## A generalized method for constructing subquadratic complexity GF(2 k ) multipliers (2004)

Venue: | IEEE Transactions on Computers |

Citations: | 22 - 0 self |

### BibTeX

@ARTICLE{Sunar04ageneralized,

author = {B. Sunar},

title = {A generalized method for constructing subquadratic complexity GF(2 k ) multipliers},

journal = {IEEE Transactions on Computers},

year = {2004},

volume = {53},

pages = {1097--1105}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce a generalized method for constructing sub-quadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication. We present a recursive construction technique that extends any d point multiplier into an n = d k point multiplier with area that is sub-quadratic and delay that is logarithmic in the bit-length n. We present a thorough analysis that establishes the exact space and time complexities of these multipliers. Using the recursive construction method we obtain six new constructions, among which one turns out to be identical to the Karatsuba multiplier. All six algorithms have sub-quadratic space complexities and two of the algorithms have significantly better time complexities than the Karatsuba algorithm. Keywords: Bit-parallel multipliers, finite fields, Winograd convolution 1

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Citation Context ...Karatsuba algorithm. Keywords: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing [1, 2], coding theory =-=[3, 4, 5]-=- and cryptography [6, 7, 8]. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field... |

418 |
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Citation Context ...complexities than the Karatsuba algorithm. Keywords: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing =-=[1, 2]-=-, coding theory [3, 4, 5] and cryptography [6, 7, 8]. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications... |

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Citation Context ...Karatsuba algorithm. Keywords: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing [1, 2], coding theory =-=[3, 4, 5]-=- and cryptography [6, 7, 8]. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field... |

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Citation Context ...rallel multipliers, finite fields, Winograd convolution. æ 1 INTRODUCTION FINITE fields have numerous applications in digital signal processing [1], [2], coding theory [3], [4], [5], and cryptography =-=[6]-=-, [7], [8]. The efficient implementation of finite field operations, i.e., addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field operations is i... |

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Citation Context ...complexities cited above [13], [14], [15], [16], [17], [18]. An alternative approach for the initial multiplication step was developed by using the polynomial version of the Karatsuba-Ofman algorithm =-=[19]-=-. The recursive splitting of polynomials and the special reassembly of the partial products drastically reduces the number of AND gates required for the multiplication operation: nlog2 3 . Although th... |

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Citation Context ...ds: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing [1, 2], coding theory [3, 4, 5] and cryptography =-=[6, 7, 8]-=-. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field operations is intimately r... |

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Citation Context ...complexities than the Karatsuba algorithm. Keywords: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing =-=[1, 2]-=-, coding theory [3, 4, 5] and cryptography [6, 7, 8]. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications... |

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Citation Context ...Karatsuba algorithm. Keywords: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing [1, 2], coding theory =-=[3, 4, 5]-=- and cryptography [6, 7, 8]. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field... |

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Citation Context ... and T⊕ denote the delays of a two input AND gate and a two input XOR gate, respectively. A variation to this theme which combines the multiplication and reduction steps was proposed by Mastrovito in =-=[12]-=-. Although, considerable research went into the design and optimization on this theme and its variations, the space and time complexity figures have not improved much further than the quadratic space ... |

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Citation Context ...ue to support arbitrary polynomial lengths and present its complexity analysis. 2 Winograd Short Convolution In this section we briefly describe the short convolution algorithm introduced by Winograd =-=[20, 21]-=-. Let a(x) and b(x) represent d − 1 degree polynomials a(x) = �d−1 i=0 aixi and b(x) = �d−1 i=0 bixi defined over an arbitrary field. The product of the polynomials is computed as where ck = c(x) = a(... |

37 | Low-Complexity Bit-Parallel Canonical and Normal Basis Multipliers for a Class of Finite Fields
- Koc, Sunar
(Show Context)
Citation Context ...e design and optimization on this theme and its variations, the space and time complexity figures have not improved much further than the quadratic space and logarithmic time complexities cited above =-=[13, 14, 15, 16, 17, 18]-=-. An alternative approach for the initial multiplication step was developed by using the polynomial version of the Karatsuba-Ofman algorithm [19]. The recursive splitting of polynomials and the specia... |

