Elliptic Curve Cryptography on FPGAs

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Elliptic Curve Cryptography on FPGAs

• Kimmo Järviinen
• Helsinki University of Technology
• Signal Processing Laboratory
• Otakaari 5A, 02150 Espoo, Finland

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Outline 1

• Introduction
• Polynomial Math over Finite Fields
• Normal Basis Mathematics
• Galois Fields
• Polynomial Math over Finite Fields
• Normal Basis Mathematics
• Elliptic Curves
• Elliptic Curves over Real Numbers
• Elliptic Curves over Galois Fields
• Multiplication over an Elliptic Curve
• Elliptic Curve Discrete Logarithm Problem

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Outline 2

• Elliptic Curve Cryptography
• Elliptic Curve Diffie-Hellman (ECDH)
• Elliptic Curve Digital Signature Algorithm
• Standards
• Patents
• Implementations of Elliptic Curve Cryptography

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Outline 3

• Implementations of Elliptic Curve Cryptography
• Published ECC Implementations on FPGAs
• Software implementations

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Introduction

• Elliptic curves for cryptography was introduced
  in 1985.
• They did not invent a new cryptographic
  algorithm, but they implemented existing
  public-key algorithms using elliptic curves.

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Advantages of Elliptic Curve Cryptography (ECC)

• Offers better security with a shorter key length.
  E.g., level of secuirty achieved with ECC using a
  173-bit key is equivalent to conventional
  public-key cryptography (e.g., RSA) using a
  1024-bit key

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Advantages of ECC 2

• Shorter key lengths mean less memory need in key
  storage
• Fewer hardware resources.

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The Mathematics of ECC

• Deeper and more difficult than mathematics used
  for conventional cryptography

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Groups

• A group consists of a set G and a binary
  operation ? such that for any elements a,b,c in G
• a?b is in G (G is closed under ?)
• (a?b)?c a?(b ?c) (? is associative)
• There exists an e in G such that
• a?e e?a for all a in G (G has an identity
  e.)
• For each a in G, there exists an a in G such
  that a?a a?a e.
• G is Abelian or commutative if a?b b ?a for all
  a and b in G

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Examples of Groups

• The set Z of integers under addition
• The set Q of rational numbers under addition
• The set Q-0 under multiplication
• The set R of real numbers under addition
• The set R-0 under multiplication

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Fields

• A field consists of a set F and two binary
  operations and such that
• F is an abelian group under
• F-0, where 0 is the identity of F under is an
  abelian group under
• is distributive over , i.e.,
• a(bc) ab ac

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Galois Fields

• A Galois field has a finite number of elements
• Example The integers modulo a prime number p
• The number of elements in a Galois field is
  always the power of a prime.
• For any power of a prime q, there is exactly one
  field with q elements and it is denoted by
  GF(q).