## Subquadratic Computational Complexity Schemes for Extended Binary Field Multiplication Using Optimal Normal Bases (2007)

Citations: | 8 - 3 self |

### BibTeX

@MISC{Fan07subquadraticcomputational,

author = {H. Fan and M. A. Hasan},

title = {Subquadratic Computational Complexity Schemes for Extended Binary Field Multiplication Using Optimal Normal Bases},

year = {2007}

}

### OpenURL

### Abstract

Based on a recently proposed Toeplitz matrix-vector product approach, a subquadratic computational complexity scheme is presented for multiplications in binary extended finite fields using Type I and II optimal normal bases. basis. Index Terms Finite field, subquadratic computational complexity multiplication, normal basis, optimal normal

### Citations

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Computational method and apparatus for finite field arithmetic
- MASSEY, OMURA
- 1986
(Show Context)
Citation Context ...s. One of the most important advances in the normal basis multiplication is the discovery of the two types (Type I and Type II) of optimal normal bases (ONB) in 1987 [1]. When the Massey-Omura scheme =-=[2]-=- is applied, the computational complexity (i.e., the number of arithmetic operations in the ground field GF(2)) of a GF(2 n ) multiplication using an ONB is O(n 2 ), while that using an arbitrary norm... |

51 |
Arithmetic complexity of computations
- Winograd
- 1980
(Show Context)
Citation Context ...i, k ≤ n − 1. Let n = 2 i (i > 0), T be an n × n Toeplitz matrix and V an n × 1 column vector. Then the following noncommutative formula can be used to compute the Toeplitz matrix-vector product 2sTV =-=[6]-=-: TV = ⎛ ⎝ T1 T0 T2 T1 ⎞ ⎛ ⎠ ⎝ V0 V1 ⎞ ⎛ P1 + P2 ⎞ ⎠ = ⎝ P0 + P2 ⎠ , (1) where T0, T1 and T2 are (n/2) × (n/2) matrices and are individually in Toeplitz form, and V0 and V1 are (n/2)×1 column vectors,... |

43 | Low-complexity bit-parallel canonical and normal basis multipliers for a class of finite fields
- KOC, SUNAR
- 1998
(Show Context)
Citation Context ...s [8]. For practical purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], [10], [11], [12], =-=[13]-=-, [14], [15] and [16]. Based on some of these properties, below we present two subquadratic computational complexity schemes for multiplications in GF(2 n ) using Type I and II ONB. 4sA. Formulation f... |

34 |
A modified Massey-Omura parallel multiplier for a class of finite fields
- HASAN, WANG, et al.
- 1993
(Show Context)
Citation Context ...urity reasons [8]. For practical purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], [10], =-=[11]-=-, [12], [13], [14], [15] and [16]. Based on some of these properties, below we present two subquadratic computational complexity schemes for multiplications in GF(2 n ) using Type I and II ONB. 4sA. F... |

29 |
Modular construction of low complexity parallel multipliers for a class of finite field GFð2mÞ
- Hasan, Wang, et al.
- 1992
(Show Context)
Citation Context ...or security reasons [8]. For practical purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], =-=[10]-=-, [11], [12], [13], [14], [15] and [16]. Based on some of these properties, below we present two subquadratic computational complexity schemes for multiplications in GF(2 n ) using Type I and II ONB. ... |

28 |
An efficient optimal normal basis type II multiplier
- SUNAR, KOC
- 2001
(Show Context)
Citation Context ...ractical purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], [10], [11], [12], [13], [14], =-=[15]-=- and [16]. Based on some of these properties, below we present two subquadratic computational complexity schemes for multiplications in GF(2 n ) using Type I and II ONB. 4sA. Formulation for Type I ON... |

23 | A generalized method for constructing subquadratic complexity GF(2k) multipliers
- Sunar
(Show Context)
Citation Context ...on over extended binary field GF(2 n ), recently there have been considerable efforts to develop practical algorithms with the computational complexity less than O(n 2 ), see for example [3], [4] and =-=[5]-=-. In [3], a subquadratic space complexity multiplier for Type I ONB is proposed. It relies on a permutation of the normal basis and then recursive applications of the Karatsuba algorithm. Recently, a ... |

22 | A new approach to sub-quadratic space complexity parallel multipliers for extended binary fields
- Fan, Hasan
- 2007
(Show Context)
Citation Context ...iplication over extended binary field GF(2 n ), recently there have been considerable efforts to develop practical algorithms with the computational complexity less than O(n 2 ), see for example [3], =-=[4]-=- and [5]. In [3], a subquadratic space complexity multiplier for Type I ONB is proposed. It relies on a permutation of the normal basis and then recursive applications of the Karatsuba algorithm. Rece... |

16 | On orders of optimal normal basis generators
- Gao, Vanstone
- 1995
(Show Context)
Citation Context ... For practical purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], [10], [11], [12], [13], =-=[14]-=-, [15] and [16]. Based on some of these properties, below we present two subquadratic computational complexity schemes for multiplications in GF(2 n ) using Type I and II ONB. 4sA. Formulation for Typ... |

12 | Normal Bases over Finite Fields
- Gao
(Show Context)
Citation Context ...ers, respectively, we note that there appears to be adequate number of optimal normal bases that are of practical interest. For example, there are 430 values of n ≤ 2000 for which there exists an ONB =-=[9]-=-. We also note that there is no limitation on n when applying the Toeplitz matrix-vector product approach to design ONB multipliers. For example, if we wish to use formula (1) but 2 does not divide n,... |

8 | Weak Fields for ECC
- Menezes, Teske, et al.
- 2004
(Show Context)
Citation Context ...lly, the Toeplitz matrix-vector product of size n + 1 is computed and the last bit of the resulting (n + 1)-bit vector is discarded. In some cryptosystems, Type I ONB are avoided for security reasons =-=[8]-=-. For practical purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], [10], [11], [12], [13],... |

6 |
A New Low Complexity Parallel Multiplier for a Class of Finite Fields
- Leone
- 2001
(Show Context)
Citation Context ... multiplication over extended binary field GF(2 n ), recently there have been considerable efforts to develop practical algorithms with the computational complexity less than O(n 2 ), see for example =-=[3]-=-, [4] and [5]. In [3], a subquadratic space complexity multiplier for Type I ONB is proposed. It relies on a permutation of the normal basis and then recursive applications of the Karatsuba algorithm.... |

4 |
Simple Multiplication Algorithm for a Class of GF(2 n
- Fan
- 1996
(Show Context)
Citation Context ...reasons [8]. For practical purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], [10], [11], =-=[12]-=-, [13], [14], [15] and [16]. Based on some of these properties, below we present two subquadratic computational complexity schemes for multiplications in GF(2 n ) using Type I and II ONB. 4sA. Formula... |

3 |
Low-complexity linear array multiplier for normal basis of type-II
- Lee, Chang
(Show Context)
Citation Context ...purposes, e.g., n < 2000, Type II ONB are more abundant than Type I ONB [9]. Properties of Type I and II ONB can be found in various references, e.g., [1], [9], [10], [11], [12], [13], [14], [15] and =-=[16]-=-. Based on some of these properties, below we present two subquadratic computational complexity schemes for multiplications in GF(2 n ) using Type I and II ONB. 4sA. Formulation for Type I ONB Let ˆ X... |