Pairings on hyperelliptic curves

We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairing-based cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss the techniques used to optimize the pairing computation on hyperelliptic curves, and present many directions for further research.

[1]  Kristin Lauter.  THE EQUIVALENCE OF THE GEOMETRIC AND ALGEBRAIC GROUP LAWS FOR JACOBIANS OF GENUS 2 CURVES , 2001 .

[2]  Jiwu Huang,et al.  A note on the Ate pairing , 2008, International Journal of Information Security.

[3]  Steven D. Galbraith,et al.  Supersingular Curves in Cryptography , 2001, ASIACRYPT.

[4]  Steven D. Galbraith,et al.  Pairings on Hyperelliptic Curves with a Real Model , 2008, Pairing.

[5]  Paulo S. L. M. Barreto,et al.  On Compressible Pairings and Their Computation , 2008, AFRICACRYPT.

[6]  Alfred Menezes,et al.  Pairing-Based Cryptography at High Security Levels , 2005, IMACC.

[7]  Yoonjin Lee,et al.  Eta pairing computation on general divisors over hyperelliptic curves y , 2007, J. Symb. Comput..

[8]  Florian Hess,et al.  Pairing Lattices , 2008, Pairing.

[9]  G. Frey.  Applications of Arithmetical Geometry to Cryptographic Constructions , 2001 .

[10]  Mitsuru Kawazoe,et al.  Pairing-friendly Hyperelliptic Curves of Type y 2 = x 5 + ax , 2008 .

[11]  Frederik Vercauteren,et al.  Ate Pairing on Hyperelliptic Curves , 2007, EUROCRYPT.

[12]  Michael Scott,et al.  Pairing Calculation on Supersingular Genus 2 Curves , 2006, Selected Areas in Cryptography.

[13]  M. Scott.  Implementing cryptographic pairings , 2007 .

[14]  Jiwu Huang,et al.  All Pairings Are in a Group , 2008, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[15]  David Mandell Freeman,et al.  A Generalized Brezing-Weng Algorithm for Constructing Pairing-Friendly Ordinary Abelian Varieties , 2008, Pairing.

[16]  Iwan M. Duursma,et al.  Tate Pairing Implementation for Hyperelliptic Curves y2 = xp-x + d , 2003, ASIACRYPT.

[17]  Eunjeong Lee,et al.  TATE PAIRING COMPUTATION ON THE DIVISORS OF HYPERELLIPTIC CURVES OF GENUS 2 , 2008 .

[18]  K. Paterson.  Advances in Elliptic Curve Cryptography: Cryptography from Pairings , 2005 .

[19]  Frederik Vercauteren,et al.  Optimal Pairings , 2010, IEEE Transactions on Information Theory.

[20]  Martijn Stam,et al.  On Small Characteristic Algebraic Tori in Pairing-Based Cryptography , 2004, IACR Cryptol. ePrint Arch..

[21]  Takakazu Satoh,et al.  Constructing pairing-friendly hyperelliptic curves using Weil restriction , 2011, IACR Cryptol. ePrint Arch..

[22]  Emanuele Cesena.  Pairing with Supersingular Trace Zero Varieties Revisited , 2008, IACR Cryptol. ePrint Arch..

[23]  Hyang-Sook Lee,et al.  Efficient and Generalized Pairing Computation on Abelian Varieties , 2009, IEEE Transactions on Information Theory.

[24]  Michael Scott,et al.  Exponentiation in Pairing-Friendly Groups Using Homomorphisms , 2008, Pairing.

[25]  David Jao,et al.  Speeding Up Pairing Computations on Genus 2 Hyperelliptic Curves with Efficiently Computable Automorphisms , 2008, Pairing.

[26]  R. Balasubramanian,et al.  The Improbability That an Elliptic Curve Has Subexponential Discrete Log Problem under the Menezes—Okamoto—Vanstone Algorithm , 1998, Journal of Cryptology.

[27]  David Mandell Freeman,et al.  Constructing Pairing-Friendly Genus 2 Curves with Ordinary Jacobians , 2007, Pairing.

[28]  Gabriel Cardona.  On the number of curves of genus 2 over a finite field , 2003 .

[29]  Matthew K. Franklin,et al.  Identity-Based Encryption from the Weil Pairing , 2001, CRYPTO.

[30]  Tatsuaki Okamoto,et al.  Homomorphic Encryption and Signatures from Vector Decomposition , 2008, Pairing.

[31]  Alice Silverberg,et al.  Supersingular Abelian Varieties in Cryptology , 2002, CRYPTO.

[32]  D. Cantor.  Computing in the Jacobian of a hyperelliptic curve , 1987 .

[33]  Arjen K. Lenstra,et al.  Unbelievable Security. Matching AES Security Using Public Key Systems , 2001, ASIACRYPT.

