Simple Asymmetric Reversal

Page #4355453 of Chapter: Crypto-for-the-rest-of-us

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Let's map the alphabet and some symbols onto the numbers 1 to 31. And let us use the number 9 as encryption key as follows: C = 9* P mod 31 (P a plaintext number and C a ciphertext number, both in the range 1-31, where we interpret 31 as o).

Accordingly we can encrypt the word "LOVE" as follows:

  • Replace the letters with their numeric value: LOVE = 12,15,22,5
  • Encrypt each plaintext letter P into its corresponding ciphertext letter C

    So 12 becomes 12*9 mod 31 = 15,
    15 becomes 15*9 mod 31 = 11
    22 becomes 22*9 mod 31 = 12
    5 becomes 5*9 mod 31 = 14
  • Replace the numeric values with their letters: 15,11,12,14 becomes: OKLN

    so the plaintext "LOVE" becomes the ciphertext "OKLN"

    If we try to use the same key for decryption we get a plaintext:

    KFOB

    which is different from the plaintext word "LOVE". So a symmetric key won't work!

    However, if we use a different key K=7 (not 9) and apply to the ciphertext then we get:


    15 becomes 15*7 mod 31 = 12
    11 becomes 11*7 mod 31 = 15
    12 becomes 12*7 mod 31 = 22
    14 becomes 14*7 mod 31 = 5

  • which translates back to the original plaintext "LOVE".

    This illustrates a simple asymmetrical cipher that does the job.


    The math behind the illustration:

    Let e be the encryption key; d, the decryption key. A letter x is encrypted to y=ex mod 31, then decrypted back to
    x = dy = dex mod 31.
    So we must have de = 1 mod 31. for e=9 and d=7, we have 7*9=63 = 31*2 + 1 = 1 mod31.

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