An Inverter Architecture for ECCGF2m Based on the Steins Algorithm

1
An Inverter Architecture for ECC-GF(2m)Based on
theSteins Algorithm
2
Objectives

• To present the development of a modular inverter
  for elliptic curves, implemented by programmable
  circuit
• To show the viability to implement this inverter
  by combinatorial circuit
• To talk about the difficulties found during the
  development of the inverter
• To explain the solutions found to allow the
  development of the inverter.

3
Justifications

• The interest about digital circuits
• The today importance of projects that work with
  cryptography implemented by hardware
• The need to create a device in order to increase
  the performance of programs that work with
  asymmetric cryptography
• Challenges speed X area
• combinatorial circuits X affine coordinates.

4
Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

5
Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

6
Introduction

• Information
• QS (symmetric cryptography)
• QP and k (asymmetric cryptography)
• Q kP
• 2P and P doubling and addition of P
• , , x2, mod, x-1 finite field arithmetic.

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A Key-Exchange Example

• cli the client side of the communication
• serv the server side of the communication
• P an element previously chosen by cli and serv
• k a private key
• QP a public key
• QS a secret key
• Q and P are elements of the same type.

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Key-Exchange

• The cli generates at random an integer number
  kcli
• The serv generates at random an integer number
  kserv
• The cli calculates QPcli kcli P
• The serv calculates QPserv kserv P
• The cli sends QPcli to the serv
• The serv sends QPserv to the cli
• The cli calculates QS kcli QPserv
• The serv calculates QS kserv QPcli.

9
Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

10
Elliptic Curves

• They are represented by the equation
• y2 xy x3 ax2 b
• for which
• x and y represent the point coordinates
• a and b define an elliptic curve over GF(2m).

11
The Main Operation of the Elliptic Curve
Algorithms

• Q kP
• for which
• k is an integer number
• P is an elliptic curve point of coordinates x and
  y
• Q is an elliptic curve point of coordinates x and
  y
• P(Px,Py) and Q(Qx,Qy) are points represented by
  affine coordinates and polynomial basis.

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Double-and-Add Algorithm
Q 10P 1010 gt ((2P)2P)2 gt 10P Q
17P 10001 gt (((2P)2)2)2P gt 17P
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Point Doubling

• S PX ((PY) / (PX)) mod p
• QX (S2 S a) mod p
• QY (S(PX QX) PY QX) mod p

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Point Addition

• S ((PY PY) / (PX PX)) mod p
  
• QX (S2 S PX PX a) mod p
• QY (S(PX QX) PY QX) mod p

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Comparison BetweenDoublings . and Additions .

• S Px (Py / Px) mod p.
• S ((Py Py) / (Px Px)) mod p.
• Qx (S2 S a) mod p.
• Qx (S2 S Px Px a) mod p.
• Qy (S(Px Qx) Py Qx) mod p.
• Qy (S(Px Qx) Py Qx) mod p.

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Unification of the Equations

• S F ((G PY) / (H PX)) mod p
• QX (S2 S PX PX a) mod p
• QY (S(PX QX) PY QX) mod p

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Attribution of Values

• Point Doubling
• F PX
• G 0
• H 0.

• Point Addition
• F 0
• G PY
• H PX.

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Finite Field Arithmetic

• Sum
• Multiplication
• Module
• Squaring
• Modular Inversion.

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Unification of the Equations

• S F ((G PY) / (H PX)) mod p
• QX (S2 S PX PX a) mod p
• QY (S(PX QX) PY QX) mod p

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Optimization of the Equations

• S F ((G PY) (H PX)-1) mod p
• QX (S2 S PX PX a) mod p
• QY (S(PX QX) PY QX) mod p

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Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

22
Modular Division Methods

• Gaussian Elimination
• Fermats Theorem
• MDC (Euclides or Stein).

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The Steins Algorithm

• (A, B, U, V) ? (Px, p, 1, 0)
• while A ! 0 and B ! 1
• if A0 1
• if deg(A) gt deg(B)
• (A, B) ? (A B, U V)
• else
• (A, B, U, V) ? (A B, A, U V, U)
• endif
• endif
• (A, U) ? (A / 2, (U / 2) mod p)
• endwhile

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The Steins Algorithm Optimized

• (A, B, U, V, DCC, Flag, slice) ? (Px, p, 1, 0,
  2, 1, 2m-1)
• while slice gt 0
• if A0 1
• if Flag 1 and DCC0 0
• (A, B, U, V, Flag) ? (A B, A, U
  V, U, 0)
• else
• (A, B) ? (A B, U V)
• endif
• endif
• (A, U) ? (A / 2, (U / 2) mod p)
• if Flag 0 and DCC0 0
• DCC ? DCC / 2
• else
• (DCC, Flag) ? ((DCC 2), 1)
• endif
• slice ? slice -1
• endwhile

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FLAG and AUX handling
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DCC handling
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A handling
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U handling
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B handling
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V handling
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Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

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A combinatorial Circuit forPoint Doubling and
Point Addition
Modular Inversion
Other Operations
Qx
Px
Px
Qy
Py
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Unification of the Equations

• S F ((G PY) (H PX)-1) mod p
• QX (S2 S PX PX a) mod p
• QY (S(PX QX) PY QX) mod p

34
Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

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The Cryptosystem
CPU
PC-board
Other Operations
Modular Inversion
EP2S180F1020C4
EP2S90F1508C3
Registers
Key-Exchange Example
P
P
Px
Qx
S
Px
Qy
PC Bus
36
Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

37
General Results
38
Modular Inversion Time for Different
Implementations
39
Scalar Multiplication Time for Different
Implementations
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Results of Our Key-Exchange Example
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Summary

• Introduction
• ECC-GF(2m)
• A combinatorial Circuit for Modular Inversion
• A combinatorial Circuit for Point Doubling and
  Point Addition
• A Cryptosystem Implemented by Programmable
  Circuits
• Results
• Conclusions.

42
Conclusions

• The Objectives was successfully achieved
• High performance to the detriment of a small
  area
• High-speed and high-density combinatorial
  circuits
• Our inverter and cryptosystem were made valid
• Our cryptosystem allows high frequency of key
  exchange.

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The End