Design and Analysis of Pairing Based Cryptographic Hardware for Prime Fields

This work deals with the design and implementation of pairing based cryptographic hardware and its security analysis against side-channel and fault attacks.

[1]  Francisco Rodríguez-Henríquez,et al.  High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves , 2010, Pairing.

[2]  Thanos Stouraitis,et al.  An RNS Implementation of an $F_{p}$ Elliptic Curve Point Multiplier , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[3]  Frédéric Valette,et al.  The Doubling Attack - Why Upwards Is Better than Downwards , 2003, CHES.

[4]  Erkay Savas,et al.  A Scalable and Unified Multiplier Architecture for Finite Fields GF(p) and GF(2m) , 2000, CHES.

[5]  Sorin A. Huss,et al.  Parallel Memory Architecture for Elliptic Curve Cryptography over GF(p) Aimed at Efficient FPGA Implementation. , 2008, SIPS 2008.

[6]  Christof Paar,et al.  A Scalable GF(p) Elliptic Curve Processor Architecture for Programmable Hardware , 2001, CHES.

[7]  Henk L. Muller,et al.  Random Register Renaming to Foil DPA , 2001, CHES.

[8]  Sanjit Chatterjee,et al.  Efficient Computation of Tate Pairing in Projective Coordinate over General Characteristic Fields , 2004, ICISC.

[9]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[10]  Hyang-Sook Lee,et al.  Efficient and Generalized Pairing Computation on Abelian Varieties , 2009, IEEE Transactions on Information Theory.

[11]  Ratna Dutta,et al.  Pairing-Based Cryptographic Protocols : A Survey , 2004, IACR Cryptol. ePrint Arch..

[12]  Marc Joye,et al.  The Jacobi Model of an Elliptic Curve and Side-Channel Analysis , 2003, AAECC.

[13]  Máire O'Neill,et al.  FPGA Montgomery multiplier architectures - a comparison , 2004, 12th Annual IEEE Symposium on Field-Programmable Custom Computing Machines.

[14]  Kouichi Itoh,et al.  A Practical Countermeasure against Address-Bit Differential Power Analysis , 2003, CHES.

[15]  Blakely.  A Computer Algorithm for Calculating the Product AB Modulo M , 1983, IEEE Transactions on Computers.

[16]  Michael Scott,et al.  Constructing Tower Extensions for the implementation of Pairing-Based Cryptography , 2009, IACR Cryptol. ePrint Arch..

[17]  Kiamal Z. Pekmestzi,et al.  FPGA-based Design of a Large Moduli Multiplier for Public-Key Cryptographic Systems , 2006, 2006 International Conference on Computer Design.

[18]  Neil Burgess,et al.  Unified radix-4 multiplier for GF(p) and GF(2^n) , 2003, Proceedings IEEE International Conference on Application-Specific Systems, Architectures, and Processors. ASAP 2003.

[19]  Russell Tessier,et al.  FPGA Architecture , 2008 .

[20]  Naofumi Takagi,et al.  A hardware algorithm for modular multiplication/division , 2005, IEEE Transactions on Computers.

[21]  N. Koblitz.  Elliptic curve cryptosystems , 1987 .

[22]  Frederik Vercauteren,et al.  Efficient Hardware Implementation of Fp-Arithmetic for Pairing-Friendly Curves , 2012, IEEE Transactions on Computers.

[23]  Dipanwita Roy Chowdhury,et al.  Single Chip Encryptor/Decryptor Core Implementation of AES Algorithm , 2008, 21st International Conference on VLSI Design (VLSID 2008).

[24]  Yehuda Lindell,et al.  Introduction to Modern Cryptography (Chapman & Hall/Crc Cryptography and Network Security Series) , 2007 .

[25]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

[26]  Tim Kerins,et al.  An FPGA implementation of a GF(p) ALU for encryption processors , 2004, Microprocess. Microsystems.

[27]  Junko Takahashi,et al.  Improved Differential Fault Analysis on CLEFIA , 2008, 2008 5th Workshop on Fault Diagnosis and Tolerance in Cryptography.

[28]  Mohammed Benaissa,et al.  Design of flexible GF(2/sup m/) elliptic curve cryptography processors , 2006, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

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

[30]  Antoine Joux,et al.  A One Round Protocol for Tripartite Diffie–Hellman , 2000, Journal of Cryptology.

