Low complexity bit-parallel polynomial basis multipliers over binary fields for special irreducible pentanomials
Finite field GF(2m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, error coding theory and computer algebra. Among finite field arithmetic operations, GF(2m) multiplication is of special interest because it is considered the most importa...
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| Published in | Integration (Amsterdam) Vol. 46; no. 2; pp. 197 - 210 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Amsterdam
Elsevier B.V
01.03.2013
Elsevier |
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| ISSN | 0167-9260 1872-7522 |
| DOI | 10.1016/j.vlsi.2011.12.006 |
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| Abstract | Finite field GF(2m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, error coding theory and computer algebra. Among finite field arithmetic operations, GF(2m) multiplication is of special interest because it is considered the most important building block. GF(2m) multipliers present reduced space and time complexities when the field is generated by some special irreducible polynomials. Among these, irreducible pentanomials of degree m are specially important because they are abundant and there are several eligible candidates for a given m. In this paper, we consider bit-parallel polynomial basis multipliers over the finite field GF(2m) generated using type 2 irreducible pentanomials, for which explicit formulas and algorithms for the computation of the products are given. In this contribution, two new subclasses of type 2 irreducible pentanomials are also introduced. The theoretical complexity analysis proves that the bit-parallel multipliers here presented have the lowest number of XOR gates known to date for similar polynomial basis multipliers based on this type of irreducible pentanomials, while the number of AND gates and the time complexity match the best known results found in the literature. |
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| AbstractList | Finite field GF (2 m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, error coding theory and computer algebra. Among finite field arithmetic operations, GF (2 m) multiplication is of special interest because it is considered the most important building block. GF (2 m) multipliers present reduced space and time complexities when the field is generated by some special irreducible polynomials. Among these, irreducible pentanomials of degree m are specially important because they are abundant and there are several eligible candidates for a given m. In this paper, we consider bit-parallel polynomial basis multipliers over the finite field GF (2 m) generated using type 2 irreducible pentanomials, for which explicit formulas and algorithms for the computation of the products are given. In this contribution, two new subclasses of type 2 irreducible pentanomials are also introduced. The theoretical complexity analysis proves that the bit-parallel multipliers here presented have the lowest number of XOR gates known to date for similar polynomial basis multipliers based on this type of irreducible pentanomials, while the number of AND gates and the time complexity match the best known results found in the literature. Finite field GF(2m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, error coding theory and computer algebra. Among finite field arithmetic operations, GF(2m) multiplication is of special interest because it is considered the most important building block. GF(2m) multipliers present reduced space and time complexities when the field is generated by some special irreducible polynomials. Among these, irreducible pentanomials of degree m are specially important because they are abundant and there are several eligible candidates for a given m. In this paper, we consider bit-parallel polynomial basis multipliers over the finite field GF(2m) generated using type 2 irreducible pentanomials, for which explicit formulas and algorithms for the computation of the products are given. In this contribution, two new subclasses of type 2 irreducible pentanomials are also introduced. The theoretical complexity analysis proves that the bit-parallel multipliers here presented have the lowest number of XOR gates known to date for similar polynomial basis multipliers based on this type of irreducible pentanomials, while the number of AND gates and the time complexity match the best known results found in the literature. |
| Author | Imaña, José L. Hermida, Román Tirado, Francisco |
| Author_xml | – sequence: 1 givenname: José L. surname: Imaña fullname: Imaña, José L. email: jluimana@dacya.ucm.es – sequence: 2 givenname: Román surname: Hermida fullname: Hermida, Román email: rhermida@dacya.ucm.es – sequence: 3 givenname: Francisco surname: Tirado fullname: Tirado, Francisco email: ptirado@dacya.ucm.es |
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| CitedBy_id | crossref_primary_10_1109_TCSI_2015_2388842 crossref_primary_10_1007_s41635_019_00087_5 crossref_primary_10_1016_j_vlsi_2013_03_001 crossref_primary_10_1007_s13389_018_0197_6 crossref_primary_10_1109_TC_2017_2778730 crossref_primary_10_1049_el_2016_0577 crossref_primary_10_1109_TCSI_2015_2495758 crossref_primary_10_1109_TCSI_2015_2500419 |
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| Keywords | GF(2m) Bit-parallel multipliers Finite field arithmetic Matrix decomposition Polynomial basis Irreducible pentanomials Complexity Finite field Arithmetic circuit XOR circuit Logic gate ) Space time Algorithm Field experiment Logic circuit Coding Integrated circuit Arithmetic operation Cryptography Computer algebra Time complexity Multiplying circuits GF |
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| SubjectTerms | Algorithms Applied sciences Arithmetic Bit-parallel multipliers Blocking Circuit properties Complexity Cryptography Design. Technologies. Operation analysis. Testing Digital circuits Electric, optical and optoelectronic circuits Electronic circuits Electronics Exact sciences and technology Finite field arithmetic formula omitted Gates Integrated circuits Irreducible pentanomials Mathematical analysis Matrix decomposition Multipliers Polynomial basis Semiconductor electronics. Microelectronics. Optoelectronics. Solid state devices |
| Title | Low complexity bit-parallel polynomial basis multipliers over binary fields for special irreducible pentanomials |
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