## Mastrovito multiplier for all trinomials (1999)

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Venue: | IEEE Transactions on Computers |

Citations: | 38 - 3 self |

### BibTeX

@ARTICLE{Sunar99mastrovitomultiplier,

author = {B. Sunar and Ç. K. Koç},

title = {Mastrovito multiplier for all trinomials},

journal = {IEEE Transactions on Computers},

year = {1999},

volume = {48},

pages = {522--527}

}

### Years of Citing Articles

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### Abstract

An efficient algorithm for the multiplication in GF(2 m) was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial x m +x+1 was given as m 2 − 1 XOR and m 2 AND gates. In this paper, we describe an architecture based on a new formulation of the multiplication matrix, and show that the Mastrovito multiplier for the generating trinomial x m + x n +1, where m � = 2n, also requires m 2 − 1 XOR and m 2 AND gates. However, m 2 − m/2 XOR gates are sufficient when the generating trinomial is of the form x m + x m/2 +1 for an even m. We also calculate the time complexity of the proposed Mastrovito multiplier, and give design examples for the irreducible trinomials x 7 + x 4 +1 and x 6 + x 3 +1.

### Citations

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Citation Context ...and hardware implementations of the basic arithmetic operations (addition, multiplication, and inversion) in the Galois field GF(2 m ) are desired in coding theory, computer algebra, and cryptography =-=[7, 4]-=-. The cryptographic applications include elliptic curve cryptosystems [8, 2], in which m is quite large, usually around several hundreds. The efficiency of an algorithm is often measured by the number... |

305 |
Elliptic Curve Public Key Cryptosystems
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Citation Context ...ultiplication, and inversion) in the Galois field GF(2 m ) are desired in coding theory, computer algebra, and cryptography [7, 4]. The cryptographic applications include elliptic curve cryptosystems =-=[8, 2]-=-, in which m is quite large, usually around several hundreds. The efficiency of an algorithm is often measured by the number of bit-level or word-level operations. In the hardware implementations, it ... |

72 |
VLSI Architectures for Computation in Galois Fields
- Mastrovito
- 1991
(Show Context)
Citation Context ... polynomial c(x). In practice, the multiplication and the reduction steps are often combined for efficiency reasons. An architecture for performing the field multiplication was proposed by Mastrovito =-=[5, 6]-=-. In this method, we represent the computation of d(x) as a matrix-vector product d = Mb, where (2m − 1) × m dimensional matrix M consists of the coefficients of the polynomial a(x). We then obtain an... |

69 | Efficient algorithms for elliptic curve cryptosystems
- Guajardo, Paar
- 1997
(Show Context)
Citation Context ...ultiplication, and inversion) in the Galois field GF(2 m ) are desired in coding theory, computer algebra, and cryptography [7, 4]. The cryptographic applications include elliptic curve cryptosystems =-=[8, 2]-=-, in which m is quite large, usually around several hundreds. The efficiency of an algorithm is often measured by the number of bit-level or word-level operations. In the hardware implementations, it ... |

62 |
Efficient VLSI Architectures for Bit-Parallel Computation in Galois Fields
- Paar
- 1994
(Show Context)
Citation Context .... The product c(x) is computed using the matrix-vector product c = Zb. The space complexity of the multiplier for the special generating trinomial x m +x+1 is shown to be m 2 −1 XOR and m 2 AND gates =-=[5, 6, 9, 10]-=-. Paar [11] conjectured that the space complexity of the Mastrovito multiplier would be the same for all trinomials x m +x n +1, where 1 ≤ n ≤ m−1. In this paper, we describe an architecture for the M... |

58 |
Applications of Finite Fields
- Menezes
(Show Context)
Citation Context ...and hardware implementations of the basic arithmetic operations (addition, multiplication, and inversion) in the Galois field GF(2 m ) are desired in coding theory, computer algebra, and cryptography =-=[7, 4]-=-. The cryptographic applications include elliptic curve cryptosystems [8, 2], in which m is quite large, usually around several hundreds. The efficiency of an algorithm is often measured by the number... |

43 | Low-complexity bit-parallel canonical and normal basis multipliers for a class of finite fields
- KOC, SUNAR
- 1998
(Show Context)
Citation Context ...is obtained by shifting down the second column m−n positions. The fourth column is obtained by shifting down the third column m − n positions, and so on. Following the construction method proposed in =-=[3]-=-, we decompose the Mastrovito matrix Z as the sum of two m × m matrices X and Y , i.e., Z = X + Y , where X is the upper m rows of the matrix M. The matrix X is an m × m Toeplitz matrix, i.e., a matri... |

16 |
VLSI architectures for multiplication over finite field GF(2m
- Mastrovito
- 1988
(Show Context)
Citation Context ... polynomial c(x). In practice, the multiplication and the reduction steps are often combined for efficiency reasons. An architecture for performing the field multiplication was proposed by Mastrovito =-=[5, 6]-=-. In this method, we represent the computation of d(x) as a matrix-vector product d = Mb, where (2m − 1) × m dimensional matrix M consists of the coefficients of the polynomial a(x). We then obtain an... |

4 |
A new architecture for a paralel finite field multiplier with low complexity based on composite fields
- Paar
- 1996
(Show Context)
Citation Context .... The product c(x) is computed using the matrix-vector product c = Zb. The space complexity of the multiplier for the special generating trinomial x m +x+1 is shown to be m 2 −1 XOR and m 2 AND gates =-=[5, 6, 9, 10]-=-. Paar [11] conjectured that the space complexity of the Mastrovito multiplier would be the same for all trinomials x m +x n +1, where 1 ≤ n ≤ m−1. In this paper, we describe an architecture for the M... |

4 | Private communication
- Paar
- 1997
(Show Context)
Citation Context ...s computed using the matrix-vector product c = Zb. The space complexity of the multiplier for the special generating trinomial x m +x+1 is shown to be m 2 −1 XOR and m 2 AND gates [5, 6, 9, 10]. Paar =-=[11]-=- conjectured that the space complexity of the Mastrovito multiplier would be the same for all trinomials x m +x n +1, where 1 ≤ n ≤ m−1. In this paper, we describe an architecture for the Mastrovito t... |

3 |
VLSI architectures for multiplication over nite eld GF(2 m
- Mastrovito
- 1988
(Show Context)
Citation Context ... and the reduction steps are often combined for e - ciency reasons. An architecture for performing the eld multiplication was proposed by Mastrovito IEEE Transactions on Computers, to appear, 1999. 1=-=[5, 6]-=-. In this method, we represent the computation of d(x) as a matrix-vector product d = Mb, where (2m , 1) m dimensional matrix M consists of the coe cients of the polynomial a(x). We then obtain an m m... |

1 |
A new architecture for a paralel nite eld multiplier with low complexity based on composite elds
- Paar
- 1996
(Show Context)
Citation Context .... The product c(x) is computed using the matrix-vector product c = Zb. The space complexity of the multiplier for the special generating trinomial x m +x+1 is shown to be m 2 , 1XOR and m 2 AND gates =-=[5, 6, 9, 10]-=-. Paar [11] conjectured that the space complexity of the Mastrovito multiplier would be the same for all trinomials x m +x n +1, where 1 n m,1. In this paper, we describe an architecture for the Mastr... |