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2. Lookup Table-Based Design of Scalar Multiplication for Elliptic Curve Cryptography.

Author

Ning, Yan-Duan, Chen, Yan-Haw, Shih, Cheng-Sin, and Chu, Shao-I

Subjects
*MULTIPLICATION, *FINITE fields, *ELLIPTIC curve cryptography, *NUMBER theory, *GROUP theory, *COMPUTATIONAL complexity
Abstract

This paper is aimed at using a lookup table method to improve the scalar multiplication performance of elliptic curve cryptography. The lookup table must be divided into two polynomials and requires two iterations of point doubling operation, for which negation operations are needed. It is well known that an inversion operation requires a lot of multiplication time. The advantage of this paper is that we are able to reduce one inverse element calculation for this problem and also improve the basic operations of finite fields through segmentation methods. If the normal basis method is used in the design of the inverse element operation, it must be converted to the normal basis through the standard basis. However, the conversion process requires a lot of matrix operations. Though the anti-element operation has good speed performance, it also increases the computational complexity. Using number theory and grouping methods will greatly improve the performance of inverse element operations. With application of the two-time point doubling operation in the hardware implementation, the developed approach reduces the computing time by 48% as compared with the conventional approach. The computational time of the scalar multiplication using the presented method is further improved by 67% over the traditional algorithm with only an area increase of 12%. Finally, the proposed lookup table-based technique can be utilized for software and hardware implementation, as the developed arithmetic operations are simple and are consistent in their execution. [ABSTRACT FROM AUTHOR]

Published
2024
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3. Lookup Table-Based Design of Scalar Multiplication for Elliptic Curve Cryptography

Author

Yan-Duan Ning, Yan-Haw Chen, Cheng-Sin Shih, and Shao-I Chu

Subjects
elliptic curve cryptography, scalar multiplication, finite field, point doubling, Technology
Abstract

This paper is aimed at using a lookup table method to improve the scalar multiplication performance of elliptic curve cryptography. The lookup table must be divided into two polynomials and requires two iterations of point doubling operation, for which negation operations are needed. It is well known that an inversion operation requires a lot of multiplication time. The advantage of this paper is that we are able to reduce one inverse element calculation for this problem and also improve the basic operations of finite fields through segmentation methods. If the normal basis method is used in the design of the inverse element operation, it must be converted to the normal basis through the standard basis. However, the conversion process requires a lot of matrix operations. Though the anti-element operation has good speed performance, it also increases the computational complexity. Using number theory and grouping methods will greatly improve the performance of inverse element operations. With application of the two-time point doubling operation in the hardware implementation, the developed approach reduces the computing time by 48% as compared with the conventional approach. The computational time of the scalar multiplication using the presented method is further improved by 67% over the traditional algorithm with only an area increase of 12%. Finally, the proposed lookup table-based technique can be utilized for software and hardware implementation, as the developed arithmetic operations are simple and are consistent in their execution.

Published
2024
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4. Efficient hardware implementations of Lopez–Dahab projective co-ordinate based scalar multiplication of ECC.

Author

Basiri, M. Mohamed Asan

Subjects
FIELD programmable gate arrays, APPLICATION-specific integrated circuits, ELLIPTIC curve cryptography, SYSTEMS on a chip, PUBLIC key cryptography, MULTIPLICATION
Abstract

Elliptic curve cryptography (ECC) is most widely used asymmetric cryptography technique used in the modern engineering applications. This article proposes efficient hardware implementations scalar multiplication of Lopez–Dahab projective co-ordinate based ECC in the platforms of application specific integrated circuit (ASIC) and field programmable gate array logic (FPGA). The configurable G F (2 163) arithmetic unit is used to design the proposed scalar multiplication in ASIC platform with 45 nm CMOS technology. The scalar multiplication includes point addition and doubling. Since the G F (2 163) operations such as addition, multiplication, fused multiply addition (FMA), and multiplicative inverse required in the point addition and doubling are performed using the configurable G F (2 163) arithmetic unit, area and power dissipation of the proposed scalar multiplication in the ASIC platform is less than various existing designs. Similarly, both the Cortex-A9 cores of Zynq 7000 system on chip (SoC) are used to perform the two scalar multiplications in parallel, where the first core performs the point addition of the scalar multiplication while the second core performs the point doubling. Here, both the cores control separate co-processors to perform the point addition or doubling. Due to this dual core implementation in FPGA, the throughput of the proposed scalar multiplication in FPGA is greater than various existing designs. [ABSTRACT FROM AUTHOR]

