Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves,

We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to (ℤ/2ℤ)3 (over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3. These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristic p>3. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels.

[1]  Joseph H. Silverman,et al.  Diophantine Geometry: An Introduction , 2000, The Mathematical Gazette.

[2]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[3]  Serge Lang,et al.  Abelian varieties , 1983 .

[4]  Pierre Cartier.  ISOGENIES AND DUALITY OF ABELIAN VARIETIES , 1960 .

[5]  Nicolas Thériault,et al.  A double large prime variation for small genus hyperelliptic index calculus , 2004, Math. Comput..

[6]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[7]  S. Lang,et al.  NUMBER OF POINTS OF VARIETIES IN FINITE FIELDS. , 1954 .

[8]  Claus Diem,et al.  An Index Calculus Algorithm for Plane Curves of Small Degree , 2006, ANTS.

[9]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[10]  Benjamin A. Smith.  Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves , 2008, EUROCRYPT.

[11]  Ravi Vakil.  Twelve points on the projective line, branched covers, and rational elliptic fibrations , 1999 .

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

[13]  Ron Livne,et al.  The arithmetic-geometric mean and isogenies for curves of higher genus , 1997 .

[14]  Herbert Lange,et al.  Complex Abelian Varieties , 1992 .

[15]  D. Mumford.  Tata Lectures on Theta I , 1982 .

[16]  Florian Hess,et al.  Computing Riemann-Roch Spaces in Algebraic Function Fields and Related Topics , 2002, J. Symb. Comput..

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

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

[19]  Kenji Ueno,et al.  Principally polarized abelian variaties dimension two or three are Jacobian varieties , 1973 .

[20]  Christophe Ritzenthaler,et al.  An Explicit Formula for the Arithmetic–Geometric Mean in Genus 3 , 2007, Exp. Math..