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Formulas for cube roots in F3m

2007
*
Discrete Applied Mathematics
*

We determine the number of nonzero coefficients (called the Hamming weight)

doi:10.1016/j.dam.2006.06.004
fatcat:ovauycrebnfcvdt7dxhabknz4a
*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] . ...##
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Low Complexity Cubing and Cube Root Computation over $\F_{3^m}$ in Polynomial Basis

2010
*
IEEE transactions on computers
*

We present low complexity

doi:10.1109/tc.2009.183
fatcat:qp3kh6qlwfg3hcwdp3n4dadhhu
*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. ...##
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Low Complexity Cubing and Cube Root Computation over $\F_{3^m}$ in Polynomial Basis

2011
*
IEEE transactions on computers
*

We present low complexity

doi:10.1109/tc.2011.37
fatcat:v7n2twa3cbdynkvorrvn45btya
*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. ...##
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Efficient Implementation of Tate Pairing on a Mobile Phone Using Java
[chapter]

2007
*
Lecture Notes in Computer Science
*

O u r optimized implementation

doi:10.1007/978-3-540-74377-4_42
fatcat:s2ehhrklkjfvjh3mwdo3f7ugce
*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 . ...##
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Efficient Implementation of Tate Pairing on a Mobile Phone using Java

2006
*
2006 International Conference on Computational Intelligence and Security
*

O u r optimized implementation

doi:10.1109/iccias.2006.295256
fatcat:7bbzi4gktzakriab3l56wdgkta
*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 . ...##
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Parallel Hardware Architectures for the Cryptographic Tate Pairing

2006
*
Third International Conference on Information Technology: New Generations (ITNG'06)
*

,co-processors,

doi:10.1109/itng.2006.107
dblp:conf/itng/BertoniBFP06
fatcat:5eti4hl4mjbtzhlh5uweqox7ze
*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 ...##
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Faster and Lower Memory Scalar Multiplication on Supersingular Curves in Characteristic Three
[chapter]

2011
*
Lecture Notes in Computer Science
*

There are of course several choices

doi:10.1007/978-3-642-19379-8_7
fatcat:5tzpfdy7bjfqpmwkfnlmektfma
*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 . ...##
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Hardware Accelerator for the Tate Pairing in Characteristic Three Based on Karatsuba-Ofman Multipliers
[chapter]

2009
*
Lecture Notes in Computer Science
*

They are included

doi:10.1007/978-3-642-04138-9_17
fatcat:wu7f4otgcfaarg6rrozqd2gv3a
*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. ...##
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Fast Architectures for the \eta_T Pairing over Small-Characteristic Supersingular Elliptic Curves

2011
*
IEEE transactions on computers
*

over F2m and

doi:10.1109/tc.2010.163
fatcat:nrduo7prxjgeti2jzk3ygzkf3q
*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. ...##
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Fast Architectures for the eta T Pairing over Small-Characteristic Supersingular Elliptic Curves
[article]

2009
*
IACR Cryptology ePrint Archive
*

over F2m and

dblp:journals/iacr/BeuchatDEOR09a
fatcat:umxkosfjoberrikckapqzju67y
*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). ...##
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Efficient Algorithms for Pairing-Based Cryptosystems
[chapter]

2002
*
Lecture Notes in Computer Science
*

We also propose faster algorithms

doi:10.1007/3-540-45708-9_23
fatcat:j5lpacqpmbfrvmb7vhexk3oox4
*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 ...##
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Multi-core Implementation of the Tate Pairing over Supersingular Elliptic Curves
[chapter]

2009
*
Lecture Notes in Computer Science
*

This paper describes the design of a fast multi-core library

doi:10.1007/978-3-642-10433-6_28
fatcat:2ujjnqnh5jhvximgqwb5vh4m7m
*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. ...##
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THE TOPOLOGICAL q-EXPANSION PRINCIPLE

1999
*
Topology
*

We show how the topological q-expansion principle can be used to equip elliptic cohomology with orientations which obey various Riemann-Roch

doi:10.1016/s0040-9383(98)00019-6
fatcat:g7eyxwojhvc45goi7hve2cmxf4
*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 , . ...##
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A Coprocessor for the Final Exponentiation of the η T Pairing in Characteristic Three
[chapter]

*
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)). ...

##
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A Treatise on the Theory of Invariants

1916
*
Nature
*

It should be remarked that the

doi:10.1038/098187a0
fatcat:ljzqthezqvg3vgxto7lf3skaom
*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. ...
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