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Formulas for cube roots in F3m
2007
Discrete Applied Mathematics
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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
2010
IEEE transactions on computers
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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
2011
IEEE transactions on computers
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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]
2007
Lecture Notes in Computer Science
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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
2006
2006 International Conference on Computational Intelligence and Security
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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
2006
Third International Conference on Information Technology: New Generations (ITNG'06)
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,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]
2011
Lecture Notes in Computer Science
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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]
2009
Lecture Notes in Computer Science
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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
2011
IEEE transactions on computers
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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]
2009
IACR Cryptology ePrint Archive
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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]
2002
Lecture Notes in Computer Science
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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]
2009
Lecture Notes in Computer Science
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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
1999
Topology
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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]
Lecture Notes in Computer Science
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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
1916
Nature
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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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