Negative exponentiation for Encryption / Decryption using Fast Exponentiation

Question

All,

I have coded Fast exponentiation algorithm to check if the encrypted value is equal to the original value for a given exponent (e) and modulo base (n). When I have a negative exponent, the values are not the same. Here is what I do:

1 > Any value base
2 > Private key prKey
3 > Public key puKey
4 > Modulo base modN

Decrypt using base ^ prKey mod modN to get decryptVal
then encrypt decryptVal ^ puKey mod modN to get encryptVal

Now encryptVal should be equal to base . In my code, this condition is satisfied for positive values of private key prKey only . For negative values of prKey , the condition is never satisfied.

Example (from my code run for various random values):

1 > base : 4092
2 > modN = 6499
3 > puKey = 5
4 > prKey = -1267

would give decryptVal = 4092 and encryptVal = 5537 != base

whereas when,

1 > base : 249
2 > modN = 6059
3 > puKey = 5
4 > prKey = 1181

gives me decryptVal = 4067 and encryptVal = 249 = base

Is this condition the expected behavior or there is a flaw in my code based on above execution results? [Note]: prKey and puKey are computed using Extended Euclidean algorithm

Answer 1

I read through your post twice and cannot quite understand what the problem is. You ask if there is a flaw in your code but you haven't shown any code. Anyway, the usual definition of exponentiation a x mod n for negative x requires that gcd(a,n) = 1. If gcd(a,n) = 1 then a ^x mod n == ( a -1 ) -x mod n . So if x is negative first compute the inverse of a and then raise that to the -x power, all mod n .