Division Algorithm and Construction of Curves with Many Points

Transcription

1 Revista Colombiana de Matemáticas Volumen , páginas Division Algorithm and Construction of Curves with Many Points Algoritmo de la división y construcción de curvas con muchos puntos Álvaro Garzón, Horacio Navarro Universidad del Valle, Cali, Colombia Abstract. We give a simple and effective method for the construction of algebraic curves over finite fields with many rational points. The curves are given as Kummer covers of the projective line. Key words and phrases. Algebraic curves, Finite fields, Rational points, Kummer etensions Mathematics Subject Classification. 14G05, 14H50. Resumen. Se presenta un simple y efectivo método para la construcción de curvas algebraicas sobre campos finitos con muchos puntos racionales. Las curvas son dadas como coberturas de Kummer de la línea proyectiva. Palabras y frases clave. Curvas algebraicas, campos finitos, puntos racionales, etensiones de Kummer. 1. Introduction Let C be a nonsingular, projective, geometrically irreducible curve defined over a finite field F q with q elements. Let C F q denote the set of Fq -rational points on C; i.e; points on C having all coordinates in F q. The Hasse-Weil bound implies #C F q q qgc, 1 where gc stands for the genus of the curve C. Curves which attains the Weil s upper bound are called maimal curves. The interest on curves over finite fields with many rational points with respect to their genera i.e; with #C F q close to known upper bounds; e.g., see the tables in [7]. was greatly renewed after Goppa s construction of linear 131

2 132 ÁLVARO GARZÓN & HORACIO NAVARRO codes with good parameters from such curves. This created a much stronger interest in the area and attracted new groups of researchers such as coding theorists and algorithmically inclined mathematicians. An added incentive was provided by the invention of elliptic-curve cryptosystem in The reader may refer to the book [4] for etensive work on this topic. The aim hence is to ehibit the construction of Kummer covers of the projective line over finite fields with many rational points. The key idea comes from [6] see also [3] and [2] and that is the construction of functions µ F q [] via Euclidean algorithm, such that for many elements = β F q, µβ is a r th power in F q with r a divisor of q 1. The paper is organized as follows: in Section 2 we present the method for the construction of good curves C and we eplain how one computes its genus and its number of rational points; in Section 3 we give some characterization of the remainder polynomial in particular cases obtained after having applied the division algorithm and in Section 4 we give several eamples based in our construction. In tables 1, 2 and 3 we summarize the good pairs gc, #C F q obtained in the eamples of Section The Method Let r > 1 be a divisor of q 1 and f, l two polynomials in F q [] such that deg f r deg l. We will denote by R l f r the remainder of the Euclidean division of f r by l, i.e., f r = lh + R l f r. 2 We will always assume that R l f r 0 i.e, f is not multiple of l. The method for the construction is then to consider the nonsingular projective model C of the curve given by the affine Kummer equation: y r = µ := R l f r. 3 As it was shown in [5, Proposition III.7.3], the genus gc can be easily derived from the multiplicities of the zeros and poles of the function µ F q. Now we will eplain how one computes the genus. If α F q is a zero of µ of multiplicity m α, then we have d α points on the curve C with first coordinate = α, where d α = gcdr, m α. These d α points have ramification inde e α := r/d α and they have different eponent equal to e α 1 by Dedekind s Eponent Theorem. The same holds for the points on the curve C above the point at infinite of the projective line i.e., for α = one defines its multiplicity as m = deg R l f r and, similarly, d = gcdr, m and e = r/d. Volumen 47, Número 2, Año 2013

3 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 133 New records q gc #C F q Former Entry Table 1 New entries q gc #CF q Table 2 Meets the record q gc #CF q Table 3 It now follows from Hurwitz s genus formula that gc = 1 + r d α. 4 2 r α F q The sum over α F q in the Formula 4 above is indeed a finite sum over only the zeros and poles of the function µ F q. In fact, if α F q is not a zero of µ, then the multiplicity is equal to zero and hence d α = r. From the genus Formula 4, we see that the genus gc is smaller as the function µ has fewer distinct zeros. So in general the inseparability of µ is desirable. Now we turn to the rational points on F q of the curve C given by Equation 3. Revista Colombiana de Matemáticas