Published
2023
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5. Low Cost Hardware Design of ECC Scalar Multiplication

Author

Inumarty, Hariveer, Mohamed Asan Basiri, M., Filipe, Joaquim, Editorial Board Member, Ghosh, Ashish, Editorial Board Member, Prates, Raquel Oliveira, Editorial Board Member, Zhou, Lizhu, Editorial Board Member, Shah, Ambika Prasad, editor, Dasgupta, Sudeb, editor, Darji, Anand, editor, and Tudu, Jaynarayan, editor

Published
2022
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7. The Techniques of Vedic Mathematics for ECC Over Weierstrass Elliptic Curve Y 2 = X 3 + Ax + B

Author

Kumar, Ankur, Gupta, Pratik, Kumar, Manoj, Angrisani, Leopoldo, Series Editor, Arteaga, Marco, Series Editor, Panigrahi, Bijaya Ketan, Series Editor, Chakraborty, Samarjit, Series Editor, Chen, Jiming, Series Editor, Chen, Shanben, Series Editor, Chen, Tan Kay, Series Editor, Dillmann, Rüdiger, Series Editor, Duan, Haibin, Series Editor, Ferrari, Gianluigi, Series Editor, Ferre, Manuel, Series Editor, Hirche, Sandra, Series Editor, Jabbari, Faryar, Series Editor, Jia, Limin, Series Editor, Kacprzyk, Janusz, Series Editor, Khamis, Alaa, Series Editor, Kroeger, Torsten, Series Editor, Liang, Qilian, Series Editor, Martín, Ferran, Series Editor, Ming, Tan Cher, Series Editor, Minker, Wolfgang, Series Editor, Misra, Pradeep, Series Editor, Möller, Sebastian, Series Editor, Mukhopadhyay, Subhas, Series Editor, Ning, Cun-Zheng, Series Editor, Nishida, Toyoaki, Series Editor, Pascucci, Federica, Series Editor, Qin, Yong, Series Editor, Seng, Gan Woon, Series Editor, Speidel, Joachim, Series Editor, Veiga, Germano, Series Editor, Wu, Haitao, Series Editor, Zhang, Junjie James, Series Editor, Hura, Gurdeep Singh, editor, Singh, Ashutosh Kumar, editor, and Siong Hoe, Lau, editor

Published
2021
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8. Elliptic Curve Cryptography: A Software Implementation

Author

Dhanda, Sumit Singh, Singh, Brahmjit, Jindal, Poonam, Bansal, Jagdish Chand, Series Editor, Deep, Kusum, Series Editor, Nagar, Atulya K., Series Editor, Singh, Brahmjit, editor, Coello Coello, Carlos A., editor, Jindal, Poonam, editor, and Verma, Pankaj, editor

Published
2021
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9. Impact of Computational Power on Cryptography

Author

Bhat, M. Iqbal, Giri, Kaiser J., Bansal, Jagdish Chand, Series Editor, Deep, Kusum, Series Editor, Nagar, Atulya K., Series Editor, Giri, Kaiser J., editor, Parah, Shabir Ahmad, editor, Bashir, Rumaan, editor, and Muhammad, Khan, editor

Published
2021
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10. The New Modified Methodology to Solve ECDLP Based on Brute Force Attack

