Here is a cipher algorithm, based on hexadecimal strings that is implemented by **XORing** the given plaintext, **N** number of times where **N** is its length. But, the catch is that every next XOR operation is done after shifting the consecutive plain text **entry to the right**. A sample operation is shown below :

Suppose the password is **‘abcd’** then the hexadecimal text is calculated as **a1d0a1d** by XORing the password with itself N times i.e. 4 times in this case.

Similarly if the password is **‘636f646572’**, then

**653cae8da8edb426052** is the hexadecimal text.

So, the problem statement is to create a decryption algorithm (in any programming language) and deduce the plain text from the given hexadecimal string.

**Examples :**

Input : a1d0a1d Output : abcd abcd once coded will return a1d0a1d Input : 653cae8da8edb426052 Output : 636f646572

**Approach : ** The key ingredient in encrypting and decrypting is in the properties of **XOR**. XOR is a bitwise operation where the result is **0** if the two possible inputs are **same** but **1** when the inputs are **different**. The XOR table is given below for reference :

Inputs | Outputs | |
---|---|---|

X | Y | Z |

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

An important and useful property of XOR that is widely popular in cryptography is that in case of multiple XORing of numbers (say **M** numbers), if we know only the **M – 1** numbers (one is unknown) along with the **XOR result** then, we can easily calculate the missing number by XORing the known numbers and the XOR result. This property is discussed with the following hexadecimal numbers :

We shall be using the above listed property the most in course of this problem. Now, if we look at the encryption diagram of **‘abcd’** at the base it is just the repeated XORing of the digits. The rightmost digit is **d** and the rightmost digit of the **‘abcd’** is **d** as well so the last digit of both plaintext and hexstring is the **same**. The next digit is **1** which is calculated by XORing the second right digit of **abcd** and the previous digit i.e. **1 = d ^ c** using the property we know the plain text digit can be deduced as **d ^ 1 = c**. Similarly, the next digit is **a** which is found by **d ^ c ^ b = a**. We only need to do this till the half of the hex string as the rest is **symmetrical so they are not required**.

**Below is the implementation of above approach : **

`# Implementation in Python 3 ` ` ` `# Hex String variable ` `hex_s ` `=` `'653cae8da8edb426052'` ` ` `# Plain text variable ` `plain ` `=` `'' ` ` ` `# variable to store the XOR ` `# of previous digits ` `x ` `=` `0` ` ` `l ` `=` `len` `(hex_s) ` ` ` `# Loop for loop from the end to ` `# the mid section of the string ` `for` `i ` `in` `range` `(l ` `-` `1` `, ` `int` `(l ` `/` `2` `) ` `-` `1` `, ` `-` `1` `): ` ` ` ` ` `# calculation of the plaintext digit ` ` ` `y ` `=` `x^` `int` `(hex_s[i], ` `16` `) ` ` ` ` ` `# calculation of XOR chain ` ` ` `x ` `=` `x^y ` ` ` `plain ` `=` `hex` `(y)[` `-` `1` `] ` `+` `plain ` ` ` `print` `(plain) ` |

**Output:**

636f646572

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