4 134 ÁLVARO GARZÓN & HORACIO NAVARRO If α F q is a zero or pole of µ with multiplicity m α, then we have d α = gcdr, m α points on C with first coordinate = α. On the other hand, if α F q, then it may be happen that the d α points on C with = α are again rational points over F q. Here is how one decides if this is the case: If α is a zero of µ, we can write y r = g α mα or y r/dα α mα/dα dα = g, 5 where g F q [] with gα 0. Then the d α points are all rational over F q if and only if gα is a d α th power of an element of F q. On the other hand, if β F q satisfies lβhβ = 0 and fβ 0. Then, the value of the function µ at β is a r th power see formula 2. This guarantees that we have r rational points on the curve C with first coordinate = β as above. So in order to have many rational points over F q we will always take l as a polynomial having all its roots in the finite field F q. Now, if we denote by l 1, the polynomial lh/ gcd lh, f, then the number of F q rational points satisfies #C F q rλ where λ = deg l 1. Observe that here we only counted the rational points coming from the roots of the polynomial l 1 in F q. Other rational points can coming from the rational solutions γ F q of the equation such that l 1 γ 0. i.e.; those elements R l f r q 1/r 1 = 0 6 γ F q such that fγ r lγhγ is a r th power of an element in F q. 7 We some times have carried out a computer search to determine the cardinality κ of the set of such elements κ := # { γ F q such that l 1 γ 0 and γ is a r-th power in F q. } 8 For each γ F q satisfying 7 we have r rational points on C with first coordinate = γ. So in practice when r is a proper divisor of q 1, after having a good candidate for a curve C with many rational points over F q with respect to its genus gc we some times have carried out a computer search to determine the cardinality κ of the set of such elements. 3. Some Characterizations of Remainder Polynomial The goal of this section is give some characterizations of the polynomial R l f and then, we will use them to construct curves with many rational points Volumen 47, Número 2, Año 2013

5 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 135 Theorem 3.1. Let m 1 be a positive integer, f and l F q [] polynomials such that: i f and l are relatively prime. ii deg l = d, a divisor of m. iii l is irreducible. Then, R l f q m 1 = 1. Proof. First suppose that deg l = m. Since gcd f, l = 1 and l is an irreducible polynomial, then f / l, and therefore the class of f, in the residue class field F := F q []/ l, is non zero. On the other hand, since the finite field F has cardinality q m, then we have that f qm 1 + l = f + l q m 1 = 1 + l and therefore, f qm 1 1 l ; consequently, there eits h F q [] such that that is to say, R l f q m 1 = 1. f qm 1 = lh + 1, Now if deg l = d with d m, then by similar arguments as above we obtain that f qd 1 = lh + 1 for some h F q [] and since then we have: f qm 1 = q m 1 = q dt 1 = q d t 1 = q d 1 q dt , f qd 1 q dt = lh + 1 q dt = l h + 1 for some h F q []. Corollary 3.2. Let m 1 be a positive integer and let f, l i F q [] with i = 1, 2 polynomials such that 1 f and l i are relatively prime, 2 deg l i = d i, a divisor of m, Revista Colombiana de Matemáticas

