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Formulas for cube roots in F3m

Omran Ahmadi, Darrel Hankerson, Alfred Menezes
2007 Discrete Applied Mathematics  
We determine the number of nonzero coefficients (called the Hamming weight) in the polynomial representation of  ...  Thus it is worthwhile to have fast algorithms for computing cube roots in F 3 m .  ...  The fastest algorithms known for pairing computations on these supersingular elliptic curves require the evaluation of cube roots in F 3 m [5, 3] .  ... 
doi:10.1016/j.dam.2006.06.004 fatcat:ovauycrebnfcvdt7dxhabknz4a

Low Complexity Cubing and Cube Root Computation over $\F_{3^m}$ in Polynomial Basis

O Ahmadi, F R Henríquez
2010 IEEE transactions on computers  
We present low complexity formulae for the computation of cubing and cube root over F 3 m constructed using special classes of irreducible trinomials, tetranomials and pentanomials.  ...  We show that for all those special classes of polynomials, field cubing and field cube root operation have the same computational complexity when implemented in hardware or software platforms.  ...  ACKNOWLEDGMENT The authors would like to thank Darrel Hankerson, Jean-Luc Beuchat and the anonymous referees for their valuable comments that greatly helped to improve the presentation of this paper.  ... 
doi:10.1109/tc.2009.183 fatcat:qp3kh6qlwfg3hcwdp3n4dadhhu

Low Complexity Cubing and Cube Root Computation over $\F_{3^m}$ in Polynomial Basis

Omran Ahmadi, Francisco Rodriguez Henriquez
2011 IEEE transactions on computers  
We present low complexity formulae for the computation of cubing and cube root over F 3 m constructed using special classes of irreducible trinomials, tetranomials and pentanomials.  ...  We show that for all those special classes of polynomials, field cubing and field cube root operation have the same computational complexity when implemented in hardware or software platforms.  ...  ACKNOWLEDGMENT The authors would like to thank Darrel Hankerson, Jean-Luc Beuchat and the anonymous referees for their valuable comments that greatly helped to improve the presentation of this paper.  ... 
doi:10.1109/tc.2011.37 fatcat:v7n2twa3cbdynkvorrvn45btya

Efficient Implementation of Tate Pairing on a Mobile Phone Using Java [chapter]

Yuto Kawahara, Tsuyoshi Takagi, Eiji Okamoto
2007 Lecture Notes in Computer Science  
O u r optimized implementation for m = 9 7 a c hieved about 0.5 seconds for computing Tate pairing over FOMA SH901iS, NTT DoCoMo.  ...  We present an e cient implementation for computing Tate pairing on a mobile phone using Java.  ...  One cube root can be computed with the speed of at most two m ultiplications in F 3 m .  ... 
doi:10.1007/978-3-540-74377-4_42 fatcat:s2ehhrklkjfvjh3mwdo3f7ugce

Efficient Implementation of Tate Pairing on a Mobile Phone using Java

Yuto Kawahara, Tsuyoshi Takagi, Eiji Okamoto
2006 2006 International Conference on Computational Intelligence and Security  
O u r optimized implementation for m = 9 7 a c hieved about 0.5 seconds for computing Tate pairing over FOMA SH901iS, NTT DoCoMo.  ...  We present an e cient implementation for computing Tate pairing on a mobile phone using Java.  ...  One cube root can be computed with the speed of at most two m ultiplications in F 3 m .  ... 
doi:10.1109/iccias.2006.295256 fatcat:7bbzi4gktzakriab3l56wdgkta

Parallel Hardware Architectures for the Cryptographic Tate Pairing

G. Bertoni, L. Breveglieri, P. Fragneto, G. Pelosi
2006 Third International Conference on Information Technology: New Generations (ITNG'06)  
,co-processors, for the Tate pairing, in the case of the Duursma-Lee algorithm in characteristic 3.  ...  by means of different figures of merit such as computation time, circuit area and combinations thereof.Comparisons with the (few) existing proposals are carried out, showing that a large space exists for  ...  [4] m F3A + 4m MX 1 (m + 1) D F3A + Multiplier [4] ((m + 2)D − 2) F3M + m/D 8m MX + FF (6m + 2D − 2) Cube Power [4] 2m F3A + 2m MX 1 Cube Root [2] (m − 1) F3A + 2m MX 1 Table  ... 
doi:10.1109/itng.2006.107 dblp:conf/itng/BertoniBFP06 fatcat:5eti4hl4mjbtzhlh5uweqox7ze

Faster and Lower Memory Scalar Multiplication on Supersingular Curves in Characteristic Three [chapter]

Roberto Avanzi, Clemens Heuberger
2011 Lecture Notes in Computer Science  
There are of course several choices for these generators: we chose generators associated to endomorphisms for which we could find efficient explicit formulae in a suitable coordinate system.  ...  We describe new algorithms for performing scalar multiplication on supersingular elliptic curves in characteristic three. These curves can be used in pairing-based cryptography.  ...  To obtain an explicit formula consider expressions (12) , first replace x, y with X/Z and Y/Z in the two formulae for x 3 and in the formula for y 3 .  ... 
doi:10.1007/978-3-642-19379-8_7 fatcat:5tzpfdy7bjfqpmwkfnlmektfma

Hardware Accelerator for the Tate Pairing in Characteristic Three Based on Karatsuba-Ofman Multipliers [chapter]

