MCPC Summer Training

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2018-07-11 19:00 UTC

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Problem C
Permutation Code

As the owner of a computer forensics company, you have just been given the following note by a new client:

I, Albert Charles Montgomery, have just discovered the most amazing cypher for encrypting messages. Let me tell you about it.

To begin, you will need to decide on a set of symbols, call it S, perhaps with the letters RATE. The size of this set must be a power of 2 and the order of the symbols in S is important. You must note that R is at position 0, A at 1, T at 2, and E at 3. You will also need one permutation P of all those symbols, say TEAR. Finally you will need an integer, call it x. Together, these make up the key. Given a key, you are now ready to convert a plaintext message M of length n (M[0], M[1]... M[n-1]), that has some but not necessarily all of the symbols in S, into a cyphertext string C, also of length n (C[0], C[1],...C[n-1]), that has some but not necessarily all of the symbols in S. The encrypting algorithm computes C as follows:

  1. Calculate an integer d as the remainder after dividing the integer part of ($n^{1.5} + x$) by $n$. This can be expressed more succinctly as $d = (int)(n^{1.5} + x) \% n$, where $\% $ is the remainder operator.

  2. Set C[d] to be the symbol in S whose position is the same as the position of M[d] in P.

  3. For each $j \ne d$ in $0..n-1$, set C[j] to be the symbol in S whose position is the value obtained by xor-ing the position of M[j] in P with the position of M[(j+1) % n] in S. Note that the bitwise xor operator is "^" in C, C++, and Java.

For example, consider this scenario where S=RATE, P=TEAR, x=102, M=TEETER, and n=6. To compute d, first calculate $6^{1.5} + 102 = 116.696938$, then take the remainder after dividing by 6. So d = 116 % 6 = 2. The following table shows the steps in filling in the cyphertext C. Note that the order of the steps is not important.

    0 1 2 3 4 5

S = R A T E 
P = T E A R
M = T E E T E R

C = E           M[0] is T, T is at P[0]. M[1] is E, E is at S[3]. C[0] = S[0 xor 3] = S[3]
    E T         M[1] is E, E is at P[1]. M[2] is E, E is at S[3]. C[1] = S[1 xor 3] = S[2]
    E T A       2 is d. M[2] is E, E is at P[1], so C[2] =  S[1]
    E T A E     M[3] is T, T is at P[0]. M[4] is E, E is at S[3]. C[3] = S[0 xor 3] = S[3]
    E T A E A   M[4] is E, E is at P[1]. M[5] is R, R is at S[0]. C[4] = S[1 xor 0] = S[1]
    E T A E A A M[5] is R, R is at P[3]. M[0] is T, T is at S[2]. C[5] = S[3 xor 2] = S[1]

I have included additional examples of encrypted messages at the end of this note for you to experiment with. However, first, I need to tell you about the decryption algorithm.

Input

The input for the decoder consists of one or more sets of {key, encrypted message} pairs. The key is on 3 separate lines. The first line contains the single integer $x$, $0 < x < 10,000$; the second line contains the string S; and the third line contains the string P, which will be a permutation of S. The length of S (and therefore P) will always be one of the following powers of two: 2, 4, 8, 16, or 32. Following the key is a line containing the encrypted message string C, which will contain at least one and at most sixty characters. The strings S, P, and C will not contain whitespace, but may contain printable characters other than letters and digits. The end of the input is a line which contains the single integer 0.

Output

For each input set print the decrypted string on a single line, as shown in the sample output.

Sample Input 1 Sample Output 1
102
RATE
TEAR
ETAEAA
32
ABCDEFGHIJKLMNOPQRSTUVWXYZ._!?,;
;ABCDEFGHIJKLMNOPQRSTUVWXYZ._!?,
MOMCUKZ,ZPD
1956
ACEHINT_
ACTN_IHE
CIANCTNAAIECIA_TAI
0
TEETER
HELLO_WORLD
THE_CAT_IN_THE_HAT