[31]  Ayman M. Bahaa Eldin,et al.  An Efficient Architecture for Interleaved Modular Multiplication , 2009 .

[32]  Thomas S. Messerges,et al.  Using Second-Order Power Analysis to Attack DPA Resistant Software , 2000, CHES.

[33]  William M. Springer.  Review of Cryptography: theory and practice, second edition by Douglas R. Stinson. CRC Press. , 2003, SIGA.

[34]  Tsuyoshi Takagi,et al.  Power Analysis Attacks and Countermeasures on ηT Pairing over Binary Fields , 2008 .

[35]  Jian Wang,et al.  High Performance Architecture for Elliptic Curve Scalar Multiplication over GF(2m) , 2008, IACR Cryptol. ePrint Arch..

[36]  Paulo S. L. M. Barreto,et al.  On Compressible Pairings and Their Computation , 2008, AFRICACRYPT.

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

[38]  David Naccache,et al.  The Sorcerer's Apprentice Guide to Fault Attacks , 2006, Proceedings of the IEEE.

[39]  Avishek Saha,et al.  Speed-area optimized FPGA implementation for Full Search Block Matching , 2007, 2007 25th International Conference on Computer Design.

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

[41]  Victor S. Miller,et al.  Use of Elliptic Curves in Cryptography , 1985, CRYPTO.

[42]  Frederik Vercauteren,et al.  Optimal Pairings , 2010, IEEE Transactions on Information Theory.

[43]  Michael Scott,et al.  A Taxonomy of Pairing-Friendly Elliptic Curves , 2010, Journal of Cryptology.

[44]  Debdeep Mukhopadhyay,et al.  Preventing the Side-Channel Leakage of Masked AES S-Box , 2007, 15th International Conference on Advanced Computing and Communications (ADCOM 2007).

[45]  Christof Paar,et al.  Area efficient GF(p) architectures for GF(p/sup m/) multipliers , 2002, The 2002 45th Midwest Symposium on Circuits and Systems, 2002. MWSCAS-2002..

[46]  François-Xavier Standaert,et al.  A Tutorial on Physical Security and Side-Channel Attacks , 2004, FOSAD.

[47]  Ricardo Dahab,et al.  Multiplication and Squaring on Pairing-Friendly Fields , 2006, IACR Cryptol. ePrint Arch..

[48]  Palash Sarkar,et al.  Pairing Computation on Twisted Edwards Form Elliptic Curves , 2008, Pairing.

[49]  Alessandro Barenghi,et al.  A FPGA Coprocessor for the Cryptographic Tate Pairing over Fp , 2008, Fifth International Conference on Information Technology: New Generations (itng 2008).

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

[51]  Koji Nakano,et al.  Redundant Radix-2r Number System for Accelerating Arithmetic Operations on the FPGAs , 2008, 2008 Ninth International Conference on Parallel and Distributed Computing, Applications and Technologies.

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

[53]  Manfred Josef Aigner,et al.  Randomized Addition-Subtraction Chains as a Countermeasure against Power Attacks , 2001, CHES.

[54]  Atsuko Miyaji,et al.  Efficient Elliptic Curve Exponentiation Using Mixed Coordinates , 1998, ASIACRYPT.

[55]  Joos Vandewalle,et al.  Hardware implementation of an elliptic curve processor over GF(p) , 2003, Proceedings IEEE International Conference on Application-Specific Systems, Architectures, and Processors. ASAP 2003.

[56]  Kouichi Sakurai,et al.  Power Analysis Breaks Elliptic Curve Cryptosystems even Secure against the Timing Attack , 2000, INDOCRYPT.

[57]  Soonhak Kwon.  Efficient Tate Pairing Computation for Supersingular Elliptic Curves over Binary Fields , 2004, IACR Cryptol. ePrint Arch..

[58]  Pradeep Kumar Mishra,et al.  Pipelined Computation of Scalar Multiplication in Elliptic Curve Cryptosystems , 2004, CHES.

[59]  JaeCheol Ha,et al.  Relative Doubling Attack Against Montgomery Ladder , 2005, ICISC.

[60]  Ingrid Verbauwhede,et al.  Extended abstract: Unified digit-serial multiplier/inverter in finite field GF(2m) , 2008, 2008 IEEE International Workshop on Hardware-Oriented Security and Trust.