36 | Mastrovito multiplier for all trinomials
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Citation Context ...e design and optimization on this theme and its variations, the space and time complexity figures have not improved much further than the quadratic space and logarithmic time complexities cited above =-=[13, 14, 15, 16, 17, 18]-=-. An alternative approach for the initial multiplication step was developed by using the polynomial version of the Karatsuba-Ofman algorithm [19]. The recursive splitting of polynomials and the specia... |

28 |
VLSI Designs for Multiplications over Finite Fields GF(2 m
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27 |
Shift Register Sequences �Holden Day
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(Show Context)
Citation Context ...ds: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing [1, 2], coding theory [3, 4, 5] and cryptography =-=[6, 7, 8]-=-. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field operations is intimately r... |

26 |
An Efficient Optimal Normal Basis Type II Multiplier
- Sunar, Koc
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25 |
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Some bilinear forms whose multiplicative complexity depends on the field of constants
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Citation Context ...ue to support arbitrary polynomial lengths and present its complexity analysis. 2 Winograd Short Convolution In this section we briefly describe the short convolution algorithm introduced by Winograd =-=[20, 21]-=-. Let a(x) and b(x) represent d − 1 degree polynomials a(x) = �d−1 i=0 aixi and b(x) = �d−1 i=0 bixi defined over an arbitrary field. The product of the polynomials is computed as where ck = c(x) = a(... |

18 |
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(Show Context)
Citation Context ...ds: Bit-parallel multipliers, finite fields, Winograd convolution 1 Introduction Finite fields have numerous applications in digital signal processing [1, 2], coding theory [3, 4, 5] and cryptography =-=[6, 7, 8]-=-. The efficient implementation of finite field operations, i.e. addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field operations is intimately r... |

18 | Solving elliptic curve discrete logarithm problems using Weil descent
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Citation Context ...ently composite bit-length would benefit from the recursive construction technique. In certain applications non-prime bit-lengths may be prohibited. In fact, an attack based on Weil descent was shown =-=[22]-=- to be effective on elliptic curve discrete logarithm problems built over certain composite extension fields. Hence, in practice composite extensions are avoided as much as possible in elliptic curve ... |

16 |
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Citation Context ...vector space, which facilitates the use of various bases for the representation of the field elements. Among the many proposed bases, the canonical basis [4], the normal basis [8], and the dual basis =-=[9, 10, 11]-=- are the most commonly used ones. In this paper, we focus our attention on canonical basis bit-parallel multiplier architectures for even characteristic field extensions, i.e. GF (2 k ). In the canoni... |

3 |
Finite Field Inversion over the Dual Basis
- Fenn, Benaissa, et al.
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Citation Context ...vector space, which facilitates the use of various bases for the representation of the field elements. Among the many proposed bases, the canonical basis [4], the normal basis [8], and the dual basis =-=[9, 10, 11]-=- are the most commonly used ones. In this paper, we focus our attention on canonical basis bit-parallel multiplier architectures for even characteristic field extensions, i.e. GF (2 k ). In the canoni... |

1 |
Error-Correcting Codes. Cambrdige
- Peterson, Jr
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(Show Context)
Citation Context ...hm. Index Terms—Bit-parallel multipliers, finite fields, Winograd convolution. æ 1 INTRODUCTION FINITE fields have numerous applications in digital signal processing [1], [2], coding theory [3], [4], =-=[5]-=-, and cryptography [6], [7], [8]. The efficient implementation of finite field operations, i.e., addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of... |

1 |
Finite Fields for Computer Scientists and
- McEliece
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(Show Context)
Citation Context ...l multipliers, finite fields, Winograd convolution. æ 1 INTRODUCTION FINITE fields have numerous applications in digital signal processing [1], [2], coding theory [3], [4], [5], and cryptography [6], =-=[7]-=-, [8]. The efficient implementation of finite field operations, i.e., addition, multiplication, and inversion, is therefore critical for many applications. The efficiency of field operations is intima... |