (1992a) for the currently best results in this area. Acknowledgements Parts of the first author s work was done on a visit to Macquarie University and during a sabbatical visit to the Institute for Scientific Computation at ETH Zurich, whose hospitality is gratefully ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, vol. 88 of Mathematics and its applications. Kluwer Academic Publishers, 1992b.

....0 = K[X] f(X) GF (3 q 0 ) with q 0 Gamma 2 log 3 n 0 and f an irreducible polynomial. Such an f can be surely found in time O(n 3 ) cf. Sho90, Theorem 4.1. where even more efficient methods are described. To carry out the computations in K[X] f(X) we use algorithms as described in [Shp92]. 3 Some lemmas In this section we collect the lemmas of more general character. 3.1 Small rank The first is from [HJ85, Section 0.4.6] Lemma 3.1 Let M be a (m Theta n) matrix over K with rg(M ) k. Then there are non singular matrices X (m Theta k) Y (k Theta n) and B (k Theta k) ....

I. Shparlinski. Computational and Algorithmic Problems in Finite Fields. Kluwer Academic Publishers, 1992.

....X j=1 ff i j ) Delta(i 1 : i w ) which vanishes exactly when P 00 1 has a positive answer. Thus, P 2 is shown to be NP complete for q m ary codes. Note that in the course of this reduction we pass from F q to its extension of degree m, where m is a part of the instance. It is known [141] that one can construct extensions (irreducible polynomials) in polynomial time. The second part is, given an instance of P 2 for q m ary codes, to present a polynomial reduction to q ary codes. This is easy since both codes have alphabets of the same characteristic. To build this reduction, we ....

I. E. Shparlinski, Computational and Algorithmic Problems in Finite Fields, Dordrecht: Kluwer (1992).

....m ) 2 32 von zur Gathen, Karpinski Shparlinski Corollary 8.3. Let m; k; t 1, q = p r , f 2 F q k [x 1 ; xm ] be t sparse. Then the image of T ffi f : F m q k F q can be calculated with (kt 1) O(mpr) evaluations of T ffi f . Proof. Using the algorithms of Shoup (1992) or Shparlinski (1990) (see also Shparlinski 1992a) we can construct in time (pkr) O(1) a set M F q k with cardinality (pkr) O(1) containing a primitive root of F q k . Setting U = f0g [ f e 2 F q k : 0 e sg F q k for 2 M , we have from Theorem 8.2 that (T ffi f) F m q k ) 2M (T ffi ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, vol. 88 of Mathematics and its applications. Kluwer Academic Publishers, 1992a.

....2 F q of sufficiently large order; certainly a primitive root is sufficient. Results about the construction, distribution and density of primitive roots can be found in Lidl Niederreiter (1983) see Shparlinski (1992b) for a survey and also von zur Gathen Giesbrecht (1990) Perel muter Shparlinski (1990) , Shoup (1992) Shparlinski (1992a) for the currently best results in this area. Acknowledgements Parts of the first author s work was done on a visit to Macquarie University and during a sabbatical visit to the Institute for Scientific Computation at ETH Zurich, whose hospitality is gratefully ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, vol. 88 of Mathematics and its applications. Kluwer Academic Publishers, 1992b.

....to a new upper bound for the number of zeros of sparse polynomials which is probably of independent interest. This new bound allows us to improve some of the results of [6] which are based on bounds for the number of solutions of exponential equations from [31, 34, 35] see also Section 3. 3 of [32]. 3 Preparations For integers a, b, and k, we denote by oe k (a; b; t) the following exponential sum oe k (a; b; t) t X x=1 e p i a# kx j e p Gamma1 (bx) We need the following upper bound, which follows quickly from the classical bound for Gauss sums. Lemma 4. Assume that gcd(a; ....

....of course) Moreover, in the special case of m = p k with fixed prime p and large integer k 1 a bound is known [19] of a very short exponential sum with # x , roughly of length exp(c log 2=3 m) with some constant c 0. Several more bounds of exponential sums with # x can be found in [18, 19, 20, 21, 26, 32]. Although it is not quite clear how to extend Lemma 7 to composite moduli, the bounds of [31, 34, 35] can be generalized, see [28] All of them can be used to obtain some analogues of the results of this paper for composite moduli. Certainly it would be interesting to study what happens on the ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, Kluwer Acad. Publ., Dordrecht, 1992.

.... the corresponding extension but it is easy reformulate all of them in a form taking into account the cots of such construction (without loosing the main features of the algorithm) Indeed, during recent years a very substantial progress in this area has been achieved (for a survey see Chapter 2 of [14]) Say for the field of Theorem 4 below we may use the probabilistic algorithm from [12] with expected number of O(l 2 log 2 l log log l l log q log l log log l) arithmetical operations over F q , as for Theorem 5 we may apply the deterministic algorithm from [11] using l 4 p 1=2 (log ....

....; um ) 0 for any (u 1 ; um ) 2 U. Proof. Firstly let us construct the field F q k where k = lm; l = dlog(d 2) log qe by using the algorithm of [12] in time O i p 1=2 (m log d log q) O(1) j (see (1) Then the algorithms of [13] and of [15] see also Chapter 2 of [14]) in time p(m log d log q) O(1) construct a set M 2 F q k of size jMj = p(m log d log q) O(1) and containing a primitive root of F q k . It follows from Theorem 2.3 of [2] that f is identical to zero if and only if f(0; 0) 0 and f( i ; iq l ; iq l(m Gamma1) ....