[61]  Marc Joye,et al.  Elliptic Curve Cryptosystems in the Presence of Permanent and Transient Faults , 2005, Des. Codes Cryptogr..

[62]  H. T. Kung,et al.  A Regular Layout for Parallel Adders , 1982, IEEE Transactions on Computers.

[63]  Ingrid Verbauwhede,et al.  A Parallel Processing Hardware Architecture for Elliptic Curve Cryptosystems , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[64]  Debdeep Mukhopadhyay,et al.  High speed Fp multipliers and adders on FPGA platform , 2010, 2010 Conference on Design and Architectures for Signal and Image Processing (DASIP).

[65]  Joos Vandewalle,et al.  Hardware implementation of a Montgomery modular multiplier in a systolic array , 2003, Proceedings International Parallel and Distributed Processing Symposium.

[66]  JaeCheol Ha,et al.  Provably Secure Countermeasure Resistant to Several Types of Power Attack for ECC , 2007, WISA.

[67]  Dipanwita Roy Chowdhury,et al.  A Robust GF(p) Parallel Arithmetic Unit for Public Key Cryptography , 2007 .

[68]  Gunnar Gaubatz.  Versatile Montgomery Multiplier Architectures , 2002 .

[69]  Frederik Vercauteren,et al.  Faster -Arithmetic for Cryptographic Pairings on Barreto-Naehrig Curves , 2009, CHES.

[70]  Tanja Lange,et al.  Twisted Edwards Curves , 2008, AFRICACRYPT.

[71]  Dipanwita Roy Chowdhury,et al.  Parallel crypto-devices for GF(p) elliptic curve multiplication resistant against side channel attacks , 2009, Comput. Electr. Eng..

[72]  P. L. Montgomery.  Modular multiplication without trial division , 1985 .

[73]  Nicolas Guillermin.  A high speed coprocessor for elliptic curve scalar multiplication over Fp , 2010 .

[74]  Frederik Vercauteren,et al.  The Eta Pairing Revisited , 2006, IEEE Transactions on Information Theory.

[75]  Sanu Mathew,et al.  An improved unified scalable radix-2 Montgomery multiplier , 2005, 17th IEEE Symposium on Computer Arithmetic (ARITH'05).

[76]  Manfred Schimmler,et al.  Area and time efficient modular multiplication of large integers , 2003, Proceedings IEEE International Conference on Application-Specific Systems, Architectures, and Processors. ASAP 2003.

[77]  Gang Chen,et al.  A High-Performance Elliptic Curve Cryptographic Processor for General Curves Over ${\rm GF}(p)$ Based on a Systolic Arithmetic Unit , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[78]  Qiang Liu,et al.  Non-interleaving architecture for hardware implementation of modular multiplication , 2005, 2005 IEEE International Symposium on Circuits and Systems.

[79]  Nicolas Estibals,et al.  Compact Hardware for Computing the Tate Pairing over 128-Bit-Security Supersingular Curves , 2010, Pairing.

[80]  Darrel HANKERSON,et al.  Software Implementation of Pairings , 2009, Identity-Based Cryptography.

[81]  Debdeep Mukhopadhyay,et al.  Effect of glitches against masked AES S-box implementation and countermeasure , 2009, IET Inf. Secur..

[82]  Elena Trichina,et al.  Implementation of Elliptic Curve Cryptography with Built-In Counter Measures against Side Channel Attacks , 2002, CHES.

[83]  Debdeep Mukhopadhyay,et al.  An Improved Fault Based Attack of the Advanced Encryption Standard , 2009, AFRICACRYPT.

[84]  Frederik Vercauteren,et al.  Fault and Side-Channel Attacks on Pairing Based Cryptography , 2004, IACR Cryptology ePrint Archive.

[85]  Moti Yung,et al.  A Remark on Implementing the Weil Pairing , 2005, CISC.

[86]  Tsuyoshi Takagi,et al.  A Fast Parallel Elliptic Curve Multiplication Resistant against Side Channel Attacks , 2002, Public Key Cryptography.

[87]  Pankaj Rohatgi,et al.  Towards Sound Approaches to Counteract Power-Analysis Attacks , 1999, CRYPTO.

[88]  Debdeep Mukhopadhyay,et al.  Fault Attack, Countermeasures on Pairing Based Cryptography , 2011, Int. J. Netw. Secur..

[89]  Kouichi Sakurai,et al.  Elliptic Curves with the Montgomery-Form and Their Cryptographic Applications , 2000, Public Key Cryptography.

