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Advanced Information Security 4 FIELD ARITHMETIC
Dr. Turki F. Al-Somani2015
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Module Outlines
Finite Field Arithmetic GF(p) Arithmetic GF(2m) Arithmetic
Polynomial basis Normal basis
Addition/subtraction Squaring Multiplication Inversion
Summary
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Finite Field Arithmetic
In abstract algebra, a finite field is a field that contains only finitely many elements.
Finite fields are important in number theory, algebraic geometry, Galois theory, coding theory, and cryptography.
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Finite Field Arithmetic (contd.)
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Finite Field Arithmetic (contd.)
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Finite Field Arithmetic (contd.)
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Finite Field Arithmetic (contd.)
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Finite Field Arithmetic (contd.)
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GF(2m) Arithmetic
The finite GF(2m) field has particular importance in cryptography since it leads to particularly efficient hardware implementations.
Elements of the field are represented in terms of a basis.
Most implementations use either a Polynomial Basis or a Normal Basis.
Normal basis is more suitable for hardware implementations than polynomial basis since operations are mainly comprised of rotation, shifting and exclusive-OR operations which can be efficiently implemented in hardware.
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Polynomial Basis
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Polynomial Basis
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Normal Basis
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Normal Basis (contd.)
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Normal Basis (contd.)
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Optimal Normal Basis
An optimal normal basis (ONB) is one with the minimum number of terms, or equivalently, the minimum possible number of nonzero λij
This value is 2m-1, and since it allows multiplication with minimum complexity, such a basis would normally lead to more efficient hardware implementations.
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Optimal Normal Basis (Contd.)
Note: Type 1 is circled.
Optimal Normal Basis Types
Now CN=2n-1
Type I:
Rule 2 means: for every i in the range [0, n-1], (2k mod n+1) must result in a unique integer in the range [1, n].
Cont.
Type II:
Rule 2a means that every 2k mod 2n+1, in the range [1 to 2n]. Therefore 2 is called the generator for all the possible locations
in the 2n+1 field Rule 2b means that even if 2k does not generate every
element in the range [1, 2n], however, half of points in the field of form by rule 2a can be hit. It is because SQR(2k) can be taken.
The points generated by rule 2b are in the form of perfect squares.
ONB Type I & II (n ≤ 230)
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Survey Paper (2006)
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NB Multiplication
Multiplication is more complicated than addition and squaring operations in finite field arithmetic.
An efficient multiplier is highly needed and is the key for efficient finite field computations.
Finite filed multipliers using normal basis can be classified into two main categories: 𝜆-matrix based multipliers Conversion based multipliers
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𝜆-matrix based multipliers
Massey and Omura Multiplier Hasan et. al. Multiplier Gao and Sobelman Multiplier Reyhani-Masoleh and Hasan Multiplier
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Example: Type I
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Example: Type II
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Massey and Omura Multiplier
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Hasan et. al. Multiplier
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Gao and Sobelman Multiplier
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Reyhani-Masoleh and Hasan Multiplier
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Comparisons
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Conversion based multipliers Sunar and Koc Multiplier Wu et. al. Multiplier
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Sunar and Koc Multiplier
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Wu et. al. Multiplier
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Comparisons
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Normal Basis Inversion
Inversion algorithms:
Standard algorithms
Exponent Decomposing algorithms
Exponent Grouping inversion algorithms
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Standard Algorithms
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Exponent Decomposing Algorithms
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Exponent Decomposing Algorithms (contd.)
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Exponent Decomposing Algorithms (contd.)
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Exponent Grouping inversion Algorithms
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Exponent Grouping inversion Algorithms (contd.)
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Exponent Grouping inversion Algorithms (contd.)
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Comparisons
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Pipelining Paper (2009)
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Pipelining Paper (2009)
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UQU Pipelining Paper (2010)
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Systolic Arrays Paper (2011)
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IEEE VLSI Systolic Arrays Paper (2014)
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Summary
Efficient computations in finite fields and their architectures are important in many applications, including coding theory, computer algebra systems, and public-key cryptosystems (e.g., elliptic curve cryptosystems (ECC).
The most commonly used basis are polynomial basis and normal basis.
Normal basis is more suitable for hardware implementations than polynomial basis since operations in normal basis representation are mainly comprised of rotation, shifting and exclusive-ORing which can be efficiently implemented in hardware.
THANKS & GOOD LUCK NEXT IS: 5 ECC CRYPTOGRAPHY
Dr. Turki F. Al-Somani2015