Author

Somsuk, Kritsanapong, Sanemueang, Chalida, Kacprzyk, Janusz, Series Editor, Pal, Nikhil R., Advisory Editor, Bello Perez, Rafael, Advisory Editor, Corchado, Emilio S., Advisory Editor, Hagras, Hani, Advisory Editor, Kóczy, László T., Advisory Editor, Kreinovich, Vladik, Advisory Editor, Lin, Chin-Teng, Advisory Editor, Lu, Jie, Advisory Editor, Melin, Patricia, Advisory Editor, Nedjah, Nadia, Advisory Editor, Nguyen, Ngoc Thanh, Advisory Editor, Wang, Jun, Advisory Editor, Unger, Herwig, editor, Sodsee, Sunantha, editor, and Meesad, Phayung, editor

Published
2019
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11. Chosen base‐point side‐channel attack on Montgomery ladder with x ‐only coordinate: with application to secp256k1.

Author

Wei, Congming, Chen, Jiazhe, Wang, An, Wang, Beibei, Shi, Hongsong, and Wang, Xiaoyun

Abstract

This study revisits the side‐channel security of the elliptic curve cryptography (ECC) scalar multiplication implemented with Montgomery ladder. Focusing on a specific implementation that does not use the y ‐coordinate for point addition (ECADD) and point doubling (ECDBL), the authors show that Montgomery ladder on Weierstrass curves is vulnerable to a chosen base‐point attack. Unlike the normal implementation with y ‐coordinate, in the scenario of this study, the chosen base‐point strategy will not lead to operations with two same inputs during the ECADD and/or ECDBL. Instead, by choosing a suitable base‐point, one will find that there are operations that share a common operand; while it is not the case if the base‐point is not chosen correctly. This results in the recovery of the secret (fixed) scalar. They also experiment the methods of shared operand detection on a real‐world SoC, where a secp256k1 dedicated Montgomery ladder scalar multiplication with x ‐only coordinate is implemented, to show the efficiency of the scalar recovery attack. Naturally, the attack can be generalised to other Weierstrass curves when they contain special points. [ABSTRACT FROM AUTHOR]

Published
2020
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12. A Fast and Compact FPGA Implementation of Elliptic Curve Cryptography Using Lambda Coordinates

Author

Gövem, Burak, Järvinen, Kimmo, Aerts, Kris, Verbauwhede, Ingrid, Mentens, Nele, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Pointcheval, David, editor, Nitaj, Abderrahmane, editor, and Rachidi, Tajjeeddine, editor

Published
2016
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13. How to Use Koblitz Curves on Small Devices?

Author

Järvinen, Kimmo, Verbauwhede, Ingrid, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Joye, Marc, editor, and Moradi, Amir, editor

Published
2015
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14. Throughput/area optimised pipelined architecture for elliptic curve crypto processor.

Author

Imran, Malik, Rashid, Muhammad, Jafri, Atif Raza, and Kashif, Muhammad

Abstract

A pipelined architecture is proposed in this work to speed up the point multiplication in elliptic curve cryptography (ECC). This is achieved, at first; by pipelining the arithmetic unit to reduce the critical path delay. Second, by reducing the number of clock cycles (latency), which is achieved through careful scheduling of computations involved in point addition and point doubling. These two factors thus, help in reducing the time for one point multiplication computation. On the other hand, the small area overhead for this design gives a higher throughput/area ratio. Consequently, the proposed architecture is synthesised on different FPGAs to compare with the state‐of‐the‐art. The synthesis results over GF(2m) show that the proposed design can work up to a frequency of 369, 357 and 337 MHz when implemented for m = 163, 233 and 283 bit key lengths, respectively, on Virtex‐7 FPGA. The corresponding throughput/slice figures are 42.22, 12.37 and 9.45, which outperform existing implementations. [ABSTRACT FROM AUTHOR]

Published
2019
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15. Efficient Implementation of NIST-Compliant Elliptic Curve Cryptography for Sensor Nodes

Author

Liu, Zhe, Seo, Hwajeong, Großschädl, Johann, Kim, Howon, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Qing, Sihan, editor, Zhou, Jianying, editor, and Liu, Dongmei, editor

