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Can someone explain with examples. Furthermore, is a key required to crack the ciphers?

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Both the columnar and and the grid transposition ciphers are certain types of transposition ciphers. Both ciphers normally require a key for enciphering / deciphering. However, you can say that these ciphers, in their simple form can be easily cracked by brute-force.

Columnar Transposition

In a columnar transposition cipher you write the message out in rows of a fixed length. Then you read out column by column, where the columns are chosen in a scrambled order. Normally the key defines the width of the rows and moreover the "scramble" part in column permutation. Take the ciphertext "Thisisaciphertextthatneedstobecracked" and the key "ZEBRAS" for example. "ZEBRAS" defines the permutation in regard to the alphabetical order of the letters, which is "6 3 2 4 1 5" here.

1 2 3 4 5 6                              6 3 2 4 1 5
T h i s i s                              T h i s i s
a c i p h e                              a c i p h e
r t e x t t  --> using key "ZEBRAS" -->  r t e x t t
h a t n e e                              h a t n e e
d s t o b e                              d s t o b e
c r a c k e                              c r a c k e
d X X X X X                              d X X X X X

You get your ciphertext by reading your columns with your new indices, i.e. "ihtebkx iiettax hctasrx spxnocx seteeex Tarhdcd".

Grid Transposition

The grid transposition cipher is similar to the columnar transposition cipher. The difference here is that you introduce row indices in addition to column indices. You first create a grid that can fit all of the letters of your ciphertext and again, fill the extra spaces with X. Take the ciphertext "hiwhatsup" and the key "BAZYFG" for example. Again the key defines the columnar permutation, but also the row permutation. "BAZ" is for the columns and would map to "2 1 3" and "YFG" is for the rows and would map to "3 1 2".

  1 2 3                                   2 1 3
1 b e s                                 3 e d s
2 t c o     --> using key "BAZYFG" -->  1 e b s
3 d e s                                 2 c t o
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