Bilinear character sums over elliptic curves

Finite Fields and Their Applications 14 (2008) 132–141 http://www.elsevier.com/locate/ffa

Bilinear character sums over elliptic curves ✩ Igor Shparlinski Department of Computing, Macquarie University, Sydney, NSW 2109, Australia Received 8 April 2006; revised 17 October 2006 Available online 17 December 2006 Communicated by Stephen D. Cohen

Abstract Let E be an elliptic curve over a finite field Fq of q elements given by an affine Weierstraß equation. Let ⊕ denote the group operation in the Abelian group of points on E. We also use x(P ) to denote the x-component of a point P = (x(P ), y(P )) ∈ E. We estimate character sums     ρ(U )ϑ(V )ψ x(U ⊕ V ) , Wρ,ϑ (ψ, U , V) = U ∈ U V ∈V

where U and V are arbitrary sets of Fq -rational points on E, ψ is a nontrivial additive character of Fq and ρ(U ) and ϑ(V ) are arbitrary bounded complex functions supported on U and V, respectively. Our bound of sums Wρ,ϑ (ψ, U , V) is nontrivial whenever # U > q 1/2+ε

and #V > q ε

for some fixed ε > 0. We also give various applications of this bound. © 2006 Elsevier Inc. All rights reserved. Keywords: Elliptic curves; Character sums

✩

This work was supported in part by ARC grant DP0556431. E-mail address: [email protected].

1071-5797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ffa.2006.12.005

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1. Introduction We fix a finite field Fq of q elements and an elliptic curve E over Fq given by an affine Weierstraß equation E: y 2 + (a1 x + a3 )y = x 3 + a2 x 2 + a4 x + a6 , with some a1 , . . . , a6 ∈ Fq , see [17]. We recall that the set of all points on E forms an Abelian group, with the point at infinity O as the neutral element, and we use ⊕ to denote the group operation. As usual we write every point P = O on E as P = (x(P ), y(P )). Let E(Fq ) denote the set of Fq -rational points on E. We estimate character sums Wρ,ϑ (ψ, U, V) =

 

  ρ(U )ϑ(V )ψ x(U ⊕ V ) ,

U ∈ U V ∈V

where U and V are arbitrary subsets of E(Fq ), ρ(U ) and ϑ(V ) are arbitrary bounded complex functions supported on U and V and ψ is a nontrivial additive character of Fq . Note, that we always assume that the values for which the corresponding summation term is not defined (for example the terms with U = −V in the sum Wρ,ϑ (ψ, U, V)) are excluded from summation. It is well known that double exponential sums 

ψ(ab)

and

a∈A b∈B



χ(a + b),

a∈A b∈B

with additive and multiplicative characters ψ and χ and arbitrary sets A, B ⊆ Fp have proved to be useful in many applications; see [5–12,15,16] as well as [18, Problem 14.a, Chapter 6], and the references therein. We also remark that slightly different double sums over points on an elliptic curve E have recently been considered in [1], namely the sum    ψ x(abP ) a∈A b∈B

for a fixed point P ∈ E(Fq ) and arbitrary sets A, B ⊆ Z/T Z, where T is the order of P . The above sums have also been considered with arbitrary weights αa and βb supported on A and B, respectively. Here we obtain an upper bound for the bilinear sums Wρ,ϑ (ψ, U, V) that is nontrivial whenever # U > q 1/2+ε for some fixed ε > 0.

and #V > q ε

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We apply this bound to several problems of different flavours: • We study an elliptic curve analogue of a problem which has been considered by A. Sárközy [15] over finite fields. • We consider Beatty sequences on elliptic curves and estimate character sums with these sequences. • We estimate certain twisted character sums. Throughout the paper, the implied constants in symbols ‘O’ and ‘’ may sometimes, where obvious, depend on an integer parameter ν  1 and a real parameter ε > 0 and are absolute otherwise (we recall that U  V and U = O(V ) are both equivalent to the inequality |U |  cV with some constant c > 0). 2. Preliminaries We consider the function field Fq (E) of the elliptic curve. The degree of a rational function f (X, Y ) ∈ Fq (E) is the maximum weighted degree of the monomials using deg X = 2 and deg Y = 3. For an s-dimensional vector c = (c1 , . . . , cs ) ∈ Fsq and a vector V = (V1 , . . . , Vs ) ∈ E(Fq )s of s pairwise distinct affine points, we consider the function fc,V (U ) ∈ Fq (E) given by fc,V (U ) =

s 

ci x(U ⊕ Vi ),

(1)

i=1

for U = ±Vi , i = 1, . . . , s. It has been shown in [14] that the explicit addition formulas for points on elliptic curves, imply the following statement. Lemma 1. For a nonzero vector c ∈ Fsq and a vector V = (V1 , . . . , Vs ) ∈ E(Fq )s of s pairwise distinct affine points, fc,V (U ) ∈ Fq (E) is a nonconstant rational function of degree O(s), which has poles at U = ±V1 , . . . , ±Vs . Also, as in [14], we note that a combination of Lemma 1 and the Bombieri bound [4] leads to the following estimate. Lemma 2. Let ψ be a nontrivial additive character. For a nonzero vector c ∈ Fsq and a vector V = (V1 , . . . , Vs ) ∈ E(Fq )s of s pairwise distinct affine points, the bound 

