How Do I Calculate 5¹²⁸ in eight easy steps?
The answer is:
293,873,587,705,571,876,992,184,134,305,561,419,454,666,389,193,021,880,377,187,926,569,604,314,863,681,793,212,890,625 [here], and I did it in eight mathematical operations.
A recent paper showed that it is possible to determine the private key of RSA by simply listening to the radio waves emitted from a mobile phone. This is because the RSA method uses multiplication and square operations, and which can be observed as the processor consumes different amounts of electrical power as it performs the calculation.
In RSA we decrypt by taking the cipher, and then raising it to the power of d:
Message=Cipher^d (mod N)
and where N is the multiplication of two prime numbers. Our decryption key is (d,N). Thus if we find d, we crack RSA. We know N, as the public key is (e,N).
In order to perform the exponent operation (Cipher^d), we normally use the square and multiply method. So 5⁴ (where 4 is the exponent) becomes:
5² = 25
25²= 625If we can to multiply 5⁸ that is 5² squared to give 5⁴, and then if we square again we get 5⁸. It has thus taken us three operations to find a power of 8. For 5⁶⁴, we will need six operations:
