Principles of CMOS VLSI design: a systems perspective
Principles of CMOS VLSI design: a systems perspective
Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A Comparison of VLSI Architecture of Finite Field Multipliers Using Dual, Normal, or Standard Bases
IEEE Transactions on Computers
Optimal normal bases in GF(pn)
Discrete Applied Mathematics
Journal of Cryptology
A VLSI Architecture for Fast Inversion in GF(2/sup m/)
IEEE Transactions on Computers
IEEE Transactions on Computers
VLSI design for exponentiation in GF(2n)
AUSCRYPT '90 Proceedings of the international conference on cryptology on Advances in cryptology
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
IEEE Transactions on Computers - Special issue on computer arithmetic
Finite field inversion over the dual basis
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Low-Energy Digit-Serial/Parallel Finite Field Multipliers
Journal of VLSI Signal Processing Systems - Special issue on application specific systems, architectures and processors
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Elliptic Curve Public Key Cryptosystems
Elliptic Curve Public Key Cryptosystems
GF(2m) Multiplication and Division Over the Dual Basis
IEEE Transactions on Computers
IEEE Transactions on Computers
VLSI Designs for Multiplication over Finite Fields GF (2m)
AAECC-6 Proceedings of the 6th International Conference, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
A Subexponential Algorithm for Discrete Logarithms over All Finite Fields
CRYPTO '93 Proceedings of the 13th Annual International Cryptology Conference on Advances in Cryptology
Efficient Algorithms for Elliptic Curve Cryptosystems
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
A Fast Software Implementation for Arithmetic Operations in GF(2n)
ASIACRYPT '96 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Error Control Coding, Second Edition
Error Control Coding, Second Edition
Fast arithmetic architectures for public-key algorithms over Galois fields GF((2n)m)
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
New Parallel Architecture for Modular Multiplication and Squaring Based on Cellular Automata
PARA '02 Proceedings of the 6th International Conference on Applied Parallel Computing Advanced Scientific Computing
Fast Normal Basis Multiplication Using General Purpose Processors
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
A High Performance Reconfigurable Elliptic Curve Processor for GF(2m)
CHES '00 Proceedings of the Second International Workshop on Cryptographic Hardware and Embedded Systems
Error Detection in Polynomial Basis Multipliers over Binary Extension Fields
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
Power-time flexible architecture for GF(2k) elliptic curve cryptosystem computation
Proceedings of the 13th ACM Great Lakes symposium on VLSI
Low Complexity Multiplication in a Finite Field Using Ring Representation
IEEE Transactions on Computers
Efficient Multiplication Beyond Optimal Normal Bases
IEEE Transactions on Computers
Constructing Composite Field Representations for Efficient Conversion
IEEE Transactions on Computers
Fast Normal Basis Multiplication Using General Purpose Processors
IEEE Transactions on Computers
Low Complexity Word-Level Sequential Normal Basis Multipliers
IEEE Transactions on Computers
A digit-serial multiplier for finite field GF(2m)
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
Fault Detection Architectures for Field Multiplication Using Polynomial Bases
IEEE Transactions on Computers
An efficient technique for synthesis and optimization of polynomials in GF(2m)
Proceedings of the 2006 IEEE/ACM international conference on Computer-aided design
A Graph-Based Unified Technique for Computing and Representing Coefficients over Finite Fields
IEEE Transactions on Computers
Multi-gigabit GCM-AES Architecture Optimized for FPGAs
CHES '07 Proceedings of the 9th international workshop on Cryptographic Hardware and Embedded Systems
Improved throughput bit-serial multiplier for GF(2m) fields
Integration, the VLSI Journal
Time-space efficient exponentiation over GF(2m)
ICCSA'03 Proceedings of the 2003 international conference on Computational science and its applications: PartI
Fast exponentiaion over GF(2m) based on cellular automata
ICCS'03 Proceedings of the 2003 international conference on Computational science: PartII
Low complexity digit serial systolic montgomery multipliers for special class of GF(2m)
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
Scalable Gaussian Normal Basis Multipliers over GF(2m) Using Hankel Matrix-Vector Representation
Journal of Signal Processing Systems
Low-power and high-speed design of a versatile bit-serial multiplier in finite fields GF(2m)
Integration, the VLSI Journal
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This contribution describes a new class of arithmetic architectures for Galois fields $GF(2^k)$. The main applications of the architecture are public-key systems which are based on the discrete logarithm problem for elliptic curves. The architectures use a representation of the field $GF(2^k)$ as $GF((2^n)^m)$, where $k=n\cdot m$. The approach explores bit parallel arithmetic in the subfield $GF(2^n)$ and serial processing for the extension field arithmetic. This mixed parallel-serial (hybrid) approach can lead to fast implementations. As the core module, a hybrid multiplier is introduced and several optimizations are discussed. We provide two different approaches to squaring. We develop exact expressions for the complexity of parallel squarers in composite fields, which can have a surprisingly low complexity. The hybrid architectures are capable of exploring the time-space trade-off paradigm in a flexible manner. In particular, the number of clock cycles for one field multiplication, which is the atomic operation in most public-key schemes, can be reduced by a factor of $n$ compared to other known realizations. The acceleration is achieved at the cost of an increased computational complexity. We describe a proof-of-concept implementation of an ASIC for multiplication and squaring in $GF((2^n)^m)$, $m$ variable.