  ψ fc,V (U )  sq 1/2

U ∈E(Fq )

holds. We recall that according to our agreement, the sum in Lemma 2 runs over all points U ∈ E(Fq ) at which the function fc,V (U ) is defined.

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3. Bilinear sums Theorem 3. Let ψ be a nontrivial additive character. Let U and V be arbitrary sets of E(Fq ) with   ρ(U )  1,

  and ϑ(V )  1,

u ∈ U,

v ∈ V.

Then for any fixed integer ν  1 we have Wρ,ϑ (ψ, U, V)  (# U)1−1/2ν (#V)1/2 q 1/2ν + (# U)1−1/2ν #Vq 1/4ν . Proof. Writing          Wρ,ϑ (ψ, U, V)  ϑ(V )ψ x(U ⊕ V )   U ∈ U V ∈V

and applying the Hölder inequality, we obtain        2ν Wρ,ϑ (ψ, U, V)2ν  (# U)2ν−1  ϑ(V )ψ x(U ⊕ V )   U ∈ U V ∈V

     2ν   ϑ(V )ψ x(U ⊕ V )  

 (# U)2ν−1

U ∈E(Fq ) V ∈V



= (# U)2ν−1

ν 

V1 ,...,V2ν ∈V i=1

×

 U ∈E(Fq )



ψ

ν  i=1

ϑ(Vi )

2ν 

ϑ(Vi )

i=ν+1

x(U ⊕ Vi ) −

2ν 

 x(U ⊕ Vi ) .

i=ν+1

In the case when (V1 , . . . , Vν ) is a permutation of (Vν+1 , . . . , V2ν ), we estimate the sum over U trivially as #E(Fq ) = O(q). Otherwise, the sum inside the character corresponds to a function of the form (1) and by Lemmas 1 and 2 we estimate the character sum for those points as O(q 1/2 ). Therefore, putting everything together we derive     Wρ,ϑ (ψ, U, V)2ν  (# U)2ν−1 (#V)ν q + (#V)2ν q 1/2 , which finishes the proof.

2

Furthermore, if for some fixed ε > 0 we have # U > q 1/2+ε and #V > q ε , then the bound of Theorem 3 takes form     Wρ,ϑ (ψ, U, V)  # U#V q 1/4ν−ε/2 + q −ε/2ν

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and taking ν = 1/ε (and recalling that ε  1/2 thus ν = 1/ε  1/ε + 1  3ε/2),   Wρ,ϑ (ψ, U, V)  # U#Vq −ε2 /3 . Also, if # U#V q 1+ε then assuming without loss of generality that # U  #V (thus # U  q 1/2+ε/2 ) and using the bound of Theorem 3 with ν = 1, we derive √   Wρ,ϑ (ψ, U, V)  # U#Vq + # U #Vq 1/4  # U#Vq −ε/4 . 4. Sárközy problem for elliptic curves A. Sárközy [15] shows that for any sets A, B, C, D ∈ Fq the number of solutions N (A, B, C, D) of the equation a + b = cd,

a ∈ A, b ∈ B, c ∈ C, d ∈ D,

satisfies     N (A, B, C, D) − #A#B#C#D   #A#B#C#Dq.   q In particular,    #A#B#C#D N (A, B, C, D) = 1 + O q −ε/2 q whenever #A#B#C#D  q 3+ε for some fixed ε > 0 and sufficiently large q. (Note that in [15] only the case of prime q is considered, but the extension to arbitrary finite fields is immediate.) Here we estimate the number of solutions M(S, T , U, V) of the equation x(S) + x(T ) = x(U ⊕ V ),

S ∈ S, T ∈ T , U ∈ U, V ∈ V,

for any sets S, T , U, V ∈ E(Fq ). However, the threshold q 3+ε has to be replaced with q 7/2+ε . Theorem 4. For every ε > 0 and arbitrary sets S, T , U, V ∈ E(Fq ) with #S#T # U#V  q 7/2+ε we have  #S#T # U#V   . M(S, T , U, V) = 1 + O q −ε/2 q Proof. Let Ψ be the set of all q additive characters of Fq , and let Ψ ∗ be the set of nontrivial characters. Since

 0, if z = 0, ψ(z) = (2) q, if z = 0, ψ∈Ψ

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we obtain  1     ψ x(S) + x(T ) − x(U ⊕ V ) q