6 136 ÁLVARO GARZÓN & HORACIO NAVARRO 3 l i is irreducible. Then R l1l 2 f q m 1 = 1. Proof. Suppose that f qm 1 l 1 l 2 h + 1 for all h F q []. By Theorem 3.1, the polynomial f qm 1 1 l 2 / l 1 and, since gcd l 1, l 2 = 1, we have that l 2 / l 1 and therefore, f qm = fqm l 2 / l 1 l 2 which is a contradiction. Corollary 3.3. If f and l F q [] are coprime and l is a proper divisor of the polynomial qm then R l f q m 1 = 1. Proof. It follows from Corollary 3.2 and the fact that the product of all monic irreducible polynomials over F q, whose degrees divide m, is precisely qm. In accordance with previous results, if we are interested in obtaining nonconstant remainders, the polynomials f and l must have at least one common factor. This fact does not guarantee that it is always possible to characterize the remainder polynomial, however with an appropriate choice of such polynomials one has a nice results as it is shown in the net theorem. Theorem 3.4. Let f = 1 and l = h 1 be polynomials in F q [] with h = q m 1 /q 1. Then R l h 1 1 qm 1 = i = h 1 i=1 Proof. Let q be the polynomial defined by Volumen 47, Número 2, Año 2013

7 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 137 q 2 q = i q 1 ih + q 1 ih 1 + q 1 ih Then, i=1 h 1 q 2 q = i i=1 h 1 h 1 q 2 = q 1h k + i i q 3 i=1 q 1 ih h 1 q 2 1 = i q 2 q ih k h i h 1 i=2 h 1 i=1 i=1 h 1 h 1. q 1 ih k q 1 ih k + 1 q ih k h h 1 q 3 = q 1h k + i + 1 i q 3 i=1 h 1 h 1 q 1 ih k q 2 h k + 1 i=1 h 1 q 3 = q 1h k + q 3 h 1 = i=0 On the other hand, h 1 q 1 ih k h h 1 q 1 ih k + 2 h k + 1 i=1 h 1 h 1 q 1 ih k + h + 2 h 1 q 1 ih k h + 2 h k + 1 k=1 h k + 1 = q 1h + + h + 2 h qm 1 = qm h + h = q 1h + + h + 2 h h = h 1 q h Remark 3.5. Observe that the polynomial R l 1 q m 1 has q m q different roots in F q m 1. In fact, first R l 1 q m 1 q+ +q m 2 q = h = 1 q 1 i. i=0 Revista Colombiana de Matemáticas

8 138 ÁLVARO GARZÓN & HORACIO NAVARRO Second, the polynomial p = q+ +q m 2 i=0 i = q+ +qm 2 is separable because if h = 1 + q + + q m 2, then the polynomial h 1 is separable and since p 1 = h 1, then p is also separable. Finally, it is easy to see that if α h 1 = 0, then α F q m 1. We end up this section with some characterizations of the remainder polynomial in the particular case when the polynomial l belongs to certain class of polynomials over finite fields, that will be used in net section for the construction of some curves with many rational points. First we begin with a definition. Let m, j be integers with m 1 and let s j 1, 2,..., m be the elementary symmetric function in m variables over F q. For all j Z and integers m 1 we define a polynomial s m,j F q [] see [1] as follows: s m,j := 0, for j < 0 s m,0 := 1 s m,1 := s 1, q,..., qm 1 = + q + + qm 1 s m,2 := s 2, q,..., qm 1 = 1+q + 1+q2 + + qm 2 +q m 1. s m,m := s m, q,..., qm 1 = 1+q+ +qm 1 s m,j := 0, for j > m. Observe that the polynomials s m,1 = + q + + qm 1 and s m,m = 1+q+ +qm 1 are nothing but the trace and norm polynomials corresponding to the finite etension F q m/f q. Theorem 3.6. If f = and l = s m,1, then R sm,1 q m 1 = q + + qm 1 = s m 1,1 q Proof. The proof is direct: Volumen 47, Número 2, Año 2013

