Montgomery's Ladder Technique
Many algorithms for exponentiation do not provide defence against side-channel attacks. Namely, an attacker observing the sequence of squarings and multiplications can (partially) recover the exponent involved in the computation. This is a problem if the exponent should remain secret, as with many public-key cryptosystems. A technique called Montgomery's Ladder addresses this concern.
Given the binary expansion of a positive, non-zero integer n=(nk-1...n0)2 with nk-1=1 we can compute xn as follows:
x1=x; x2=x2 for i=k-2 to 0 do If ni=0 then x2=x1*x2; x1=x12 else x1=x1*x2; x2=x22 return x1The algorithm performs a fixed sequence of operations (up to log n): a multiplication and squaring takes place for each bit in the exponent, regardless of the bit's specific value.
Read more about this topic: Exponentiation By Squaring
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