Published
2013
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17. Efficient Java Implementation of Elliptic Curve Cryptography for J2ME-Enabled Mobile Devices

Author

Großschädl, Johann, Page, Dan, Tillich, Stefan, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Askoxylakis, Ioannis, editor, Pöhls, Henrich C., editor, and Posegga, Joachim, editor

Published
2012
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20. Evaluation Metrics of Physical Non-invasive Security

Author

Li, Huiyun, Wu, Keke, Yu, Fengqi, Yuan, Hai, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Samarati, Pierangela, editor, Tunstall, Michael, editor, Posegga, Joachim, editor, Markantonakis, Konstantinos, editor, and Sauveron, Damien, editor

Published
2010
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22. New Elliptic Curve Multi-scalar Multiplication Algorithm for a Pair of Integers to Resist SPA

Author

Liu, Duo, Tan, Zhiyong, Dai, Yiqi, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Yung, Moti, editor, Liu, Peng, editor, and Lin, Dongdai, editor

Published
2009
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23. A Hardware Analysis of Twisted Edwards Curves for an Elliptic Curve Cryptosystem

Author

Baldwin, Brian, Moloney, Richard, Byrne, Andrew, McGuire, Gary, Marnane, William P., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Becker, Jürgen, editor, Woods, Roger, editor, Athanas, Peter, editor, and Morgan, Fearghal, editor

Published
2009
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24. High Speed Compact Elliptic Curve Cryptoprocessor for FPGA Platforms

Author

Rebeiro, Chester, Mukhopadhyay, Debdeep, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Chowdhury, Dipanwita Roy, editor, Rijmen, Vincent, editor, and Das, Abhijit, editor

Published
2008
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27. Efficient Window-Based Scalar Multiplication on Elliptic Curves Using Double-Base Number System

Author

Barua, Rana, Pandey, Sumit Kumar, Pankaj, Ravi, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Srinathan, K., editor, Rangan, C. Pandu, editor, and Yung, Moti, editor

Published
2007
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28. Efficient Implementation of Public Key Cryptosystems on Mote Sensors (Short Paper)

Author

Wang, Haodong, Li, Qun, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Ning, Peng, editor, Qing, Sihan, editor, and Li, Ninghui, editor

Published
2006
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29. SPA-Resistant Simultaneous Scalar Multiplication

Author

Lee, Mun-Kyu, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Gervasi, Osvaldo, editor, Gavrilova, Marina L., editor, Kumar, Vipin, editor, Laganà, Antonio, editor, Lee, Heow Pueh, editor, Mun, Youngsong, editor, Taniar, David, editor, and Tan, Chih Jeng Kenneth, editor

Published
2005
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30. Two New Fast Methods for Simultaneous Scalar Multiplication in Elliptic Curve Cryptosystems

Author

Shi, Runhua, Cheng, Jiaxing, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Lu, Xicheng, editor, and Zhao, Wei, editor

Published
2005
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32. On Distributed Computing on Elliptic Curves

Author

Meskanen, Tommi, Renvall, Ari, Steinby, Paula, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Karhumäki, Juhani, editor, Maurer, Hermann, editor, Păun, Gheorghe, editor, and Rozenberg, Grzegorz, editor

Published
2004
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33. Enhanced Montgomery Multiplication

Author

Gueron, Shay, Goos, Gerhard, editor, Hartmanis, Juris, editor, van Leeuwen, Jan, editor, Kaliski, Burton S., editor, Koç, çetin K., editor, and Paar, Christof, editor

Published
2003
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40. An exploration of affine group laws for elliptic curves

Author

Hisil Huseyin, Wong Kenneth Koon-Ho, Carter Gary, and Dawson Ed

Subjects
elliptic curve, group law, point addition, point doubling, projective coordinates, rational maps, birational equivalence, riemann–roch theorem, rational simplification, scalar multiplication, elliptic curve cryptography, Mathematics, QA1-939
Abstract

Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.

