ECC: Do We Need to Count?

A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves’ cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming.

[1]  H. W. Lenstra,et al.  Factoring integers with elliptic curves , 1987 .

[2]  Alfred Menezes,et al.  Elliptic curve public key cryptosystems , 1993, The Kluwer international series in engineering and computer science.

[3]  Everett W. Howe On the group orders of elliptic curves over finite fields , 2001, math/0110262.

[4]  P Erdős,et al.  On the number of positive integers . . . , 1966 .

[5]  R. Schoof Journal de Theorie des Nombres de Bordeaux 7 (1995), 219{254 , 2022 .

[6]  Stephen C. Pohlig,et al.  An Improved Algorithm for Computing Logarithms over GF(p) and Its Cryptographic Significance , 2022, IEEE Trans. Inf. Theory.

[7]  R. Lercier,et al.  "Computing isogenies in F_ ," ANTS-II , 1996 .

[8]  Reynald Lercier,et al.  Computing Isogenies in F2n , 1996, ANTS.

[9]  J. Couveignes Isogeny cycles and the Schoof-Elkies-Atkin algorithm , 1996 .

[10]  N. Koblitz PRIMALITY OF THE NUMBER OF POINTS ON AN ELLIPTIC CURVE OVER A FINITE FIELD , 1988 .

[11]  H. Halberstam,et al.  On Integers All of Whose Prime Factors are Small , 1970 .

[12]  Horst G. Zimmer,et al.  Constructing elliptic curves with given group order over large finite fields , 1994, ANTS.

[13]  Reynald Lercier,et al.  Counting the Number of Points on Elliptic Curves over Finite Fields: Strategies and Performance , 1995, EUROCRYPT.

[14]  K. Dickman On the frequency of numbers containing prime factors of a certain relative magnitude , 1930 .

[15]  François Morain,et al.  Schoof's algorithm and isogeny cycles , 1994, ANTS.

[16]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[17]  R. Schoof Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[18]  J. Dixon Asymptotically fast factorization of integers , 1981 .

[19]  R. Zuccherato,et al.  Counting Points on Elliptic Curves Over F2m , 1993 .

[20]  de Ng Dick Bruijn On the number of positive integers $\leq x$ and free of prime factors $>y$ , 1951 .

[21]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[22]  Martin E. Hellman,et al.  An improved algorithm for computing logarithms over GF(p) and its cryptographic significance (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[23]  Kenji Koyama,et al.  Equivalence of Counting the Number of Points on Elliptic Curve over the Ring Zn and Factoring n , 1998, EUROCRYPT.

[24]  Jacques Stern,et al.  Security Analysis of a Practical "on the fly" Authentication and Signature Generation , 1998, EUROCRYPT.