Jean-Luc Beuchat, Jérémie Detrey, Nicolas Estibals, Eiji Okamoto, Francisco Rodríguez-Henríquez
2009 Lecture Notes in Computer Science  
They are included in a VHDL code generator allowing for the exploration of a wide range of operators. Then, we outline the architecture of a coprocessor for the Tate pairing over F3m .  ...  This paper is devoted to the design of fast parallel accelerators for the cryptographic Tate pairing in characteristic three over supersingular elliptic curves.  ...  Acknowledgments The authors would like to thank Nidia Cortez-Duarte and the anonymous referees for their valuable comments.  ... 
doi:10.1007/978-3-642-04138-9_17 fatcat:wu7f4otgcfaarg6rrozqd2gv3a

Fast Architectures for the \eta_T Pairing over Small-Characteristic Supersingular Elliptic Curves

Jean-Luc Beuchat, Jeremie Detrey, Nicolas Estibals, Eiji Okamoto, Francisco Rodriguez Henriquez
2011 IEEE transactions on computers  
over F2m and F3m .  ...  A final exponentiation is still required to obtain a unique value, which is desirable in most cryptographic protocols.  ...  ACKNOWLEDGMENTS The authors would like to thank Nidia Cortez-Duarte along with the anonymous referees for their valuable comments.  ... 
doi:10.1109/tc.2010.163 fatcat:nrduo7prxjgeti2jzk3ygzkf3q

Fast Architectures for the eta T Pairing over Small-Characteristic Supersingular Elliptic Curves [article]

Jean-Luc Beuchat, Jérémie Detrey, Nicolas Estibals, Eiji Okamoto, Francisco Rodríguez-Henríquez
2009 IACR Cryptology ePrint Archive  
over F2m and F3m .  ...  A final exponentiation is still required to obtain a unique value, which is desirable in most cryptographic protocols.  ...  ACKNOWLEDGMENTS The authors would like to thank Nidia Cortez-Duarte for her valuable comments. This work was supported by the Japan-France Integrated Action Program (Ayame Junior/Sakura).  ... 
dblp:journals/iacr/BeuchatDEOR09a fatcat:umxkosfjoberrikckapqzju67y

Efficient Algorithms for Pairing-Based Cryptosystems [chapter]

Paulo S. L. M. Barreto, Hae Y. Kim, Ben Lynn, Michael Scott
2002 Lecture Notes in Computer Science  
We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm , the latter technique being also useful in contexts other than that of pairing-based  ...  In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger  ...  Indeed, since the cubing operation is linear in characteristic 3, given P = (x, y) one computes 3P = (x 3 , y 3 ) with the formulas: x 3 = (x 3 ) 3 − b y 3 = −(y 3 ) 3 These formulas are derived from the  ... 
doi:10.1007/3-540-45708-9_23 fatcat:j5lpacqpmbfrvmb7vhexk3oox4

Multi-core Implementation of the Tate Pairing over Supersingular Elliptic Curves [chapter]

Jean-Luc Beuchat, Emmanuel López-Trejo, Luis Martínez-Ramos, Shigeo Mitsunari, Francisco Rodríguez-Henríquez
2009 Lecture Notes in Computer Science  
This paper describes the design of a fast multi-core library for the cryptographic Tate pairing over supersingular elliptic curves.  ...  We also try to answer one important design question that surges: how many cores should be utilized for a given application?  ...  Acknowledgments The authors would like to thank Jérémie Detrey for his valuable comments.  ... 
doi:10.1007/978-3-642-10433-6_28 fatcat:2ujjnqnh5jhvximgqwb5vh4m7m

THE TOPOLOGICAL q-EXPANSION PRINCIPLE

Gerd Laures
1999 Topology  
We show how the topological q-expansion principle can be used to equip elliptic cohomology with orientations which obey various Riemann-Roch formulas.  ...  in Katz's [23] ring of divided congruences.  ...  Let R be a ring in which N is invertible and which contains a primitive Nth root of unity , .  ... 
doi:10.1016/s0040-9383(98)00019-6 fatcat:g7eyxwojhvc45goi7hve2cmxf4

A Coprocessor for the Final Exponentiation of the η T Pairing in Characteristic Three [chapter]

Jean-Luc Beuchat, Nicolas Brisebarre, Masaaki Shirase, Tsuyoshi Takagi, Eiji Okamoto
Lecture Notes in Computer Science  
In this paper, we propose a novel arithmetic operator implementing addition, cubing, and multiplication over F 3 97 and show that a coprocessor based on a single such operator meets this timing constraint  ...  Beuchat et al. proposed for instance a coprocessor which computes the characteristic three ηT pairing, from which the Tate pairing can easily be derived, in 33 µs on a Cyclone II FPGA.  ...  An Operator for Multiplication, Addition, and Cubing over F 3 97 Consider again the closed formula for cubing over F 3 [x]/(x 97 + x 12 + 2) (Equation (1)).  ... 
doi:10.1007/978-3-540-73074-3_4 fatcat:2zrx2ze6vbfunemcvonvsui5fu

A Treatise on the Theory of Invariants

G. B. M.
1916 Nature  
It should be remarked that the formula (90) for the general rth transvectant of Chapter III, Section 2 may be employed to great advantage in representing concomitants in terms of the roots.  ...  These formulas hold true for zero exponents. Employing them as recursion formulas we have immediately for Lemma 2.  ... 
doi:10.1038/098187a0 fatcat:ljzqthezqvg3vgxto7lf3skaom
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