[34]  Don Coppersmith,et al.  Fast evaluation of logarithms in fields of characteristic two , 1984, IEEE Trans. Inf. Theory.

[35]  Andrew M. Odlyzko,et al.  Discrete Logarithms in Finite Fields and Their Cryptographic Significance , 1985, EUROCRYPT.

[36]  Igor E. Shparlinski,et al.  MOV attack in various subgroups on elliptic curves , 2004 .

[37]  Gabriel Cardona,et al.  Curves of genus two over fields of even characteristic , 2002 .

[38]  Nigel P. Smart,et al.  High Security Pairing-Based Cryptography Revisited , 2006, ANTS.

[39]  Joseph H. Silverman,et al.  The arithmetic of elliptic curves , 1986, Graduate texts in mathematics.

[40]  Christof Paar,et al.  Cantor versus Harley: optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems , 2005, IEEE Transactions on Computers.

[41]  Fangguo Zhang.  Twisted Ate pairing on hyperelliptic curves and applications , 2010, Science China Information Sciences.

[42]  Paulo S. L. M. Barreto,et al.  Efficient pairing computation on supersingular Abelian varieties , 2007, IACR Cryptol. ePrint Arch..

[43]  Steven D. Galbraith,et al.  Computing pairings using x-coordinates only , 2009, Des. Codes Cryptogr..

[44]  YoungJu Choie,et al.  Implementation of Tate Pairing on Hyperelliptic Curves of Genus 2 , 2003, ICISC.

[45]  Laura Hitt.  On the Minimal Embedding Field , 2007, Pairing.

[46]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

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

[48]  David Mandell Freeman,et al.  Abelian varieties with prescribed embedding degree , 2008, IACR Cryptol. ePrint Arch..

[49]  Annegret Weng,et al.  Constructing hyperelliptic curves of genus 2 suitable for cryptography , 2003, Math. Comput..

[50]  Frederik Vercauteren,et al.  Hyperelliptic Pairings , 2007, Pairing.

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

[52]  Tanja Lange,et al.  Handbook of Elliptic and Hyperelliptic Curve Cryptography , 2005 .

[53]  Tsuyoshi Takagi,et al.  Novel Efficient Implementations of Hyperelliptic Curve Cryptosystems Using Degenerate Divisors , 2004, WISA.

[54]  Hovav Shacham,et al.  Short Signatures from the Weil Pairing , 2001, J. Cryptol..

[55]  Paul C. van Oorschot,et al.  Parallel Collision Search with Cryptanalytic Applications , 2013, Journal of Cryptology.

[56]  晋輝 趙,et al.  H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen and F. Vercauteren (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Math. Appl. (Boca Raton)., Chapman & Hall/CRC, 2006年，xxxiv + 808ページ． , 2009 .

[57]  Antoine Joux,et al.  A One Round Protocol for Tripartite Diffie–Hellman , 2000, Journal of Cryptology.

[58]  Victor S. Miller,et al.  The Weil Pairing, and Its Efficient Calculation , 2004, Journal of Cryptology.

[59]  Michael Scott,et al.  A Taxonomy of Pairing-Friendly Elliptic Curves , 2010, Journal of Cryptology.

[60]  Steven D. Galbraith,et al.  Ordinary abelian varieties having small embedding degree , 2007, Finite Fields Their Appl..

[61]  William E. Burr,et al.  Recommendation for Key Management, Part 1: General (Revision 3) , 2006 .

[62]  이윤진,et al.  Eta pairing computation on general divisors over hyperelliptic curves y2 = xp - x + d , 2008 .

[63]  Ian F. Blake,et al.  Advances in Elliptic Curve Cryptography: Frontmatter , 2005 .

[64]  Steven D. Galbraith,et al.  Distortion maps for genus two curves , 2006, IACR Cryptol. ePrint Arch..

[65]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[66]  Tanja Lange,et al.  Formulae for Arithmetic on Genus 2 Hyperelliptic Curves , 2005, Applicable Algebra in Engineering, Communication and Computing.

[67]  Neal Koblitz,et al.  Hyperelliptic cryptosystems , 1989, Journal of Cryptology.

[68]  Tanja Lange,et al.  Fast Bilinear Maps from the Tate-Lichtenbaum Pairing on Hyperelliptic Curves , 2006, ANTS.

[69]  Annegret Weng,et al.  Elliptic Curves Suitable for Pairing Based Cryptography , 2005, Des. Codes Cryptogr..

[70]  Frederik Vercauteren,et al.  The Eta Pairing Revisited , 2006, IEEE Transactions on Information Theory.

[71]  Kristin E. Lauter,et al.  Improved Weil and Tate Pairings for Elliptic and Hyperelliptic Curves , 2004, ANTS.