9 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 139 qm 1 = qm q + + q m 1 + qm q + + qm 1 = + q + + qm 1 q q + + qm 1 = + q + + qm 1 q q + + qm 1 = = s m,1 s m,1 q 1 s m 1,1 q Theorem 3.7. Let f = + 1, τ m := m 1 j=0 s m,j and h = q m 1 /q 1 then R τm + 1 h = + 1τ m 1 q Proof. See [2] Lemma 3.3. Remark 3.8. With notations as Theorem 3.7, we have: i The polynomial τ m is separable, its roots belong to F q m and has degree h 1. ii The polynomial R τm + 1 h has q m q + 2 different roots in F q m 1. iii s m,m + 1 = τ m + s m,m. Proof. See [2] Lemmas 3.2 and Constructions In this section we will construct curves over finite fields using the characterizations obtained in section above. Theorem 4.1. Let h = q m 1/q 1. The nonsingular complete geometrically irreducible curve C over F q m defined by the Kummer equation y r = R l 1 q m 1 = h 1 with r > 1 a divisor of q m 1, has genus gc = 2 + r 1 m 2 i=0 qi e 1 e 2 Revista Colombiana de Matemáticas

10 140 ÁLVARO GARZÓN & HORACIO NAVARRO with e 1 = gcdr, q 1 and e = gcd r, 1 + q + + q m 2. The number of rational points satisfies #C F q rh 1 = r q m q Proof. By Remark 3.5, the polynomial R l 1 q m 1 has q m 2 + +q 2 +2 different roots in F q m 1 and one can write R l 1 q m 1 as R l 1 q m 1 q+ +q m 2 q = 1 q 1 i. In this case, we have q m q points totally ramified in C with ramification inde r, namely = 0 and = α with α F q m 1 corresponding to the roots of R l 1 q m 1 in F q m 1 {0, 1}. The point with first coordinate = 1 has ramification inde e 1 = gcdr, q 1, and the point at infinity has ramification inde e = gcdr, 1 + q + + q m 2. Then, the formula for the genus follows by 4. The claim about the number of rational points is clear. i=0 Corollary 4.2. If m = 2 and r = q 2 1, then the curve C over F q 2 the Kummer equation defined by is the Hermitian curve. y q2 1 = R l 1 q 2 1 = 1 q 1 Proof. Observe that the points with first coordinate = 0, = 1 and = have ramification inde e 0 = q 2 1, e 1 = q + 1 and e = q 2 1 respectively. Now it is easy to see that gc = q 2 q /2. For the rational points, by Theorem 4.1, we have #CF q 2 q 2 1q. The points above = 0 and = are rational and those q 1 above = 1 are also rational. Therefore #CF q 2 = q 2 1q+q 1+2 = q As an application of Theorem 4.1 we have: Eample 4.3. In this eample we will construct a curve C over F 5 3 with gc = 7 and 220 rational points. This is a new entry in [7]. The curve C is defined by the Kummer equation y 4 = µ := = In this case we have that h = 5 3 1/4 = 31 and therefore by Theorem 4.1, the polynomial l = 31 1 provides r h 1 = 4 30 = 120 rational points. Other points could be found if we analyze the solutions of Equation 6. Volumen 47, Número 2, Año 2013

11 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 141 In this particular case the eact value of κ in 8 is 24, namely, the roots of the polynomial , and hence we have 4 24 = 96 additional points. Finally, we will analyze the ramification points of the function µ. First, observe that µ has three roots in F 5 F 5 3, namely = 0, = 1 and = 1, the other four roots those roots of the polynomials and belongs to F 5 2. By 5, the points with first coordinate = 0, = 1 and = are rational, while the points above = 1 are not rational. Hence we have d 0 + d 1 + d = = 4 etra rational points. Summarizing, the curve C has = 220 rational points. The genus follows from Theorem 4.1. We now summarize the results obtained in the particular case when q = p 3 Table 4. In this case r a divisor of p 3 1, h = p 2 + p + 1 and the curves are defined by the Kummer equation: y r = p2 +p. We omit the details. q 3 r gc #C F q 3 old entry Table 4. Eamples of curves with many points using Theorem 4.1. Now we will construct curves over F q m using the Theorem 3.6. Theorem 4.4. The nonsingular complete geometrically irreducible curve C over F q m defined by the Kummer equation y r = R sm,1 q m 1 = s m 1,1 q Revista Colombiana de Matemáticas

