On Pairing Inversion Problems

In many aspects, cryptanalyses of pairing based cryptography consider protocol level security and take difficulties of primitives for granted. In this survey, we consider pairing inversion. At the time this manuscript was written(April 2007), to the best of the author's knowledge, there are neither known feasible algorithms for pairing inversions nor published proofs that the problem is unfeasible.

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

[2]  Takakazu Satoh On Degrees of Polynomial Interpolations Related to Elliptic Curve Cryptography , 2005, WCC.

[3]  Paulo S. L. M. Barreto,et al.  Efficient Algorithms for Pairing-Based Cryptosystems , 2002, CRYPTO.

[4]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1991, STOC '91.

[5]  Tanja Lange,et al.  Interpolation of the Discrete Logarithm in Fq by Boolean Functions and by Polynomials in Several Variables Modulo a Divisor of Q-1 , 2003, Discret. Appl. Math..

[6]  T. Elgamal A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, CRYPTO 1984.

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

[8]  Takakazu Satoh,et al.  On Polynomial Interpolations related to Verheul Homomorphisms , 2006 .

[9]  Ueli Maurer,et al.  The Diffie–Hellman Protocol , 2000, Des. Codes Cryptogr..

[10]  A. Miyaji,et al.  New Explicit Conditions of Elliptic Curve Traces for FR-Reduction , 2001 .

[11]  Eike Kiltz,et al.  On the interpolation of bivariate polynomials related to the Diffie-Hellman mapping , 2004, Bulletin of the Australian Mathematical Society.

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

[13]  Moti Yung,et al.  Advances in Cryptology — CRYPTO 2002 , 2002, Lecture Notes in Computer Science.

[14]  Paulo S. L. M. Barreto,et al.  Pairing-Friendly Elliptic Curves of Prime Order , 2005, Selected Areas in Cryptography.

[15]  Tanja Lange,et al.  Polynomial Interpolation of the Elliptic Curve and XTR Discrete Logarithm , 2002, COCOON.

[16]  Clifford C. Cocks An Identity Based Encryption Scheme Based on Quadratic Residues , 2001, IMACC.

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

[18]  Igor E. Shparlinski,et al.  Elliptic Curves with Low Embedding Degree , 2006, Journal of Cryptology.

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

[20]  Aggelos Kiayias,et al.  Polynomial Reconstruction Based Cryptography , 2001, Selected Areas in Cryptography.

[21]  Antoine Joux A One Round Protocol for Tripartite Diffie-Hellman , 2000, ANTS.

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

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

[24]  Eric R. Verheul,et al.  Evidence that XTR Is More Secure than Supersingular Elliptic Curve Cryptosystems , 2001, EUROCRYPT.

[25]  Ueli Maurer,et al.  The Relationship Between Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms , 1999, SIAM J. Comput..

[26]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1993, IEEE Trans. Inf. Theory.

[27]  David Mandell Freeman,et al.  Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10 , 2006, ANTS.

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

[29]  Steven D. Galbraith,et al.  Simplified pairing computation and security implications , 2007, J. Math. Cryptol..

[30]  Chi Sung Laih,et al.  Advances in Cryptology - ASIACRYPT 2003 , 2003 .

[31]  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.

[32]  Jeffrey Shallit,et al.  Algorithmic Number Theory , 1996, Lecture Notes in Computer Science.

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

[34]  Antoine Joux,et al.  The Weil and Tate Pairings as Building Blocks for Public Key Cryptosystems , 2002, ANTS.

[35]  Colin Boyd,et al.  Cryptography and Coding , 1995, Lecture Notes in Computer Science.