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Introduction Randomized Algorithms Input Driven Hidden Markov Models Analysis of the Oswald Aigner Randomizations Conclusion

Hidden Markov Model CryptanalysisC. Karlof, D. Wagner

presented by: Stephan Neumann

January 23th, 2009


Outline

1 IntroductionMotivationAttacker Model

2 Randomized AlgorithmsBinary algorithm for scalar multiplicationRandomized binary algorithm for scalar multiplication

3 Input Driven Hidden Markov ModelsDefinitionKey Inference ProblemKey Inference Problem Approaches

4 Analysis of the Oswald Aigner RandomizationsOswald Aigner RandomizationResults

5 Conclusion


Motivation

Motivation

Side channel attacks exploit physical characteristics

Side channel attacks often successful because of high correlationbetween physical characteristics and key.

Randomization as technique to disassociate the side channelinformation from the key.

No evaluation techniques for randomized countermeasures

Goal of the paper

Evaluation technique for randomized countermeasures against side channelattacks


Attacker Model

The Attacker

Capabilities:Able to observe physical output of a system (by power consumption,time consumption, etc.)

Able to observe outputs of multiple executions of an algorithm with thesame key

Key inference problem:Given a sequence of observation chains, the attacker tries to infer theused key of the algorithm


Attacker Model

General Approach

1 Develop generic side channel attack against randomized algorithms2 Apply attack to randomized countermeasures against side channel

attacks

Interpretation

A countermeasure is good if the attack performs poorly


Outline





5 Conclusion


Binary algorithm for scalar multiplication

Binary algorithm for scalar multiplication

s0/D s1/AD

0:(1)1:(1)

1:(1)

0:(1)

Assumption

Attacker can distinguish between Addition + Doubling (AD) and Doubling (D)

Attacker can distinguish between k(i) = 0 and k(i) = 1.


Randomized binary algorithm for scalar multiplication

Naive randomized binary multiplication algorithm

s0/D

s1/AD

s2/AD0:(0.5)

0:(0.5)

1:(1)

0:(0.5)

0:(0.5)

1:(1)

1:(1)

0:(0.5)

0:(0.5)

Assumption


No one-to-one correspondence between key and output, but someinformation leaks.




s0/D

s1/AD

s2/AD0:(0.5)

0:(0.5)

1:(1)

0:(0.5)

0:(0.5)

1:(1)

1:(1)

0:(0.5)

0:(0.5)

Assumption


No one-to-one correspondence between key and output

, but someinformation leaks.




s0/D

s1/AD

s2/AD0:(0.5)

0:(0.5)

1:(1)

0:(0.5)

0:(0.5)

1:(1)

1:(1)

0:(0.5)

0:(0.5)

Assumption


No one-to-one correspondence between key and output, but someinformation leaks.



Tasks

1 Efficient inference algorithms are needed.2 Measurement errors need to be modelled. (Side channel

measurements may be noisy.)3 A model is needed that handles input.4 Multiple traces need to be handled. (more powerful attacks)


Outline





5 Conclusion


Definition

Input Driven Hidden Markov Model

Definition

A Input Driven Hidden Markov Model is a septuple

M = (S, I, O, A, B, C, s0)

S: finite set of internal statesI: finite set of input symbolsO: finite set of symbols that represent operations observable over the sidechannelA : |S|× |I|× |S| transition matrix where Aijk = Pr [Q l

n = sk |Q ln−1 = si , Kn = kj ]

B : |S| × |O| output matrix where Bij = Pr [Y ln = oj |Q l

n = si ]D : n × |I| key distribution matrix where Cjk = Pr [Kj = ik ]s0 ∈ S: initial state

Note

Probabilistic finite state machines can be transformed into Input DrivenHidden Markov Models in a straight-forward manner.


Key Inference Problem

Key Inference Problem for IDHMM (single trace)

Key Inference Problem (single trace)

Let H be an Input Driven Hidden Markov Model and k a N bit key

Generate an execution q of H on k

Let y be the output of the corresponding trace

The single trace Key Inference Problem is to find k given H and y .

There exists an efficient algorithm for the single trace key inference problem.



Key Inference Problem for IDHMM


Let H be an Input Driven Hidden MarkovModel and k a N bit key

Generate L executionsq = (q1, q2, ..., qL) of H on k

Let y = (y1, y2, ..., yL) be the outputof the corresponding L traces

The Key Inference Problem is to find kgiven H and y .