12 142 ÁLVARO GARZÓN & HORACIO NAVARRO with r > 1, a divisor of q m 1, has genus gc = r 1 q m 1 2 2e where e 0 = gcdr, q 1, and the number of rational points satisfies #CF q r q m 1 1. Proof. First observe that the polynomial s m 1,1 is separable and has its roots in F q m 1. Therefore we have eactly q m 2 1 points totally ramified in C with ramification inde r. The points = 0 and = have ramification inde e 0 = gcdr, q 1. Here we write the polynomial R sm,1 q m 1 as R sm,1 q m 1 = s m 1,1 q = q q qm 2 1 q. Now the genus follows from 4. The affirmation corresponding to the number of rational points is clear. Eample 4.5. In this eample we will construct a curve C over F 5 3 with gc = 3 and 192 rational points. This is the better value known in [7]. The curve C is defined by the Kummer equation F 5 3 y 4 = = Since the polynomial l = has all its roots in F 5 3, then we have r h 1 = 4 24 = 96 rational points. The value κ in 8 for this case is κ = 24, namely, the roots of the polynomial hence we have 4 24 = 96 additional points. Finally, by 5, the points with first coordinate = 0, = and = α F 5 3, which are roots of the polynomials and 3 + 2, are not rational. The genus follows from Theorem 4.4. In Table 5 we summarize some results obtained using Theorem 4.4 in the particular case q = p and m = 3. The curves C are defined by the Kummer equation y r = p 1 + p2 1 Now, we will construct curves over F q m using Theorem 3.7. Theorem 4.6. Let m 2. The nonsingular complete geometrically irreducible curve C over F q m defined by the Kummer equation y r = R τm + 1 h = + 1τ m 1 q Volumen 47, Número 2, Año 2013

13 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 143 q 3 r gc #C F q 3 old entry Table 5. Eamples of curves with many points using Theorem 4.4. with r d, has genus gc = r 1 m 2 i=1 qi + r e 2 where e = gcdr, q 1 and the number rational points satisfies #CF q rh 1 = r q + q q m 1. Proof. By Remark 3.8, the polynomial R lm + 1 h has q m q + 2 different roots α F q m 1, so, each point on C with first coordinate = α has ramification inde r. The point at has ramification inde e = gcdr, q 1. Now the genus follows from 4. To obtain a bound to the number of rational points, observe that since τ m = s m,m + 1 s m,m, then τ m 1 = 1 m+1 0; so, the polynomials + 1 and τ m are coprimes and hence we conclude that the number rational points satisfies #CF q rh 1. Corollary 4.7. If m = 2 and r = q + 1, then the curve C over F q 2 the Kummer equation defined by y q+1 = R τ2 + 1 h = + 1 is maimal. Revista Colombiana de Matemáticas

