### Abstract

In this paper we consider a special type of dual basis for finite fields, GF(2^{m}), where the variants of m are presented in section 2. We introduce our field representing method for efficient field arithmetic(such as field multiplication and field inversion). It reveals a very effective role for both software and hardware(VLSI) implementations, but the aspect of hardware design of its structure is out of this manuscript and so, here, we deal only the case of its software implementation(the efficiency of hardware implementation is appeared in another article submitted to IEEE Transactions on Computers). A brief description of several advantageous characteristics of our method is that (1) the field multiplication in GF(2^{m}) can be constructed only by m + 1 vector rotations and the same amount of vector XOR operations, (2) there is required no additional work load such as basis changing(from standard to dual basis or from dual basis to standard basis as the conventional dual based arithmetic does), (3) the field squaring is only bit-by-bit permutation and it has a good regularity for its implementation, and (4) the field inversion process is available to both cases of its implementation using Fermat’s Theorem and using almost inverse algorithm[14], especially the case of using the almost inverse algorithm has an additional advantage in find- ing(computing) its complete inverse element(i.e., there is required no pre-computed table of the values, x^{-k}, k = 1, 2,…).

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 12-28 |

Number of pages | 17 |

Volume | 1560 |

ISBN (Print) | 3540656448, 9783540656449 |

Publication status | Published - 1999 |

Event | 2nd International Workshop on Practice and Theory in Public Key Cryptography, PKC 1999 - Kamakura, Japan Duration: 1999 Mar 1 → 1999 Mar 3 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1560 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 2nd International Workshop on Practice and Theory in Public Key Cryptography, PKC 1999 |
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Country | Japan |

City | Kamakura |

Period | 99/3/1 → 99/3/3 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 1560, pp. 12-28). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1560). Springer Verlag.

**A new aspect of dual basis for efficient field arithmetic.** / Lee, Chang Hyi; Lim, Jong In.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 1560, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1560, Springer Verlag, pp. 12-28, 2nd International Workshop on Practice and Theory in Public Key Cryptography, PKC 1999, Kamakura, Japan, 99/3/1.

}

TY - GEN

T1 - A new aspect of dual basis for efficient field arithmetic

AU - Lee, Chang Hyi

AU - Lim, Jong In

PY - 1999

Y1 - 1999

N2 - In this paper we consider a special type of dual basis for finite fields, GF(2m), where the variants of m are presented in section 2. We introduce our field representing method for efficient field arithmetic(such as field multiplication and field inversion). It reveals a very effective role for both software and hardware(VLSI) implementations, but the aspect of hardware design of its structure is out of this manuscript and so, here, we deal only the case of its software implementation(the efficiency of hardware implementation is appeared in another article submitted to IEEE Transactions on Computers). A brief description of several advantageous characteristics of our method is that (1) the field multiplication in GF(2m) can be constructed only by m + 1 vector rotations and the same amount of vector XOR operations, (2) there is required no additional work load such as basis changing(from standard to dual basis or from dual basis to standard basis as the conventional dual based arithmetic does), (3) the field squaring is only bit-by-bit permutation and it has a good regularity for its implementation, and (4) the field inversion process is available to both cases of its implementation using Fermat’s Theorem and using almost inverse algorithm[14], especially the case of using the almost inverse algorithm has an additional advantage in find- ing(computing) its complete inverse element(i.e., there is required no pre-computed table of the values, x-k, k = 1, 2,…).

AB - In this paper we consider a special type of dual basis for finite fields, GF(2m), where the variants of m are presented in section 2. We introduce our field representing method for efficient field arithmetic(such as field multiplication and field inversion). It reveals a very effective role for both software and hardware(VLSI) implementations, but the aspect of hardware design of its structure is out of this manuscript and so, here, we deal only the case of its software implementation(the efficiency of hardware implementation is appeared in another article submitted to IEEE Transactions on Computers). A brief description of several advantageous characteristics of our method is that (1) the field multiplication in GF(2m) can be constructed only by m + 1 vector rotations and the same amount of vector XOR operations, (2) there is required no additional work load such as basis changing(from standard to dual basis or from dual basis to standard basis as the conventional dual based arithmetic does), (3) the field squaring is only bit-by-bit permutation and it has a good regularity for its implementation, and (4) the field inversion process is available to both cases of its implementation using Fermat’s Theorem and using almost inverse algorithm[14], especially the case of using the almost inverse algorithm has an additional advantage in find- ing(computing) its complete inverse element(i.e., there is required no pre-computed table of the values, x-k, k = 1, 2,…).

UR - http://www.scopus.com/inward/record.url?scp=84945290573&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945290573&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84945290573

SN - 3540656448

SN - 9783540656449

VL - 1560

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 12

EP - 28

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -