Distinguishing Exponent Digits by Observing Modular Subtractions

We analyse timing variations in an implementation of modular multiplication which has certain standard characteristics. This shows that squarings and multiplications behave differently when averaged over a number of random observations. Since power analysis can reveal such data, secret RSA exponents can be deduced if a standard square and multiply exponentiation algorithm is used. No knowledge of the modulus or input is required to do this. The technique generalises to the m-ary and sliding windows exponentiation methods since different multipliers can be distinguished. Moreover, only a small number of observations (independent of the key size and well under 1k) are required to perform the cryptanalysis successfully. Thus, if the modular multiplication algorithm cannot be made any safer, the exponent must be modified on every use.