[90]  Gerd Ascheid,et al.  Designing an ASIP for Cryptographic Pairings over Barreto-Naehrig Curves , 2009, CHES.

[91]  Christof Paar,et al.  Efficient hardware architectures for modular multiplication on FPGAs , 2005, International Conference on Field Programmable Logic and Applications, 2005..

[92]  Tanja Lange,et al.  Faster Addition and Doubling on Elliptic Curves , 2007, ASIACRYPT.

[93]  Marc Joye,et al.  Weierstraß Elliptic Curves and Side-Channel Attacks , 2002, Public Key Cryptography.

[94]  Yehuda Lindell,et al.  Introduction to Modern Cryptography , 2004 .

[95]  David Narh Amanor,et al.  Efficient Hardware Architectures for Modular Multiplication , 2005 .

[96]  Ingrid Verbauwhede,et al.  A fast dual-field modular arithmetic logic unit and its hardware implementation , 2006, 2006 IEEE International Symposium on Circuits and Systems.

[97]  Patrick Schaumont,et al.  State-of-the-art of secure ECC implementations: a survey on known side-channel attacks and countermeasures , 2010, 2010 IEEE International Symposium on Hardware-Oriented Security and Trust (HOST).

[98]  Marc Joye,et al.  The Montgomery Powering Ladder , 2002, CHES.

[99]  G. R. Blakley,et al.  A Computer Algorithm for Calculating the Product AB Modulo M , 1983, IEEE Trans. Computers.

[100]  Ingrid Verbauwhede,et al.  Efficient pipelining for modular multiplication architectures in prime fields , 2007, GLSVLSI '07.

[101]  Neal Koblitz,et al.  CM-Curves with Good Cryptographic Properties , 1991, CRYPTO.

[102]  Douglas R. Stinson,et al.  Cryptography: Theory and Practice,Second Edition , 2002 .

[103]  Kazuo Ohta,et al.  Improved countermeasure against Address-bit DPA for ECC scalar multiplication , 2010, 2010 Design, Automation & Test in Europe Conference & Exhibition (DATE 2010).

[104]  Nigel P. Smart,et al.  The Hessian Form of an Elliptic Curve , 2001, CHES.

[105]  Thomas S. Messerges,et al.  Investigations of Power Analysis Attacks on Smartcards , 1999, Smartcard.

[106]  Florian Hess,et al.  Pairing Lattices , 2008, Pairing.

[107]  Mohammed Benaissa,et al.  Fast Elliptic Curve Cryptography on FPGA , 2008, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[108]  Scott Hauck,et al.  High-performance carry chains for FPGAs , 1998, FPGA '98.

[109]  Chester Rebeiro,et al.  High Speed Compact Elliptic Curve Cryptoprocessor for FPGA Platforms , 2008, INDOCRYPT.

[110]  Francisco Rodríguez-Henríquez,et al.  Hardware Accelerator for the Tate Pairing in Characteristic Three Based on Karatsuba-Ofman Multipliers , 2009, CHES.

[111]  Alfred Menezes,et al.  Guide to Elliptic Curve Cryptography , 2004, Springer Professional Computing.

[112]  Francisco Rodríguez-Henríquez,et al.  Multi-core Implementation of the Tate Pairing over Supersingular Elliptic Curves , 2009, CANS.

[113]  Marc Joye,et al.  Protections against Differential Analysis for Elliptic Curve Cryptography , 2001, CHES.

[114]  David Chaum,et al.  Security without identification: transaction systems to make big brother obsolete , 1985, CACM.

[115]  Antoine Joux.  A One Round Protocol for Tripartite Diffie-Hellman , 2000, ANTS.

[116]  Jerome A. Solinas,et al.  ID-based Digital Signature Algorithms , 2003 .

[117]  Elaine B. Barker,et al.  SP 800-57. Recommendation for Key Management, Part 1: General (revised) , 2007 .

[118]  Alfred Menezes,et al.  Weak Fields for ECC , 2004, CT-RSA.

[119]  Chih-Tsun Huang,et al.  Elixir: High-Throughput Cost-Effective Dual-Field Processors and the Design Framework for Elliptic Curve Cryptography , 2008, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[120]  Louis Goubin,et al.  A Refined Power-Analysis Attack on Elliptic Curve Cryptosystems , 2003, Public Key Cryptography.