14 144 ÁLVARO GARZÓN & HORACIO NAVARRO Proof. The points = 0, = 1 and the point at have ramification inde e 0 = q + 1, e 1 = q + 1 and e = q + 1/ gcdq + 1, 2 respectively. Now it is easy to see that gc = q + 1 gcdq + 1, 2 /2. For the rational points, observe that since τ 2 = q + + 1, then by Theorem 4.6, we have at least q + 1q rational points. The points on C with first coordinate = 0, = 1 are rational and the gcdq + 1, 2 points above are also rational. On the other hand, the eact value of κ in 8 is { q + 1q 2, if q is even; κ = q + 1q 3, if q is odd. Therefore #C {2q 2 + 1, if q is even; F q 2 = 2q 2 q + 1, if q is odd. As an application of Theorem 4.6 we have: Eample 4.8. In this eample we will construct a curve C over F 2 3 with gc = 9 and #C F 2 3 = 45. This is the better value known in [7]. We defined the curve C over F 2 3 by equation y 7 = µ := In this case we have that h = /1 = 7 and therefore by Theorem 4.6, the polynomial τ 2 = provides r h 1 = 7 6 = 42 rational points. Other points could come from the analysis of the solutions of Equation 6. In this case the eact value of κ in 8 is 0. Finally, we will analyze the ramification points of the function µ. First, observe that µ has two roots in F 2 F 2 3, namely = 0 and = 1. The other two roots those roots of the polynomial belongs to F 2 2. By 5 the points with first coordinate = 0, = 1 and = are rational, and we have eactly d 0 + d 1 + d = = 3 etra rational points. Summarizing, the curve C has = 45 rational points. The genus follows from Theorem 4.6. In Table 6 we summarize some eamples using Theorem 4.6. In the particular case q = p and m = 3 we have that h = q + 1, r is a divisor of q + 1 and the Kummer equation y r = + 1τ 2 q = + 1 q q Volumen 47, Número 2, Año 2013

15 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 145 Eample q r gc #C F q old entry Table 6. Eamples of curves with many points using Theorem Others Eamples In general, given two polynomials f, l F q [], it is very hard to give a characterization of the remainder polynomial R l f. However, although we do not give an eplicit formula for the remainder, we are going to give 4 eamples of good curves over the finite field F 243. These curves are defined as always by Kummer equations of the kind y n = R l f, with r and n divisors of q 1. In this particular case we will take f = and the polynomial l as a product of irreducible polynomials form the list below. l 1 = 1 + l 2 = 2 + l 3 = l 4 = l 5 = l 6 = l 7 = l 8 = l 9 = l 10 = l 11 = Eample 5.1. Taking l = l 3 l 4 l 5 l 6, r = 22 and n = 2, we obtain R l 22 = Now it is easy to verify that the curve C defined by the equation y 2 = has gc = 4 and #C F 3 5 = 306. Eample 5.2. Taking l = l 7 l 8 l 9 l 10, r = 22 and n = 2, we gets the curve C defined by the equation y 2 = , Revista Colombiana de Matemáticas

16 146 ÁLVARO GARZÓN & HORACIO NAVARRO which has gc = 6 and 345 rational points. Eample 5.3. The curve C given by y 11 = has gc = 5 and #C F 3 5 = 364. Here we take r = n = 11 and l = l 1 2 l 2. Then R l 11 = Eample 5.4. The curve C defined by y 11 = has gc = 15 and #C F 3 5 = 519. We here take r = n = 11 and l = l 3 l 11. Hence R l 11 = = We summarize the results obtained in this section in Table 7. q r gc #C F q old entry Table 7. Eamples of curves with many points over F 243. References [1] A García and H. Stichtenoth, A Class of Polynomials over Finite Fields, Finite Fields and Their Applications , [2] A. Garzón, Euclidean Algorithm and Kummer Covers with many Points, Revista Colombiana de Matemáticas , no. 1, [3] A. Garzón and A. García, On Kummer Covers with many Points over Finite Fields, Journal of Pure and Applied Algebra , [4] H Niederreiter and C.P Xing, Rational Points on Curves over Finite Fields. Theory and Applications, LMS Lecture Note Series 285, Cambridge University Press, [5] H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin, Germany, [6] G. Van der Geer and M. Van der Vlugt, Kummers Covers with many Points, Finite Fields and their Appl. 2000, [7], Tables with many Points, 2013, available in Volumen 47, Número 2, Año 2013

17 DIVISION ALGORITHM AND CONSTRUCTION OF CURVES 147 Recibido en noviembre de Aceptado en septiembre de 2013 Departamento de Matemáticas Universidad del Valle Facultad de Ciencias Carrera 13 No Cali, Colombia Revista Colombiana de Matemáticas

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