M(S, T , U, V) =

S∈S T ∈T U ∈ U V ∈V ψ∈Ψ

#S#T # U#V + Δ, q

= where

   1        |Δ|   ψ x(S) + x(T ) − x(U ⊕ V )  q ∗ ψ∈Ψ S∈S T ∈T U ∈ U V ∈V

            1          ψ x(S)  ψ x(T )  ψ x(U ⊕ V ) .  q ∗ ψ∈Ψ

S∈S

T ∈T

U ∈ U V ∈V

Using Theorem 3 with ν = 1 and then the Cauchy inequality, we obtain |Δ| 



# U#Vq −1

   √         −3/4    + # U #Vq ψ x(S)  ψ x(T )   ψ∈Ψ ∗ S∈S



√   # U#Vq −1 + # U #Vq −3/4 

 

   2 

   × ψ x(S)   ψ∈Ψ ∗ S∈S

T ∈T

    2   . ψ x(T )  

ψ∈Ψ ∗ T ∈T

Now, by (2) we conclude that         2      2      = qR, ψ x(S) ψ x(S)    

ψ∈Ψ ∗ S∈S

ψ∈Ψ S∈S

where R is the number of pairs S1 , S2 ∈ S with x(S1 ) = x(S2 ). Since there are at most two points on E(Fq ) with the same x-coordinate, we have R  2#S. Using the same argument for the sum over T ∈ T , we obtain the bound 

√ √ # U #Vq 1/4 #S#T √ = #S#T # U#Vq + #S#T # U #Vq 1/4 .

|Δ| 

# U#Vq +

It is obvious that for #S#T # U#V  q 7/2+ε  q 3+ε we have √ #S#T # U#Vq q 3/2  q −ε/2 . = √ #S#T # U#Vq −1 #S#T # U#V

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Clearly, we can assume that # U  #V. Then #S#T 

q 7/2+ε q 7/2+ε .  # U#V (# U)2

Therefore √ #U #S#T # U #Vq 1/4 q 5/4  =  q −ε/2 , √ 1/2+ε/2 #S#T # U#Vq −1 q #S#T # U

√

which concludes the proof.

2

5. Beatty sequences on elliptic curves For two fixed real numbers α and β, the corresponding non-homogeneous Beatty sequence is the sequence of integer parts αn + β , n = 1, 2, 3, . . . . Recently, character sums with Beatty sequences have been estimated in [2,3] (see also references therein to a variety of other works on Beatty sequences). Here we apply Theorem 3 to estimate the sums Sα,β (ψ, N, P ) =

N     ψ x αn + β P n=1

for a given point P ∈ Eq (Fq ). Recall that the discrepancy D(A) of a sequence A = (a1 , . . . , aN ) of N (not necessarily distinct) real numbers in the unit interval [0, 1] is defined by the relation     I (A, γ )  D(A) = sup  − γ , N 0γ 1 where I (A, γ ) is the number of positive integers n  N such that for which an < γ . We denote by Δα (N ) the discrepancy of the sequence of the fractional parts {αn}, n = 1, . . . , N . Theorem 5. Let ψ be a nontrivial additive character and let α and β be a fixed real numbers. Then for any N  T , where T is the order P ∈ Eq (Fq ), the following bound holds: Sα,β (ψ, N, P )  N 1/2 q 1/4 + N Dα (N ). Proof. Let K  N be a positive integer, and let Δ be a real number in the interval (0, 1]. For every real number γ , let   Nγ = 1  n  N : {αn + β − γ } < 1 − Δ ,   Kγ = 1  k  K: {αk + γ } < Δ ,

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and put Nγc = {1, 2, . . . , N} \ Nγ . From the definition of the discrepancy, we immediately conclude that   #Nγc = N Δ + O N Dα,β (N ) .

(3)

As in [2,3], we note that, by the pigeonhole principle, #Kγ  0.5KΔ

(4)

for some choice of γ ∈ (0, 1]. Fix γ with this property, and put N = Nγ , N c = Nγc and K = Kγ . We have for every k ∈ K: Sα,β (ψ, N, P ) =



    ψ x α(n + k) + β P + O(k)

nN

=



    ψ x α(n + k) + β P + O(K)

nN

=



      ψ x α(n + k) + β P + O K + #N c .

n∈N

Therefore, Sα,β (ψ, N, P ) =

  W + O K + #N c , #K

(5)

where W=

      ψ x α(n + k) + β P . n∈N k∈K

For any n ∈ N and k ∈ K, we have 

   α(n + k) + β = α(n + k) + β − α(n + k) + β = (αn + β − γ ) + (αk + γ ) − {αn + β − γ } − {αk + γ } = αn + β − γ + αk + γ .

