Sliding Windows Succumbs to Big Mac Attack

1
Sliding Windows Succumbs to Big Mac Attack

• Colin D. Walter
• www.co.umist.ac.uk

2
Aims

• Re-think the power of DPA
• Use it on a single exponentiation
• Longer keys are more unsafe!

3
DPA Attack on RSA

• Summary Differential Power Analysis
  (DPA) is used to determine the secret exponent
  in an embedded RSA cryptosystem.
• Assumption The
  implementation uses a small multiplier whose
  power consumption is data dependent and
  measurable.

4
History

• P. Kocher, J. Jaffe B. Jun
  Introduction to Differential Power Analysis
  and Related Attacks Crypto 99
• T. S. Messerges, E.A. Dabbish R.H. Sloan Power
  Analysis Attacks of Modular Exponentiation in
  Smartcards CHES 99

5
Multipliers

• Switching a gate in the H/W requires more power
  than not doing so
• On average, a Mult-Acc opn abc has data
  dependent contributions roughly linear in the
  Hamming weights of a and b
• Variation occurs because of the initial state set
  up by the previous mult-acc opn.

6
First Results

• This theory was checked by simulation and found
  to be broadly correct
• Refinements were made to this model (which will
  be reported elsewhere)
• These give a more precise detailed partial
  ordering.

7
Combining Traces I

• The long integer product AB in an exponentiation
  contains a large number of small digit
  multiply-accumulates aibjck
• Identify the power subtraces of each aibjck
  from the power trace of AB
• Average the power traces for fixed i as j varies
  this gives a trace tri which depends on ai but
  only the average of the digits of B.

8
Combining Traces
a0?b0
a0?b1
a0?b2
a0?b3
9
Combining Traces
a0?b0
10
Combining Traces
a0?b1
a0?b0
11
Combining Traces
a0?b2
a0?b1
a0?b0
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Combining Traces
a0?b3
a0?b2
a0?b1
a0?b0
13
Combining Traces
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Combining Traces
Average the traces
a0?(b0b1b2b3)/4
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Combining Traces
_

• b is effectively an average random digit
• So trace is characteristic of a0 only, not B.
• tr0

_
a0?b
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Combining Traces II

• The dependence of tri on B is minimal if
  B has enough digits
• Concatenate the average traces tri for each ai
  to obtain a trace trA which reflects properties
  of A much more strongly than those of B
• The smaller the multiplier or the larger the
  number of digits (or both) then the more
  characteristic trA will be.

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Combining Traces
tr0
18
Combining Traces
tr0
tr1
19
Combining Traces
tr0
tr1
tr2
20
Combining Traces
tr0
tr1
tr2
tr3
21
Combining Traces

• Question Is the trace trA sufficiently
  characteristic to determine repeated use of a
  multiplier A in an exponentiation routine?

trA
22
Distinguish Digits?

• Averaging over the digits of B has reduced the
  noise level
• In m-ary exponentiation we only need to
  distinguish
• squares from multiplies
• the multipliers A(1), A(2), A(3), ,
  A(m1)
• For small enough m and large enough number of
  digits they can be distinguished in a simulation
  of clean data.

23
Distances between Traces
power
tr0
tr1
i
n
0
n
d(0,1) ( ? i0(tr0(i)?tr1(i))2 )½
24
Simulation
gate switch count
tr0
tr1
i
n
0
n
d(0,1) ( ? i0(tr0(i)?tr1(i))2 )½
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Simulation Results

• 16-bit multiplier, 4-ary expn, 512-bit modulus.
• d(i,j) distance between traces for ith and
  jth
  multiplications of expn.
• Av d for same multipliers 2428 gates
• SD for same multipliers 1183
• Av d for different multipliers 23475 gates
• SD for different multipliers 481

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Simulation Results

• Equal exponent digits can be identified their
  traces are close
• Unequal exponent digit traces are not close
• Squares can be distinguished from multns their
  traces are not close to any other traces
• There are very few errors for typical cases.

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Expnt Digit Values

• Pre-computations A(i1) ? A ? A(i) mod M
  provide traces for known multipliers. So
• We can determine which multive opns are squares
• We can determine the exp digit for each multn
• Minor extra detail for i 0, 1 and m1
• This can be done independently for each opn.

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Some Conclusions

• The independence means attack time proportional
  to secret key length
• Longer modulus means better discrimination
  between traces
• No greater safety against this attack from longer
  keys.

29
Warning

• With the usual DPA averaging already done, it may
  be possible to use a single exponentiation to
  obtain the secret key
• So using expnt drf(M) with random r may be
  no defence.

30
Final Conclusions

• Sliding Windows expn method may be broken in this
  way
• Like a Big Mac, you can nibble away at each
  secret exponent digit in turn and enjoy finding
  out its value.