Published
2011
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42. Um sistema criptografico para curvas elipticas sobre GF(2m) implementado em circuitos programaveis

Author

Dias, Mauricio Araujo, Oliveira, José Raimundo de, 1950, Dahab, Ricardo, Damiani, Furio, Tatsch, Peter Jürgen, Marranghello, Norian, Universidade Estadual de Campinas. Faculdade de Engenharia Elétrica e de Computação, Programa de Pós-Graduação em Engenharia Elétrica, and UNIVERSIDADE ESTADUAL DE CAMPINAS

Subjects
Hardware, Curvas elípticas, Elliptic curve, Cryptography, Point doubling, Criptografia, VHDL (Linguagem descritiva de hardware), Circuitos integrados, Circuitos digitais, Point addition, Combinatorial circuit, FPGA
Abstract

Orientador: Jose Raimundo de Oliveira Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação Resumo: Este trabalho propõe um sistema criptográfico para Criptografia baseada em Curvas Elípticas (ECC). ECC é usada alternativamente a outros sistemas criptográficos, como o algoritmo RSA (Rivest-Shamir-Adleman), por oferecer a menor chave e a maior segurança por bit. Ele realiza multiplicação de pontos (Q = kP) para curvas elípticas sobre corpos finitos binários. Trata-se de um criptosistema programável e configurável. Graças às propriedades do circuito programável (FPGA) é possível encontrar soluções otimizadas para diferentes curvas elípticas, corpos finitos e algoritmos. A característica principal deste criptosistema é o uso de um circuito combinacional para calcular duplicações e adições de pontos, por meio da aritmética sobre corpos finitos. Os resultados deste trabalho mostram que um programa de troca de chaves fica aproximadamente 20.483 vezes mais rápido com a ajuda do nosso sistema criptográfico. Para desenvolver este projeto, nós consideramos que o alto desempenho tem prioridade sobre a área ocupada pelos seus circuitos. Assim, nós recomendamos o uso deste circuito para os casos em que não sejam impostas restrições de área, mas seja exigido alto desempenho do sistema Abstract: This work proposes a cryptosystem for Elliptic Curve Cryptography (ECC). ECC has been used as an alternative to other public-key cryptosystems such as the RSA (Rivest-Shamir-Adleman algorithm) by offering the smallest key size and the highest strength per bit. The cryptosystem performs point multiplication (Q = kP) for elliptic curves over binary polynomial fields (GF(2m)). This is a programmable and scalable cryptosystem. It uses the abilities of reconfigurable hardware (FPGA) to make possible optimized circuitry solutions for different elliptic curves, finite fields and algorithms. The main feature of this cryptosystem is the use of a combinatorial circuit to calculate point doublings and point additions, through finite field arithmetic. The results of this work show that the execution of a key-exchange program is, approximately, 20,483 times faster with the help of our cryptosystem. To develop this project we considered that high-performance has priority over area occupied by its circuit. Thus, we recommend the use of this circuit in the cases for which no area constraints are imposed but high performance systems are required. Doutorado Engenharia de Computação Doutor em Engenharia Elétrica

Published
2021

43. Arithmetic Operations on Elliptic Curves Defined over Un-conventional Element Finite Fields.

Author

Daikpor, Michael Naseimo and Adegbenro, Oluwole

Abstract

This paper presents arithmetic operations on elliptic curves defined over an un-conventional number system character digit element finite field. A high speed elliptic curve scalar multiplication algorithm in this special field domain is also presented. The heuristically-obtained results show the efficient nature of the proposed elliptic curve arithmetic operations. [ABSTRACT FROM PUBLISHER]

Published
2012
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44. Hardware Designs and Architectures for Projective Montgomery ECC Over GF (p) Benefiting from Mapping Elliptic Curve Computations to Different Degrees of Parallelism.

