A password rainbow table stores hash chains for various passwords. A password chain is computed, starting from the original password, P, via a reduce(compression) function R and a hash function H. A recurrence relation exists where Xi+1 = R(H(Xi)), X0 = P. Then the hash chain of length n for the original password P can be formed: X1, X2, X3, ... , Xn-2, Xn-1, Xn, H(Xn). P and H(Xn) are then stored together in the rainbow table. Constructing the rainbow tables takes a very long time and is computationally expensive. A separate table needs to be constructed for the various hash algorithms (e.g. SHA1, MD5, etc.). However, once a rainbow table is computed, it can be very effective in cracking the passwords that have been hashed without the use of salt.