Final answer:
To find the encryption matrix in a Hill cipher, convert plaintext and ciphertext to numeric form, set them in matrices, multiply the inverse of the plaintext matrix by the ciphertext matrix mod 26, and solve for the encryption matrix.
Step-by-step explanation:
The student has asked how to find the encryption matrix used in the Hill cipher given a ciphertext 'ELNI' and a plaintext 'dont'. In the Hill cipher, plaintext letters are converted into numerical values (A=0, B=1, ..., Z=25), set into matrices, and multiplied by an encryption matrix. The mod 26 of the product gives us the ciphertext matrices. To determine the encryption matrix, we must set up the equations based on the plaintext and ciphertext.
Plaintext 'dont' corresponds to numeric values d=3, o=14, n=13, t=19, so our plaintext matrix is
P = | 3 14 |
| 13 19 |. The ciphertext 'ELNI' corresponds to numeric values E=4, L=11, N=13, I=8, so our ciphertext matrix is
C = | 4 11 |
| 13 8 |.
The equation we need to solve is C = P × K mod 26, where K represents the encryption matrix. We need to find K such that K × P mod 26 = C. Since matrix multiplication is not commutative, we need to multiply the inverse of P by C (P-1 × C mod 26) to find K. Calculating the inverse of the matrix P in modulo 26 is a multi-step process, involving finding the determinant, then the adjugate, and finally multiplying by the reciprocal of the determinant mod 26. Once you have P-1, it can be multipled by the matrix C to find the encryption matrix K.