[121]  Bodo Möller,et al.  Securing Elliptic Curve Point Multiplication against Side-Channel Attacks , 2001, ISC.

[122]  Tsuyoshi Takagi,et al.  Zero-Value Register Attack on Elliptic Curve Cryptosystem , 2005, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[123]  Francis M. Crowe,et al.  A scalable dual mode arithmetic unit for public key cryptosystems , 2005, International Conference on Information Technology: Coding and Computing (ITCC'05) - Volume II.

[124]  Ingrid Verbauwhede,et al.  Superscalar Coprocessor for High-Speed Curve-Based Cryptography , 2006, CHES.

[125]  Jean-Sébastien Coron,et al.  Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems , 1999, CHES.

[126]  Arjen K. Lenstra,et al.  Efficient Subgroup Exponentiation in Quadratic and Sixth Degree Extensions , 2002, CHES.

[127]  VerbauwhedeIngrid,et al.  Multicore Curve-Based Cryptoprocessor with Reconfigurable Modular Arithmetic Logic Units over GF(2^n) , 2007 .

[128]  Johann Großschädl,et al.  On Software Parallel Implementation of Cryptographic Pairings , 2008, IACR Cryptol. ePrint Arch..

[129]  Kwok-Wo Wong,et al.  A random number generator based on elliptic curve operations , 2004 .

[130]  Paulo S. L. M. Barreto,et al.  Compressed Pairings , 2004, CRYPTO.

[131]  Jean-Jacques Quisquater,et al.  A Practical Implementation of the Timing Attack , 1998, CARDIS.

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

[133]  Nigel P. Smart,et al.  Preventing SPA/DPA in ECC Systems Using the Jacobi Form , 2001, CHES.

[134]  Vipul Gupta,et al.  Accelerating next-generation public-key cryptosystems on general-purpose CPUs , 2005, IEEE Micro.

[135]  Jorge Guajardo Thomas Wollinger Christof Paar.  AREA EFFICIENT GF ( p ) ARCHITECTURES FOR GF ( p m ) MULTIPLIERS , 2002 .

[136]  Paulo S. L. M. Barreto,et al.  Pairing-Friendly Elliptic Curves of Prime Order , 2005, Selected Areas in Cryptography.

[137]  Wu Shuhua,et al.  A timing-and-area tradeoff GF(p) elliptic curve processor architecture for FPGA , 2005, Proceedings. 2005 International Conference on Communications, Circuits and Systems, 2005..

[138]  Julio César López-Hernández,et al.  High-Speed Parallel Software Implementation of the ηT Pairing , 2010, CT-RSA.

[139]  Jean-Pierre Seifert,et al.  Parallel scalar multiplication on general elliptic curves over Fp hedged against Non-Differential Side-Channel Attacks , 2002, IACR Cryptol. ePrint Arch..

[140]  Ricardo Dahab,et al.  Implementing Cryptographic Pairings over Barreto-Naehrig Curves , 2007, Pairing.

[141]  Chih-Tsun Huang,et al.  A Highly Efficient Cipher Processor for Dual-Field Elliptic Curve Cryptography , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[142]  Berk Sunar,et al.  Low-Power Elliptic Curve Cryptography Using Scaled Modular Arithmetic , 2004, CHES.

[143]  Francisco Rodríguez-Henríquez,et al.  A parallel architecture for fast computation of elliptic curve scalar multiplication over GF(2/sup m/) , 2004, 18th International Parallel and Distributed Processing Symposium, 2004. Proceedings..

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

[145]  Kendall Ananyi,et al.  Flexible Hardware Processor for Elliptic Curve Cryptography Over NIST Prime Fields , 2009, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[146]  Jean-Jacques Quisquater,et al.  High-speed hardware implementations of Elliptic Curve Cryptography: A survey , 2007, J. Syst. Archit..

[147]  Akashi Satoh,et al.  A Scalable Dual-Field Elliptic Curve Cryptographic Processor , 2003, IEEE Trans. Computers.

[148]  M. Anwar Hasan,et al.  High-Performance Architecture of Elliptic Curve Scalar Multiplication , 2008, IEEE Transactions on Computers.

[149]  Hubert Kaeslin,et al.  Digital Integrated Circuit Design: From VLSI Architectures to CMOS Fabrication , 2008 .