Key Inference Problem Approaches

Key Inference Problem for IDHMM

Formally, we seek k with

k = argmaxk∈{0,1}N Pr [K = k |Y = (y1, y2, ..., yL)]

Reformulation:

Infer each key bit separately

ki = argmaxb∈{0,1}Pr [Ki = b|Y = (y1, y2, ..., yL)]

Use the approximated key bits to infer the entire key

k = (kN , kN−1, ..., k2, k1)

Algorithms to compute ki exponential in L, the number of traces.

Approximation technique needed



Belief propagation

Idea

Reduce the multiple trace key inference problem to L single trace inferenceproblems, thereby improving the key distribution approximation

In the processing of a trace, the trace and the prior distribution are usedto compute the new key distribution

Dj = Pr [ki = 1|yj , Dj−1]



Belief propagation cont’d

Find:Pr [ki |y j ]

Abbreviation:We will abbreviate (qi , ki) by xi

We set:

f (x0, ..., xN) = Pr [x0, ..., xN , y j ] =NY

n=1

gn(xn−1, xn)

whereg1(x0, x1) = Pr [x0] · Pr [x1|x0] · Pr [y j

1|x1]

gn(xn−1, xn) = Pr [xn|xn−1] · Pr [y jn|xn]

and

ti(xi) =P

x0...

Pxi−1

Pxi+1

...P

xNf (x0, ..., xN)

= Pr [xi , y j ]



Belief propagation cont’d

We can compute:

Pr [ki |y j ] =X

qi

ti(xi)Pxi(ti(xi))

How to compute ti(xi) efficiently?

Transform ti(xi)ti(xi) = ri(xi) · si(xi)

whereri(xi) =

Xxi−1

gi(xi−1, xi) · ri−1(xi−1)

si(xi) =Xxi+1

gi+1(xi , xi+1) · si+1(xi+1)

with r0(x0) = 1 and sN(xN) = 1



Key Inference Problem for IDHMM cont’d

Final Approximation Approach

Initially, the keys are equally distributed

Pr [DO(ki) = 1] := 0.5

Apply belief propagation on the trace and the prior key distributionapproximation

Dj = BP(yj , Dj−1)

After processing all traces, the key bits are set according to distributionDL.

If Pr [DL(ki) = 1] > 0.5 then ki := 1, else ki := 0.

By applying a modified single trace inference algorithm, the complexity of thecomplete algorithm is O(|S|2 · N · L)


Outline





5 Conclusion


Oswald Aigner Randomization

Oswald Aigner Randomized Algorithms

Basis:Earlier presented binary scalar multiplication algorithm

Morain Olivos transformations

Morain Olivos transformations

If a multiplication factor k contains blocks of the form 01a or 01a01b, thecorresponding factor parts are transformed as follows:

01a 7→ 10a−11 (MO1)

01a01b 7→ 10a10b−11 (MO2)

Algorithms:

Oswald Aigner Randomized Algorithms

If transformation MO1 (resp. MO2) is applicable to key k within multiplicationk ·M, a coin is flipped if MO1 (resp. MO2) is applied to k .


Oswald Aigner Randomization

Oswald Aigner Randomization (OA1)

Figure: Randomized state machine representing the OA1 binary scalar multiplicationalgorithm


Results

Results of the Attacks against OA1 and OA2

The following table shows the results of the attack against the OA1 and OA2algorithms using a 192 bit key.

Remark

Results highly depending on the order of the processed traces


Outline





5 Conclusion


Conclusion

Contribution of the paper:Introduction of Hidden Markov Models as generic evaluation techniquefor randomized countermeasures

Belief propagation as reasonable approximation to handle the multipletrace key inference problem complexity

Integration of output observation errors

The randomized algorithms proposed by Oswald and Aigner failcompletely


References

C. Karlof, D. Wagner. Hidden Markov model cryptanalysis., InCryptographic Hardware and Embedded Systems – CHES 2003,Springer-Verlag LNCS 2779, 17–34, 2003

P.J Green, R. Noad, N.P. Smart. Further Hidden Markov ModelCryptanalysis, In Cryptographic Hardware and Embedded Systems -CHES 2005, 61-74

F. Morain, J. Olivos. Speeding up the computation on an ellipticcurve using addition-subtraction chains., Inform. Theory Appl. 24(1990), 531-543

E. Oswald, M. Aigner. Randomized addition-subtraction chains as acounter- measure against power attacks., In Cryptographic Hardwareand Embedded Systems – CHES 2001, Springer-Verlag LNCS 2162,39–50, 2001.

E. Oswald. Side-Channel Analysis., In Advances in Elliptic Curve Cryp-tography. Cambridge University Press, 69–86, 2005.

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