Sliding Windows Succumbs to Big Mac Attack

Sliding Windows Succumbs to

Big Mac Attack

Colin D. Walterwww.co.umist.ac.uk

CHES 2001 C.D. Walter, UMIST 2

Aims

• Re-think the power of DPA;

• Use it on a single exponentiation;

• Longer keys are more unsafe!


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.


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


Multipliers

• Switching a gate in the H/W requires more power than not doing so;

• On average, a Mult-Acc opn a×b+c 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.


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.


Combining Traces I

• The long integer product A×B in an exponentiation contains a large number of small digit multiply-accumulates: ai×bj+ck

• Identify the power subtraces of each ai×bj+ck

from the power trace of A×B;

• 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.


Combining Traces

a0b0 a0b1 a0b2 a0b3


Combining Traces

a0b0


Combining Traces

a0b0

a0b1


Combining Traces

a0b0

a0b1

a0b2


Combining Traces

a0b0

a0b1

a0b2

a0b3


Combining Traces


Combining Traces

a0(b0+b1+b2+b3)/4

Average the traces:


• b is effectively an average random digit;

• So trace is characteristic of a0 only, not B.

tr0

Combining Traces

a0b_

_


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.


Combining Traces

tr0


Combining Traces

tr0 tr1


Combining Traces

tr0 tr1 tr2


Combining Traces

tr0 tr1 tr2 tr3


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

Combining Traces

trA


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(m–1)

• For small enough m and large enough number of digits they can be distinguished in a simulation of clean data.


Distances between Traces

tr0

tr1

d(0,1) = ( i=0(tr0(i)tr1(i))2 )½ n

in0

power


Simulation

tr0

tr1

d(0,1) = ( i=0(tr0(i)tr1(i))2 )½ n

in0

gate switch count


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


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.


Expnt Digit Values

• Pre-computations A(i+1) 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 m–1;

• This can be done independently for each opn.


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.


WarningWarning

• With the usual DPA averaging

already done, it may be possible

to use a single exponentiationsingle exponentiation to

obtain the secret key;

• So using expSo using expntnt dd++rrφ(φ(MM) with ) with

random random rr may be no defence. may be no defence.


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.

Sliding Windows Succumbs to Big Mac Attack

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