## Parallel Multipliers Based on Special Irreducible Pentanomials (2003)

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Venue: | IEEE Transactions on Computers |

Citations: | 18 - 0 self |

### BibTeX

@ARTICLE{Rodríguez-henríquez03parallelmultipliers,

author = {F. Rodríguez-henríquez and Ç. K. Koç},

title = {Parallel Multipliers Based on Special Irreducible Pentanomials},

journal = {IEEE Transactions on Computers},

year = {2003},

volume = {52},

pages = {1535--1542}

}

### Years of Citing Articles

### OpenURL

### Abstract

The state-of-the-art Galois field GF(2m)multipliers offer advantageous space and time complexities when the field is generated by some special irreducible polynomial. To date, the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equally-space polynomial (ESP). Unfortunately, there exist only a few irreducible ESPs in the range of interest for most of the applications, e.g., error-correcting codes, computer algebra, and elliptic curve cryptography. Furthermore, it is not always possible to find an irreducible trinomial of degree m in this range. For those cases, where neither an irreducible trinomial or an irreducible ESP exists, the use of irreducible pentanomials has been suggested. Irreducible pentanomials are abundant, 2and there are several eligible candidates for a given m. Inthis paper, we promote the use of two special types of irreducible pentanomials. We propose new Mastrovito and dual basis multiplier architectures based on these special irreducible pentanomials, and give rigorous analyses of their space and time complexity. Index Terms: Finite fields arithmetic, parallel multipliers, pentanomials, multipliers for GF(2m). 1

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Citation Context ...1 Introduction Efficient hardware implementations of the arithmetic operations in the Galois field GF(2 m ) are frequently desired in coding theory, computer algebra, and elliptic curve cryptosystems =-=[9, 10]-=-. For these implementations, the measure of efficiency is the space complexity, i.e., the number of XOR and AND gates, and the time complexity, i.e., the total gate delay of the circuit. The represent... |

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Citation Context ...ve been reported when the irreducible polynomial used to construct the field is either an equally-spaced polynomial defined as, p(x)=x m + x (k−1)d + ···+ x 2d + x d +1, (1) where m = kd,oratrinomial =-=[7, 8, 17, 15, 3]-=-. Unfortunately, irreducible equally-spaced polynomials (ESP) are very rare. There are only 81 m values less than 1024, such that an irreducible ESP of degree m exists [17]. On the other hand, an irre... |

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Citation Context ...ations. Several architectures have been reported for multiplication in GF(2 m ). For example, efficient bit-parallel multipliers for both polynomial and normal basis representation have been proposed =-=[4, 12, 6]-=-, including the Mastrovito multiplier [15]. Another technique which was first suggested in [1] is known as the dual basis multiplier [11, 2, 17, 18]. Conventional dual basis multipliers have the prope... |

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Citation Context ...1 Introduction Efficient hardware implementations of the arithmetic operations in the Galois field GF(2 m ) are frequently desired in coding theory, computer algebra, and elliptic curve cryptosystems =-=[9, 10]-=-. For these implementations, the measure of efficiency is the space complexity, i.e., the number of XOR and AND gates, and the time complexity, i.e., the total gate delay of the circuit. The represent... |

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Citation Context ...ations. Several architectures have been reported for multiplication in GF(2 m ). For example, efficient bit-parallel multipliers for both polynomial and normal basis representation have been proposed =-=[4, 12, 6]-=-, including the Mastrovito multiplier [15]. Another technique which was first suggested in [1] is known as the dual basis multiplier [11, 2, 17, 18]. Conventional dual basis multipliers have the prope... |

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Citation Context ... for multiplication in GF(2 m ). For example, efficient bit-parallel multipliers for both polynomial and normal basis representation have been proposed [4, 12, 6], including the Mastrovito multiplier =-=[15]-=-. Another technique which was first suggested in [1] is known as the dual basis multiplier [11, 2, 17, 18]. Conventional dual basis multipliers have the property that one of the input operands is give... |

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Citation Context ...x 2 + x +1 m 2 + m +2 TA +(3+⌈log 2 m⌉)TX Dual Basis x m + x n+2 + x n+1 + x n +1 m 2 +2m −⌈ m−2 2 ⌉ +3n − 4 TA +(3+⌈log 2 m⌉)TX Dual Basis x m + x n3 + x n2 + x n1 +1 m 2 +2m − 3 TA +(3+⌈log 2 m⌉)TX =-=[16]-=- While the multipliers based on trinomials and ESPs offer more advantageous designs, we have no choice but to consider other irreducible polynomials whenever irreducible trinomials or EPSs do not exis... |

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Citation Context ...omial and normal basis representation have been proposed [4, 12, 6], including the Mastrovito multiplier [15]. Another technique which was first suggested in [1] is known as the dual basis multiplier =-=[11, 2, 17, 18]-=-. Conventional dual basis multipliers have the property that one of the input operands is given in the polynomial basis while the other input is in the dual basis. The product is then obtained in the ... |

16 |
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Citation Context ...ve been reported when the irreducible polynomial used to construct the field is either an equally-spaced polynomial defined as, p(x)=x m + x (k−1)d + ···+ x 2d + x d +1, (1) where m = kd,oratrinomial =-=[7, 8, 17, 15, 3]-=-. Unfortunately, irreducible equally-spaced polynomials (ESP) are very rare. There are only 81 m values less than 1024, such that an irreducible ESP of degree m exists [17]. On the other hand, an irre... |

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Citation Context ...omial and normal basis representation have been proposed [4, 12, 6], including the Mastrovito multiplier [15]. Another technique which was first suggested in [1] is known as the dual basis multiplier =-=[11, 2, 17, 18]-=-. Conventional dual basis multipliers have the property that one of the input operands is given in the polynomial basis while the other input is in the dual basis. The product is then obtained in the ... |

4 |
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3 |
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Citation Context ... polynomial basis while the other input is in the dual basis. The product is then obtained in the dual basis [1]. In this paper, we use a new approach for dual basis multipliers that was suggested in =-=[13]-=-. In contrast to the conventional approach, the technique proposed in [13] assumes that both operands are given in the polynomial basis. This assumption yields less time and space complexity for certa... |

1 |
A new approach for dual basis multiplication
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Citation Context ...e polynomial basis while the other input is in the dual basis. The product is then obtained in the dual basis [1]. In this paper we use a new approach for dual basis multipliers that was suggested in =-=[13]-=-. In contrast to the conventional approach, the technique proposed in [13] assumes that both operands are given in the polynomial basis. These assumption yields less time and space complexity ∗ IEEE T... |

1 |
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Citation Context ...NTRODUCTION EFFICIENT hardware implementations of the arithmetic operations in the Galois field GFð2mÞ are frequently desired in coding theory, computer algebra, and elliptic curve cryptosystems [9], =-=[10]-=-. For these implementations, the measure of efficiency is the space complexity, i.e., the number of XOR and AND gates, and the time complexity, i.e., the total gate delay of the circuit. The represent... |