Use p 5 q 7 and e 5 in the RSA cryptography system Encode

Use p = 5, q = 7 and e = 5 in the RSA cryptography system. Encode REINHARDT. Note that (p - 1)(q - 1) = 24. Find an inverse of 5 mod 24. Decode 10 1 7 17 10 24

Solution

5.   We are given p = 5, q = 7 and e = 5 for the RSA cryptolgraphy.

     To encode a message using the RSA code follow the steps below:

     1) Choose prime numbers p and q (here we are given p=5 and q=7)

     2) We multiply these 2 numbers together (7×5 = 35). This is the public key (m), So m = 35.

     3) Now we need to use an encryption key (e). Here we are given that e = 5.

        ( e must actually be relatively prime to (p-1)(q-1) )

     4) Now we are ready to encode something.

         First we can assign 00 = A, 01 = B, 02 = C, 03 = D, 04 = E etc. all the way to 25 = Z.

     5) We now use the formula: C = y^e (mod m) where y is the letter we want to encode.

          Here we have m = pq = 7 x 5 = 35 so we use (mod 35) for encoding

          Note : (mod 35 simply mean we look at the remainder when we divide by 35).

a)     So for the letters RIENHARDT, we encode each alphbet as follows

          R = 17⁵ = 1419857 (mod 35) which is equivalent to 12

E = 04⁵ = 1024 (mod 35) which is equivalent to 09

        I = 08⁵ = 32768 (mod 35) which is equivalent to 08

N = 13⁵ = 371293 (mod 35) which is equivalent to 13

H = 07⁵ = 16807 (mod 35) which is equivalent to 07

A = 00⁵ = 0 (mod 35) which is equivalent to 00

R = 17⁵ = 1419857 (mod 35) which is equivalent to 12

D = 03⁵ = 243 (mod 35) which is equivalent to 33

T = 19⁵ = 2476099 (mod 35) which is equivalent to 24

So the word \"REINHARDT\" encoded to \"10   09   08   13   07 00   12   33    24\" .

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b) Now we have to find inverse of 5 mod 24, we are given (p-1)(q-1) = 24    

     In this RSA encryption we are given both m and e. These are public keys. We are given that m = 35

     and e = 5. We need to find the two prime numbers that multiply to give 35. These are p = 7 and q = 5.

   Calculate (p-1)(q-1). In this case this it is (7-1)(5-1) = 24. Call this number theta.

   Calculate a value \"d\" such that d x e = 1 (mod theta). We already know that e is 5.

   Therefore we want 5d = 1 (mod 24). Clearly when d = 5 we have 5×5 = 25 which is 1 (mod 24).

   So the inverse of 5 (mod) 24 has given decryption key d = 5.

    That means we can now use this decryption key d = 5 to decode or decipher.

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c)     Decode 10     1 7    17 10 24

To decipher this code we need to follow the steps given below.

Now we can decipher using the formula: y = C^d (mod m),

Where, C is the codeword, m = 35 and d = 5. (d = deciper key =5 that we got in part b).

So for the cipher text 10     1 7    17 10 24 can be written as

     For 10 : y = 10⁵ = 100000 (mod 35) = 05.

   For    1 :   y = 1⁵ = 01 (mod 35) = 01.

     For   7 :   y = 7⁵ = 16807 (mod 35) = 07.

   For 17 : y = 17⁵ = 1419857 (mod 35) = 12.

     For 10 : y = 10⁵ = 100000 (mod 35) = 05.

     For 24 : y = 24⁵ = 7962624 (mod 35) = 19.

          We can now convert these numbers back to letters using A = 00, B = 01 etc. all the way Z=25.

                 05        01       07       12     05      19

                  F         B        H         M        F        T

   This gives the decoded word as: \"FBHMFT\"