Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms

  title={Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms},
  author={Daniel V. Bailey and Christof Paar},
  • D. BaileyC. Paar
  • Published in CRYPTO 23 August 1998
  • Computer Science, Mathematics
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… 

Computation in Optimal Extension Fields

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

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

  • M. O. SanuE. Swartzlander
  • Mathematics, Computer Science
    2005 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

  • T. Yanik
  • 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

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

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

  • K. GreenanE. L. MillerT. Schwarz
  • Computer Science
    2008 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

  • E. Morales
  • Computer Science, Mathematics
    ACM 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

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

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.



A Fast Software Implementation for Arithmetic Operations in GF(2n)

A software implementation of arithmetic operations in a finite field GF(2n), based on an alternative representation of the field elements that results in rather simple routines matching the structure of computer memory very well.

Efficient Algorithms for Elliptic Curve Cryptosystems

An entirely new approach which accelerates the multiplications of points which is the core operation in elliptic curve public-key systems and which proofs to be faster than traditional point multiplication methods is described.

Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method

The proposed method is based on pre-computation to generate an adequate addition-subtraction chain for multiplier the d, and by increasing the average length of zero runs in a signed binary representation of d, it can speed up the window method.

Elliptic curves in cryptography

This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems, to help engineers and computer scientists wishing (or needing) to actually implement such systems.

Elliptic curve cryptosystems

The question of primitive points on an elliptic curve modulo p is discussed, and a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point is given.

Use of Elliptic Curves in Cryptography

  • V. Miller
  • Computer Science, Mathematics
  • 1985
An analogue of the Diffie-Hellmann key exchange protocol is proposed which appears to be immune from attacks of the style of Western, Miller, and Adleman.

Fast Key Exchange with Elliptic Curve Systems

The Diffie-Hellman key exchange algorithm is implemented using the group of points on an elliptic curve over the field F2n to achieve computation rates that are slightly faster than non-elliptic curve versions with a similar level of security.

Public-Key Cryptosystems with Very Small Key Length

The purpose of this paper is to hivestigate the security and practicality of elliptic curve cryptosystems with small key sizes of about 100 bits.

Handbook of Applied Cryptography

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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