# Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms

@inproceedings{Bailey1998OptimalEF, title={Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms}, author={Daniel V. Bailey and Christof Paar}, booktitle={CRYPTO}, year={1998} }

This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of public-key cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported optimizations focus on finite fields of the form GF(p) and GF(2 m ), an OEF is the class of fields GF(p m ), for p a prime of special form and m a positive integer. Modern RISC workstation processors are…

## 210 Citations

### Computation in Optimal Extension Fields

- Mathematics, Computer Science
- 2000

Results show that OEFs when used with the new inversion and multiplication algorithms provide a substantial performance increase over other reported methods.

### Efficient Arithmetic in Finite Field Extensions with Application in Elliptic Curve Cryptography

- Mathematics, Computer ScienceJournal of Cryptology
- 2015

Results show that OEFs when used with the new inversion and multiplication algorithms provide a substantial performance increase over other reported methods.

### Multiply-accumulate architecture for a special class of optimal extension fields

- Mathematics, Computer Science2005 IEEE International Conference on Application-Specific Systems, Architecture Processors (ASAP'05)
- 2005

The Type II OEF multiplier presented uses merged arithmetic to combine multiple multiply and addition operations together, and unlike previous work, the multiplier also performs subfield and extension field reduction in parallel for this class of finite fields.

### New methods for finite field arithmetic

- Computer Science, Mathematics
- 2001

A new method for obtaining fast software implementations of the modular multiplication operation with an arbitrary prime modulus p, which has less bit-length than the word-length of a microprocessor and an arbitrary generator polynomial is described.

### Architectural support for arithmetic in optimal extension fields

- Computer Science, MathematicsProceedings. 15th IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2004.
- 2004

This work introduces two custom instructions to accelerate the reduction modulo a PM prime and shows that the multiplication in an optimal extension field can take advantage of a multiply/accumulate unit with a wide accumulator so that a certain number of 64-bit products can be summed up without overflow.

### An alternative class of irreducible polynomials for optimal extension fields

- MathematicsDes. Codes Cryptogr.
- 2011

This work proposes a new type of irreducible polynomials that are more abundant and still efficient for field multiplication and takes the advantage of polynomial residue arithmetic to achieve high performance for fieldmultiplication.

### Optimizing Galois Field Arithmetic for Diverse Processor Architectures and Applications

- Computer Science2008 IEEE International Symposium on Modeling, Analysis and Simulation of Computers and Telecommunication Systems
- 2008

This paper first anaylze existing table-based implementation and optimization techniques for multiplication in fields of the form GF(21), and proposes the use of techniques in composite fields: extensions of GF( 21) in which multiplications are performed in GF (21) and efficiently combined.

### On fast implementations of elliptic curve point multiplication

- Computer Science, MathematicsACM Southeast Regional Conference
- 2022

This work proposes a new FPGA implementation of elliptic curve point multiplication over an optimal extension field defined by an irreducible binomial of degree three and the operations on the ground field are modulo a pseudo Mersenne prime.

### Efficient Exponentiation in GF(pm) Using the Frobenius Map

- Computer Science, MathematicsICCSA
- 2006

This paper presents an efficient exponentiation algorithm in optimal extension field (OEF) GF(pm), which uses the fact that the Frobenius map is very efficient in OEFs, and shows that the new algorithm is twice as fast as the conventional square-and-multiply exponentiation.

### A State-of-the-art Elliptic Curve Cryptographic Processor Operating in the Frequency Domain

- Computer Science, MathematicsMob. Networks Appl.
- 2007

The work at hand presents the firstHardware implementation of a frequency domain multiplier suitable for ECC and the first hardware implementation of ECC in the frequency domain.

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From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of…

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• Any Java programming language book • Data Abstraction and Problem Solving with C++, 5th Edition, by Frank Carrano, Addison Wesley 2007 • Object, Abstraction, Data Structures and Design Using Java,…