Shparlinski, I., Computational and Algorithmic Problems in Finite Fields, Kluwer AP, The Netherlands, 1992.

....( g a (U m ) 6= f0g ( a 2 (T ffi f) U m ) 2 Corollary 8.3. Let m; k; t 1, q = p r , f 2 F q k [x 1 ; xm ] be t sparse. Then the image of T ffi f : F m q k F q can be calculated with (kt 1) O(mpr) evaluations of T ffi f . Proof. Using the algorithms of Shoup (1992) or Shparlinski (1990) (see also Shparlinski 1992a) we can construct in time (pkr) O(1) a set M F q k with cardinality (pkr) O(1) containing a primitive root of F q k . Setting U = f0g [ f e 2 F q k : 0 e sg F q k for 2 M , we have from Theorem 8.2 that (T ffi f) F m q k ) 2M (T ffi ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, vol. 88 of Mathematics and its applications. Kluwer Academic Publishers, 1992a.

....( g a (U m ) 6= f0g ( a 2 (T ffi f) U m ) 2 Corollary 8.3. Let m; k; t 1, q = p r , f 2 F q k [x 1 ; xm ] be t sparse. Then the image of T ffi f : F m q k F q can be calculated with (kt 1) O(mpr) evaluations of T ffi f . Proof. Using the algorithms of Shoup (1992) or Shparlinski (1990) (see also Shparlinski 1992a) we can construct in time (pkr) O(1) a set M F q k with cardinality (pkr) O(1) containing a primitive root of F q k . Setting U = f0g [ f e 2 F q k : 0 e sg F q k for 2 M , we have from Theorem 8.2 that (T ffi f) F m q k ) 2M (T ffi ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, vol. 88 of Mathematics and its applications. Kluwer Academic Publishers, 1992a.

....m Psi. Step 5 To decrypt the message, Alice merely computes Phi(a 1 ; a k ) m. It is obvious that the computational cost of this algorithm is polynomial in t, s, k and log q. More precisely, let us denote by M(q) the bit cost of an arithmetic operation over IF q . It is known [1, 6, 7] that one can take M(q) O(log q log log q) We also have to perform additions and subtractions of O(log q) bit integers. Each such arithmetic operation cost O(log q) bit operations and therefore can be estimated by M(q) as well. Indeed, although the precise value of M(q) is not known, it is ....

.... is to try to find a solution to the system of equations f i (x 1 ; x k ) 0; i = 1; k: However, all known algorithms to solve systems of polynomial equations of total degree n require (regardless of sparsity) time polynomial in n, which is exponentially large in our setting, cf. [1, 6, 7]. Another possible attack is to guess a solution. However, one expects that a system of k sparse polynomial equations in k variables of high degree over IF q has few zeroes over IF q . Thus the probability that such a random guess gives a solution is, apparently, very small. The best known ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, Kluwer Acad. Publ., Dordrecht, 1992.

.... brute force approach to computing all points on C via finding, for each a 2 F q , all b 2 F q with f(a; b) 0, takes O (n 2 q 3=2 ) operations in F q , using the fastest known deterministic algorithms to factor the univariate polynomial f(a; y) for all a 2 F q (Shoup 1990; Section 1. 1 of Shparlinski 1992, von zur Gathen Shoup 1992) We present in Section 3 a deterministic method that uses O (n 5 q) operations, i.e. polynomial time per point. The central tool for our estimates is a bound of Perel muter s (1969) on a certain exponential sum. In order to use this, we have to study in Section 4 ....

I. E. Shparlinski, Computational and algorithmic problems in finite fields, vol. 88 of Mathematics and its applications. Kluwer Academic Publishers, 1992.

....such as [36] or [17] explains these notions, and Appendix C in the latter text gives a highly readable exposition of Weil s results and their far reaching generalizations. Some small improvements on the Weil estimate exploiting the algebraic nature of the Frobenius roots are in [34, 35] and [39], Chapter 5. 43] show N 1 (C) g(q 1=2 Gamma 1) o(g) for curves of large genus over a fixed field; this is twice better than the Weil estimate. 9] 1986, Theorem 4.9) and [2] 1993) show that some variants of (2) hold for absolutely irreducible projective curves even if they are ....

....in the literature for this factorization: 5] 23] and [12] The latter paper gives O (n 7 log 2 q(n 5 log q) bit operations. These algorithms are probabilistic of the Las Vegas type, i.e. they never return an incorrect answer, but they may fail, with controllably small probability. [39], Theorem 1.7, gives a deterministic method whose cost is O(n 3:7 log q) for almost all input polynomials. We will assume from now on, without loss of generality, that f is squarefree. Next we determine for each f i whether it is absolutely irreducible, i.e. irreducible over an algebraic ....

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Shparlinski, I.E., Computational and Algorithmic Problems in Finite Fields Mathematics and its Applications 88 (1992), Kluwer Academic Publishers.

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Shparlinski I E (1992), Computational and Algorithmic Problems in Finite Fields, Kluwer, Dordrecht.

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I. E. Shparlinski, Computational and Algorithmic Problems in Finite Fields, Kluwer, 1992. 12

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