Some Side Channel Attacks On Elliptic Curve Cryptosystem

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Some Side Channel AttacksOn Elliptic Curve
Cryptosystem

• Tsuyoshi Takagi
• Technische Universtät Darmstadt
• Fachbereich Informatik
• http//www.informatik.tu-darmstadt.de/KP/

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Overview

• Elliptic Curve Cryptosystems (ECC)
• Power Analysis against ECC
• Goubins Attack
• Zero-Value Point Attack
• Smarts Isogeny Defense

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Elliptic Curve

• Elliptic curve on binary field
• Elliptic curve on prime field

All points satisfying and infinity point
Abelian group by the following addition
group identity
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Addition Formulae on EC

• EC Doubling (ECDBL)

• EC Addition (ECADD)

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Addition Formulae on EC(Jacobian Coordinates)

• ECDBL

• ECADD

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Scalar Multiplication on EC

• Scalar Multiplication
• Binary Method
• For downto
• ECDBL
• if , ECADD
• Return

binary representation
Ex.
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Power Analysis

• Simple Power Analysis (SPA)
• Observe the power consumption of devices in a
  single computation and detect the secret key
• Differential Power Analysis (DPA)
• Observe many power consumptions and analyze
  these information together with statistic tools

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SPA against ECC (Coron 1999)

• Binary method
• For downto
• ECDBL
• if , ECADD
• Return

• ECDBL
• ECADD

Ex.
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SPA Countermeasure (Coron 1999)

• Scalar Multiplication
• Double-and-add-always method
• For downto
• ECDBL
• ECADD
• Return

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Double-and-add-always method(Coron 1999)
Ex.
dummy
dummy
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Experiment by Coron (CHES1999)

• We gather many power consumption of computing
  4Pi.
• 4Pi is computed if and only if the most 2nd
  bit of d is 0.
• (2) Let si be any specific bit of 4Pi. We use the
  following
• correlation function g(t) Power(si0)
  Power(si 1)

Cited from Coron, Resistant against Resistance
against Differential Power Analysis for Elliptic
Curve Cryptosystems, CHES 1999, pp.292-302,
1999.
If 4Pi is computed, there is a difference
between Power(si0) and Power(si1).
If point 4Pi is never computed, there is no spike
in the graph.
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DPA against Double-and-add-always method (Coron
1999)

• is fixed and the attacker can choose
• ElGamal encryption, single-pass ECDH
• Power consumption of double-and-add-always method
  for each input looks same, but is slightly
  different.
• Power consumption is correlated to any bit of
  processing point.

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DPA Countermeasure (Coron 1999)

• Randomize point representation in Jacobian
  coordinates
• Scalar Multiplication
• Choose randomly
• Compute
• Return

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DPA Countermeasure(Joye-Tymen 2001)

• Use a random isomorphic curve to the original
  curve
• Scalar multiplication
• Choose randomly
• and
• Compute on
• Return

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Goubins Attack (Goubin 2003)

• Cannot randomize the points and
• Assume
• Input

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Condition of Goubins Attack

• point
• Not exist in elliptic curve
  of prime order.
• If exist, the input can be discarded.
• point
• is quadratic residue modulo

Order is 2
If b is random, this probability is about 50
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Goubins Points on Standard Curves

• SECG Curves

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Isogeny of Elliptic Curve

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Smarts Isogeny Defense(Smart 2003)

• Countermeasure against Goubins attack
• Isogeny of degree

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Smarts Isogeny Defense against Goubins attack
efficient curve
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ZVP Attack (Akishita-Takagi 2003)

• Zero-value point attack
• Generalization of Goubins attack
• Goubins attack pays attention to only
  representation of processing points.
• We consider that intermediate values of addition
  formulae are equal to 0.
• If the point has no zero-value coordinate, the
  intermediate values might become zero.

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ZVP in ECDBL

• ECDBL

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ZVP Attack

• that satisfy
• Assume
• Input

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ZVP in ECDBL

• (ED1)
• (ED2)
• (ED3) the order of P is equal to 3
• (ED4)
• (ED5)

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ZVP in ECADDJ

• ECADDJ

division polynomial (can solve for only small c)
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ZVP on Standard Curves

• SECG Curves

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Isogeny Defense against ZVP Attack is not secure
(Akishita-Takagi 2004)

• We cannot find preferred isogeny degree less than
  107 for three curves. Why?

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Smarts Isogeny Defense against ZVP attack

• (ED1)

efficient curve
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Choice of the base field ?

• A class of curves that satisfies the following
  three conditions is insecure against the ZVP
  attack.
• E has prime order