Therefore W=

    ψ x αn + β − γ P ⊕ αk + γ P . n∈N k∈K

We also remark that, since N  p, we have   # n  N: αn + β − γ ≡ s (mod T ) = O(1) uniformly for all s ∈ Z. Thus, up to a bounded scaling factor the sum W is of the type Wρ,ϑ (ψ, U, V) with # U  N , #V  #K.

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Applying Theorem 3 with G = P  (the cyclic subgroup generated by P ) and ν = 1, we obtain W



N#Kq +

√

N #Kq 1/4 .

Hence, substituting the bound in (5) and recalling (3) and (4), we derive Sα,β (ψ, N, P )  N 1/2 K −1/2 q 1/2 Δ−1/2 + N 1/2 q 1/4 + K + N Δ + N Dα,β (N ). We now choose   K = q 1/4 N 1/2 and the result follows.

and Δ = q 1/4 N −1/2 ,

2

Clearly the bound is nontrivial if T  N  q 1/2+ε for some fixed ε > 0. As in [2,3] we remark that Dα (N ) = o(1) for any irrational α (see [13, Example 2.1, Chapter 1]) and if more Diophantine information about α is known then stronger bounds on Dα (N ) are available (see [13, Theorem 3.2, Chapter 2]). 6. Twisted sums Finally, given a point P ∈ Eq (Fq ) we consider the twisted sums Tα (ψ, N, P ) =

N    ψ x(nP ) exp(2πiαn). n=1

Theorem 6. Let ψ be a nontrivial additive character and let α be a fixed real number. Then for any N  T , where T is the order P ∈ Eq (Fq ) and any real α the following bound holds: Tα (ψ, N, P )  N 1/2 q 1/4 . Proof. We have for every k  0: N       Tα (ψ, N, P ) = ψ x (n + k)P exp 2πiα(n + k) + O(K). n=1

Therefore, Tα (ψ, N, P ) =

W + O(K), K

where W=

N  K      ψ x(nP ⊕ kP ) exp 2πiα(n + k) n=1 k=1

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to which we apply Theorem 3 with ν = 1. This yields the bound Tα (ψ, N, P )  N 1/2 K −1/2 q 1/2 + N 1/2 q 1/4 + K. We now choose   K = q 1/3 N 1/3 to derive Tα (ψ, N, P )  N 1/3 q 1/3 + N 1/2 q 1/4 . It remains to note that the bound is trivial for N  q 1/2 and for N > q 1/2 the first term never dominates. 2 References [1] W.D. Banks, J.B. Friedlander, M.Z. Garaev, I.E. Shparlinski, Double character sums over elliptic curves and finite fields, Pure Appl. Math. Quart. 2 (2006) 179–197. [2] W. Banks, I.E. Shparlinski, Non-residues and primitive roots in Beatty sequences, Bull. Austral. Math. Soc. 73 (2006) 433–443. [3] W. Banks, I.E. Shparlinski, Short character sums with Beatty sequences, Math. Res. Lett. 13 (2006) 539–547. [4] E. Bombieri, On exponential sums in finite fields, Amer. J. Math. 88 (1966) 71–105. [5] J. Bourgain, More on the sum-product phenomenon in prime fields and its applications, Internat. J. Number Theory 1 (2005) 1–32. [6] F.R.K. Chung, Several generalizations of Weil sums, J. Number Theory 49 (1994) 95–106. [7] W. Duke, J.B. Friedlander, H. Iwaniec, Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997) 23–43. [8] J.B. Friedlander, H. Iwaniec, Estimates for character sums, Proc. Amer. Math. Soc. 119 (1993) 363–372. [9] J.B. Friedlander, I.E. Shparlinski, Double exponential sums over thin sets, Proc. Amer. Math. Soc. 129 (2001) 1617–1621. [10] M.Z. Garaev, Double exponential sums related to Diffie–Hellman distributions, Int. Math. Res. Not. 2005 (17) (2005) 1005–1014. [11] H. Iwaniec, A. Sárközy, On a multiplicative hybrid problem, J. Number Theory 26 (1987) 89–95. [12] A.A. Karatsuba, The distribution of values of Dirichlet characters on additive sequences, Dokl. Acad. Sci. USSR 319 (1991) 543–545 (in Russian). [13] L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences, Wiley–Interscience, 1974. [14] T. Lange, I.E. Shparlinski, Distribution of some sequences of points on elliptic curves, J. Math. Cryptology 1 (2007), in press. [15] A. Sárközy, On sums and products of residues modulo p, Acta Arith. 118 (2005) 403–409. [16] I.E. Shparlinski, On the distribution of primitive and irreducible polynomials modulo a prime, Diskret. Mat. 1 (1) (1989) 117–124 (in Russian). [17] J.H. Silverman, The Arithmetic of Elliptic Curves, Springer, Berlin, 1995. [18] I.M. Vinogradov, Elements of Number Theory, Dover, NY, 1954.