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Math Help - Cryptography....

  1. #1
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    Post Cryptography....

    Hello...

    I need some help in solving this problem...please help!

    "Z c s f f b e u a s s o x c z w u x b s n b h a o g y m q h m x z l r e c z m s i c x j k z f c x z e x q c h s k y f n w u v a y j b h t h c g f u a z t d p m t q g p r x g d s a d c y x n q t k h i v f o e b k l e b a l r j e s p b w i "

    was encrypted with a (4 X 4) Hill cipher and the first 16 bits decrypted to:
    " a r c h i m e d s e g o t s o e"


    1) We need to find the (4 X 4 ) matrices A and B such that A.M = B, where M is the encryption matrix...

    2) Use M to decrypt the entire message...


    thanks for the help!
    Last edited by Vedicmaths; September 13th 2009 at 08:54 AM.
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  2. #2
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    Quote Originally Posted by Vedicmaths View Post
    Hello...
    I need some help in solving this problem...please help!

    "Z c s f f b e u a s s o x c z w u x b s n b h a o g y m q h m x z l r e c z m s i c x j k z f c x z e x q c h s k y f n w u v a y j b h t h c g f u a z t d p m t q g p r x g d s a d c y x n q t k h i v f o e b k l e b a l r j e s p b w i "

    was encrypted with a (4 X 4) Hill cipher and the first 16 bits decrypted to:
    " a r c h i m e d s e g o t s o e"

    1) We need to find the (4 X 4 ) matrices A and B such that A.M = B, where M is the encryption matrix...

    2) Use M to decrypt the entire message...
    !
    The Hill Cipher was developed and published in 1929. Mr. Hill (& company) invented a hardware device -- for which he got a patent. The product did not become a killer app -- not because it had problems -- but in the 1930 depression era, data security was not a high priority.

    The Hill cipher can use matrices of 2x2, 3x3, 4x4, 5x5, 6x6, etc. for the encryption/decryption process.

    In this example above, the fundamental pieces of information about the Hill cipher are given.
    1) A 4x4 encryption matrix.
    2) The plain text and the matching cipher text.
    3) Only 26 letters in the system [using MOD 26]

    Some of the information supplied:
    Ciphertext= "zcsffbeuassoxczwuxbsnbhaogymqhmxzlreczmsicxjkzfcx zexqchskyfnwuvayjbhthc..."
    Plaintext = "archimedsegotsoe..."
    --------------
    4x4 ciphertext in columns of 4 row depth
    zfax
    cbsc
    sesz
    fuow

    The ciphertext letters (ascii code) transformed to numeric values a=0, z=25 (mod 26 matrix).

    C = \begin{bmatrix}25 & 05 & 00 & 23 \\<br />
02 & 01 & 18 & 02 \\<br />
18 & 04 & 18 & 25 \\<br />
05 & 20 & 14 & 22 \\<br />
\end{bmatrix}



    -------------
    4x4 plaintext
    aist
    rmes
    cego
    hdoe

    & the associated plain text numeric matrix

    P= \begin{bmatrix}00 & 08 & 04 & 19\\17 & 12 & 18 & 18\\02 & 04 & 06 & 14\\07 & 03 & 14 & 04\\ \end{bmatrix}


    [The detailed explanation has been abridged]*
    With a transpose of the 4x4 Ciphertext adjacent to a 4x4 transpose of the plaintext,
    making a 4 row x 8 column matrix. The Ciphertext was row-reduced modulo 26 to reduced
    echelon form, then forming an identity matrix for the left half, thus the right half
    was a transpose of the inverse matrix.
    That matrix was used to decrypt the data supplied.
    Something didn't work as expected.
    There was a typo in the supplied PLAINTEXT data.
    The 9th & 10th letters of the plain text appeared to be swapped, since the
    accepted spelling of the greek math man is "ARCHIMEDES".
    I changed the plaintext & recomputed. This is the resulting decryption matrix and
    the supplied ciphertext decoded.
    ===================
    A decryption matrix

    M= \begin{bmatrix}24 & 02 & 17 & 00\\ 23 & 11 & 23 & 04\\ 21 & 05 & 05 & 21\\19 & 16 & 15 & 02 \end{bmatrix}

    The decrypted message:
    archimedesgotsoexcitrdbecausehesblveqtheproblemtua thnzfjjlezpnrnnazfkmyvetcxpfni
    with spacing:
    archimedes got so excitrd because he sblveq the problem tuat hnzfjjlezpnrnnazfkmyvetcxpfni
    From this it appears that the ciphertext was not proof read when entered.

    *It makes little sense to have a full explanation of how it works when the data is invalid.
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  3. #3
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    WoW !

    Thanks a lot for the help Sir!
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