### Abstract

We introduce a new class of irreducible pentanomials over F
_{2} of the form f(x) = x
^{2}
^{b}
^{+}
^{c}+ x
^{b}
^{+}
^{c}+ x
^{b}+ x
^{c}+ 1. Let m= 2 b+ c and use f to define the finite field extension of degree m. We give the exact number of operations required for computing the reduction modulo f. We also provide a multiplier based on Karatsuba algorithm in F
_{2}[x] combined with our reduction process. We give the total cost of the multiplier and found that the bit-parallel multiplier defined by this new class of polynomials has improved XOR and AND complexity. Our multiplier has comparable time delay when compared to other multipliers based on Karatsuba algorithm.

Original language | English |
---|---|

Pages (from-to) | 359–373 |

Number of pages | 15 |

Journal | Journal of Cryptographic Engineering |

Volume | 9 |

Issue number | 4 |

Early online date | 9 Nov 2018 |

DOIs | |

Publication status | Published - 1 Nov 2019 |

### Fingerprint

### Keywords

- Finite fields
- Irreducible pentanomials
- Modular reduction
- Polynomial multiplication

### Cite this

*Journal of Cryptographic Engineering*,

*9*(4), 359–373. https://doi.org/10.1007/s13389-018-0197-6

}

*Journal of Cryptographic Engineering*, vol. 9, no. 4, pp. 359–373. https://doi.org/10.1007/s13389-018-0197-6

**A new class of irreducible pentanomials for polynomial-based multipliers in binary fields.** / Banegas, Gustavo (Corresponding author); Custódio, Ricardo; Panario, Daniel.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - A new class of irreducible pentanomials for polynomial-based multipliers in binary fields

AU - Banegas, Gustavo

AU - Custódio, Ricardo

AU - Panario, Daniel

PY - 2019/11/1

Y1 - 2019/11/1

N2 - We introduce a new class of irreducible pentanomials over F 2 of the form f(x) = x 2 b + c+ x b + c+ x b+ x c+ 1. Let m= 2 b+ c and use f to define the finite field extension of degree m. We give the exact number of operations required for computing the reduction modulo f. We also provide a multiplier based on Karatsuba algorithm in F 2[x] combined with our reduction process. We give the total cost of the multiplier and found that the bit-parallel multiplier defined by this new class of polynomials has improved XOR and AND complexity. Our multiplier has comparable time delay when compared to other multipliers based on Karatsuba algorithm.

AB - We introduce a new class of irreducible pentanomials over F 2 of the form f(x) = x 2 b + c+ x b + c+ x b+ x c+ 1. Let m= 2 b+ c and use f to define the finite field extension of degree m. We give the exact number of operations required for computing the reduction modulo f. We also provide a multiplier based on Karatsuba algorithm in F 2[x] combined with our reduction process. We give the total cost of the multiplier and found that the bit-parallel multiplier defined by this new class of polynomials has improved XOR and AND complexity. Our multiplier has comparable time delay when compared to other multipliers based on Karatsuba algorithm.

KW - Finite fields

KW - Irreducible pentanomials

KW - Modular reduction

KW - Polynomial multiplication

UR - http://www.scopus.com/inward/record.url?scp=85073577324&partnerID=8YFLogxK

U2 - 10.1007/s13389-018-0197-6

DO - 10.1007/s13389-018-0197-6

M3 - Article

VL - 9

SP - 359

EP - 373

JO - Journal of Cryptographic Engineering

JF - Journal of Cryptographic Engineering

SN - 2190-8508

IS - 4

ER -