On the relationship between squared pairings and plain pairings

In this paper, we investigate the relationship between the squared Weil/Tate pairing and the plain Weil/Tate pairing. Along these lines, we first show that the squared pairing for an arbitrary chosen point can be transformed into the plain pairing for a trace zero point which has a special form to compute them more efficiently. Then the optimizations made for computing squared pairings are combined with the computation of pairings on these trace zero points, to achieve even better performance for the computation of the 4th powered Weil pairing.

[1]  Paulo S. L. M. Barreto,et al.  Generating More MNT Elliptic Curves , 2006, Des. Codes Cryptogr..

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

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

[4]  Paulo S. L. M. Barreto,et al.  On the Selection of Pairing-Friendly Groups , 2003, Selected Areas in Cryptography.

[5]  Michael Scott.  Scaling security in pairing-based protocols , 2005, IACR Cryptol. ePrint Arch..

[6]  Tsuyoshi Takagi,et al.  Efficient Computations of the Tate Pairingfor the Large MOV Degrees , 2002, ICISC.

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

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

[9]  Steven D. Galbraith,et al.  Implementing the Tate Pairing , 2002, ANTS.

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

[11]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

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

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

[14]  Frederik Vercauteren,et al.  A comparison of MNT curves and supersingular curves , 2006, Applicable Algebra in Engineering, Communication and Computing.

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

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

[17]  Paulo S. L. M. Barreto,et al.  Efficient Implementation of Pairing-Based Cryptosystems , 2004, Journal of Cryptology.

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

[19]  Colm O'hEigeartaigh.  Speeding Up Pairing Computation , 2005, IACR Cryptol. ePrint Arch..