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Introduction Arithmetic in GF ( 2 m ) Summary - results, comments, future prospects Arithmetic operators on GF ( 2 m ) for cryptographic applications: performance - power consumption - security tradeoffs Danuta Pamua 17th December 2012


  1. Introduction Arithmetic in GF ( 2 m ) Summary - results, comments, future prospects Arithmetic operators on GF ( 2 m ) for cryptographic applications: performance - power consumption - security tradeoffs Danuta Pamuła 17th December 2012 Arithmetic operators on GF ( 2 m ) Danuta Pamuła

  2. Arithmetic operators on GF ( 2 m ) - applications, requirements Introduction Arithmetic in GF ( 2 m ) Arithmetics in GF ( 2 m ) and ECC Summary - results, comments, future prospects Thesis statement 1. Introduction Arithmetic operators on GF ( 2 m ) - application, requirements Arithmetics in GF ( 2 m ) and elliptic curve cryptography Formulated thesis Arithmetic operators on GF ( 2 m ) Danuta Pamuła

  3. Arithmetic operators on GF ( 2 m ) - applications, requirements Introduction Arithmetic in GF ( 2 m ) Arithmetics in GF ( 2 m ) and ECC Summary - results, comments, future prospects Thesis statement Arithmetic operators on GF ( 2 m ) - applications Cryptography : symmetric: AES, ... assymetric: RSA, ... , Elliptic Curve Cryptography (ECC) . error correcting codes computational biology (e.g. modelisation of genetic network) computational and algorithmic aspects of commutative algebra digital signal processing ... Arithmetic operators on GF ( 2 m ) Danuta Pamuła

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  7. Arithmetic operators on GF ( 2 m ) - applications, requirements Introduction Arithmetic in GF ( 2 m ) Arithmetics in GF ( 2 m ) and ECC Summary - results, comments, future prospects Thesis statement Thesis It is possible to create efficient and secure against some side-channel power analysis attacks GF ( 2 m ) arithmetic operators dedicated to reconfigurable hardware. Arithmetic operators on GF ( 2 m ) Danuta Pamuła

  8. Introduction Basics Arithmetic in GF ( 2 m ) Addition Summary - results, comments, future prospects Multiplication 2. Arithmetic in GF ( 2 m ) - efficient and secure hardware solutions Basics Addition Multiplication Proposed solutions Arithmetic operators on GF ( 2 m ) Danuta Pamuła

  9. Introduction Basics Arithmetic in GF ( 2 m ) Addition Summary - results, comments, future prospects Multiplication Arithmetics in GF ( 2 m ) PARAMETERS ւ ↓ ց irreducible basis (element polynomial f ( x ) field representation) (field generator) size m ↓ ց ւ standard NIST, SECG normal, GNB, ONB, cryptographic standards dual (FIPS 186-3, SEC 1, SEC 2) GNB, ONB - Gaussian/Optimal Normal Basis, NIST - National Institute of Standards and Technology, SECG - Standards for Efficient Cryptography Group Arithmetic operators on GF ( 2 m ) Danuta Pamuła

  10. Introduction Basics Arithmetic in GF ( 2 m ) Addition Summary - results, comments, future prospects Multiplication Addition in GF ( 2 m ) Addition = XOR of binary polynomials ✞ ☎ c = a XOR b ✝ ✆ Propositions (data in processor are passed in words (16, 32-bit): [ 1 / 2 ] Add every two incoming words of a , b , accumulate partial results in register c (1) or in BlockRAM (2); [ 3 ] Wait for all words of a , b , add a and b ; field size (1)(Virtex-6) (2)(Virtex-6) m [LUT] [MHz] [LUT] [MHz] 163 21 771 26 562 233 21 771 26 562 283 22 767 28 560 409 22 767 28 560 571 24 578 31 558 Arithmetic operators on GF ( 2 m ) Danuta Pamuła

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  12. Introduction Basics Arithmetic in GF ( 2 m ) Addition Summary - results, comments, future prospects Multiplication Multiplication - Mastrovito matrix approach Idea: c = Mb , where M is a m × m Mastrovito matrix Problems: 1 Size of matrix M ( m = 163 , 233 , 283 , 409 , 571 ) 2 Construction of matrix M (iterative algorithm, combination of matrices A and R ) 3 Storing matrix M 4 Multiplication of matrix M by vector b Arithmetic operators on GF ( 2 m ) Danuta Pamuła

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