Author

Al-Khatib, Mohammad, Jaafar, Azmi, Zukarnain, Zuriati, and Rushdan, Mohammad

Subjects
ELLIPTIC curve cryptography, COMPUTER input-output equipment, COMPUTER architecture, PERFORMANCE evaluation, COMPUTATIONAL complexity, INFORMATION resources
Abstract

This paper proposes several hardware design schemas for Montgomery ECC over GF (p). Three known projective coordinates were used to apply ECC computations, in order to eliminate the long time inversion operation, which are Homogeneous, López-Dahab, and Jacobean coordinates. Unlike the usual serial design implementations, our proposed designs utilize the inherent parallelism in ECC computations by using parallel hardware design, in which the ECC computations are performed in parallel to improve the performance of ECC. In addition to the performance, other factors affecting the design of ECC such as area, resources-consuming, system utilization, AT2, and AT factors were studied and enhanced. This was achieved via proposing different design schemas for Montgomery ECC by varying the degree of parallelism for ECC computations. Our proposed designs provide attractive trade-off between mentioned factors, which helps in selecting the most efficient design choices for different security applications based on the requirements and availability of resources for particular application. The performance level of homogeneous coordinates when applied using 4-PM design was five multiplication cycles, which overcomes the other coordinates systems, as well as the known serial design. The 2-PM design for homogeneous projection, on the other hand, improves the system utilization considerably with less area and resources in comparison to the designs with higher degree of parallelism. [ABSTRACT FROM AUTHOR]

Published
2011

45. Implementation of Text Encryption using Elliptic Curve Cryptography

Author

Khumanthem Manglem Singh and Laiphrakpam Dolendro Singh

Subjects
Theoretical computer science, Computer science, Point doubling, Cryptography, Encryption, Hardware_ARITHMETICANDLOGICSTRUCTURES, Elliptic curve cryptography, Arithmetic, Computer Science::Cryptography and Security, General Environmental Science, Key size, Neural cryptography, Post-quantum cryptography, business.industry, Elliptic curve Diffie–Hellman, Elliptic Curve Digital Signature Algorithm, Schoof–Elkies–Atkin algorithm, Elliptic curve, Elliptic curve point multiplication, Jacobian curve, Curve25519, Counting points on elliptic curves, General Earth and Planetary Sciences, Hyperelliptic curve cryptography, Lattice-based cryptography, business, Point addition, Tripling-oriented Doche–Icart–Kohel curve
Abstract

Elliptic Curve Cryptography has been a recent research area in the field of Cryptography. It provides higher level of security with lesser key size compared to other Cryptographic techniques. A new technique has been proposed in this paper where the classic technique of mapping the characters to affine points in the elliptic curve has been removed. The corresponding ASCII values of the plain text are paired up. The paired values serve as input for the Elliptic curve cryptography. This new technique avoids the costly operation of mapping and the need to share the common lookup table between the sender and the receiver. The algorithm is designed in such a way that it can be used to encrypt or decrypt any type of script with defined ASCII values.

Published
2015
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46. An exploration of affine group laws for elliptic curves

Author

Kenneth Koon-Ho Wong, Ed Dawson, Huseyin Hisil, and Gary Carter

Subjects
Edwards curve, rational simplification, point doubling, birational equivalence, point addition, Affine group, QA1-939, Elliptic curve cryptography, Mathematics, Applied Mathematics, Supersingular elliptic curve, Computer Science Applications, group law, Computational Mathematics, Elliptic curve, rational maps, Affine geometry of curves, projective coordinates, Law, elliptic curve cryptography, scalar multiplication, riemann–roch theorem, Schoof's algorithm, Tripling-oriented Doche–Icart–Kohel curve, elliptic curve
Abstract

Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.

Published
2011

47. An exploration of affine group laws for elliptic curves

Author

Hisil, Huseyin, Wong, Kenneth, Carter, Gary, Dawson, Ed, Hisil, Huseyin, Wong, Kenneth, Carter, Gary, and Dawson, Ed

Abstract

Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.

Published
2011

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