Consider the English alphabet: A(0) B(1) C(2) D(3) E(4) F(5) G(6) H(7) I(8) J(9)
ID: 3282902 • Letter: C
Question
Consider the English alphabet:
A(0) B(1) C(2) D(3) E(4) F(5) G(6) H(7) I(8) J(9) K(10) L(11) M(12) N(13) O(14) P(15) Q(16) R(17) S(18) T(19) U(20) V(21) W(22) X(23) Y(24) Z(25)
The number associated with each letter corresponds to its original position as it appears in the alphabet.
(a) If you attempted to use the decimation cipher with the key 13, how would the word MESSAGE be encrypted? Note: answering this question should make it clear why an encryption code of 13 should not be allowed.
(b) Use the decimation cipher with the key 5 to encrypt RETREAT.
(c) The message ADDAOS was encrypted using the decimation cipher with the key 7. Decrypt it.
Explanation / Answer
ANSWER:
a) In order for an encrypted message to be decipherable, the encryption value must be a unit modulo 26.
Encryption value ,k must be one of the: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25.
In other words, we must have gcd(k,26)=1.
encrypted value = kX(numerical equivalent of corresponding letter) mod 26
k is chosen with some restrictions from 0,1,…,m-1
when encrypted the word message will be like :
MESSAGE
M =13X12 mod 26 = 156 mod 26 =0 =A
E = 13X4 mod 26 =52 mod 26 = 0 = A
S =13X18 mod 26 = 234 mod 26 =0= A
S =13X18 mod 26 = 234 mod 26 =0= A
A = 13X0 mod 26 = 0 mod 26 =0 = A
G = 13X 6 mod 26 = 78 mod 26 =0 =A
E = 13X4 mod 26 =52 mod 26 = 0 = A
b)
k=5
R = 5X17mod 26 =85 mod 26= 7 = H
E = 5X4 mod 26 =20 mod 26 = 20 = U
T= 5X19 mod 26 = 95 mod 26 = 17=R
R = 5X17mod 26 =85 mod 26= 7 = H
E = 5X4 mod 26 =20 mod 26 = 20 = U
A = 5X0 mod 26 = 0 mod 26 =0 = A
T= 5X19 mod 26 = 95 mod 26 = 17=R
c)
k=7
A = 7X0 mod 26 = 0 mod 26 = 0= A
D = 7X3 mod 26 =21 mod 26 = 21 = V
D = 7X3 mod 26 =21 mod 26 = 21 = V
A = 7X0 mod 26 = 0 mod 26 = 0= A
O = 7X14 mod 26 = 98 mod 26 = 20 =U
S = 7X18 mod 26= 126 mod 26 = 22 = W
The encrypted message is AVVAUW
we know that in decryption A=A
so we have to find decrypted letters for V,U,W.
for V:
k=7 we know
so 7X__ mod 26 = 21 (value of V)
7X__ =21
hence the value is 3 = D
for U:
7X__ mod 26 = 20
from modular multiplicative inverse we found that the value is 14 =O
for W:
similarly as above the letter is S
Hence Decrypted.
ADDAOS
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