[150]  Bernd Meyer,et al.  Differential Fault Attacks on Elliptic Curve Cryptosystems , 2000, CRYPTO.

[151]  Peter Schwabe,et al.  New Software Speed Records for Cryptographic Pairings , 2010, LATINCRYPT.

[152]  William E. Burr,et al.  Recommendation for Key Management, Part 1: General (Revision 3) , 2006 .

[153]  Ingrid Verbauwhede,et al.  Multicore Curve-Based Cryptoprocessor with Reconfigurable Modular Arithmetic Logic Units over GF(2^n) , 2007, IEEE Transactions on Computers.

[154]  Dipanwita Roy Chowdhury,et al.  A GF(p) elliptic curve group operator resistant against side channel attacks , 2008, GLSVLSI '08.

[155]  Indranil Sengupta,et al.  An area optimized reconfigurable encryptor for AES-Rijndael , 2007 .

[156]  Michael Scott,et al.  Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions , 2009, IACR Cryptol. ePrint Arch..

[157]  Nicolas Guillermin.  A High Speed Coprocessor for Elliptic Curve Scalar Multiplications over \mathbbFp\mathbb{F}_p , 2010, CHES.

[158]  Tung-Chou Chen,et al.  Arithmetic Unit for Finite Field ${\rm GF}(2^{m})$ , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[159]  H. Edwards.  A normal form for elliptic curves , 2007 .

[160]  Christophe Clavier,et al.  Universal Exponentiation Algorithm , 2001, CHES.

[161]  Ross J. Anderson,et al.  Optical Fault Induction Attacks , 2002, CHES.

[162]  Máire O'Neill,et al.  Hardware Elliptic Curve Cryptographic Processor Over$rm GF(p)$ , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[163]  Francisco Rodríguez-Henríquez,et al.  Cryptographic Algorithms on Reconfigurable Hardware , 2010 .

[164]  Marc Joye,et al.  Low-cost solutions for preventing simple side-channel analysis: side-channel atomicity , 2004, IEEE Transactions on Computers.

[165]  Marc Joye,et al.  Hessian Elliptic Curves and Side-Channel Attacks , 2001, CHES.

[166]  Ingrid Verbauwhede,et al.  Elliptic curve cryptography on embedded multicore systems , 2008, Des. Autom. Embed. Syst..

[167]  Paul C. Kocher,et al.  Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems , 1996, CRYPTO.

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

[169]  Wayne Luk,et al.  Customizable elliptic curve cryptosystems , 2005, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[170]  Michael Scott,et al.  Side Channel Analysis of Practical Pairing Implementations: Which Path Is More Secure? , 2006, VIETCRYPT.

[171]  Frederik Vercauteren,et al.  A Fault Attack on Pairing-Based Cryptography , 2006, IEEE Transactions on Computers.

[172]  Daniel J. Bernstein,et al.  Cache-timing attacks on AES , 2005 .

[173]  Siva Sai Yerubandi,et al.  Differential Power Analysis , 2002 .

[174]  Atsuko Miyaji,et al.  Secure Elliptic Curve Exponentiation against RPA, ZRA, DPA, and SPA , 2006, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

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

[176]  Ricardo Dahab,et al.  Fast Multiplication on Elliptic Curves over GF(2m) without Precomputation , 1999, CHES.

[177]  Antoine Joux,et al.  Another Approach to Pairing Computation in Edwards Coordinates , 2008, INDOCRYPT.

[178]  Michael Scott,et al.  On the Final Exponentiation for Calculating Pairings on Ordinary Elliptic Curves , 2009, Pairing.

[179]  David Mandell Freeman,et al.  Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10 , 2006, ANTS.

[180]  Ze-mao Zhao.  ID-based Weak Blind Signature From Bilinear Pairings , 2008, Int. J. Netw. Secur..

[181]  Kenneth R. Sloan.  Comments on "A Computer Algorithm for Calculating the Product AB Modulo M" , 1985, IEEE Trans. Computers.

[182]  P. L. Montgomery.  Speeding the Pollard and elliptic curve methods of factorization , 1987 .

[183]  Kris Gaj,et al.  FPGA accelerated tate pairing based cryptosystems over binary fields , 2006, 2006 IEEE International Conference on Field Programmable Technology.

[184]  Huapeng Wu.  Montgomery Multiplier and Squarer for a Class of Finite Fields , 2002, IEEE Trans. Computers.