Cryptography

cryptosystem(P, E, K, ε, D):

• P is a finite set of possible plaintexts;

• E is a finite set of possible ciphertexts;

• K is a finite set of possible keys;

• For each k ∈ K, there is an encryption rule ek ∈ ε corresponding decryption rule dk ∈ D. each ek: P→ E and dk: E → P are functions such that: d_k(e_k(x)) = x for every element plain text x ∈ P.

Cryptography methods

• Shift Cipher

  • P = E = K = [0:26]

  • for k in K define: e_k(x) = (x + k) % 26 and d_k(y) = (y - k) % 26

  • example: wewillmeetatmidnight and k = 11 encrypt to hphtwwxppelextoytrse

• Substitution Cipher

  • P = E = [0:26]

  • K consists of all possible permutation of the 26 symobols [0:26]

  • for each permutation π ∈ K, define: e_π(x) = π(x) and d_π(y) = arcπ(y)

  • example: wewillmeetatmidnight and k = 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 encrypt to hphtwwxppelextoytrse

• Affine Cipher

  • P = E = [0:26]

  • K = {(a, b) ∈ [0:25] * [0:26] : gcd(a, 26) = 1}

  • for K = (a, b) ∈ K, define: e_k(x) = (ax + b) % 26 and d_k(y) = arca(y - b)

  • example: wewillmeetatmidnight and k = 1 11 encrypt to hphtwwxppelextoytrse

• Vigenere Cipher

  • P = E = K = [0:26] ^ m

  • k = (k1, k2, k3, ..., km) ∈ K

  • for k ∈ K, define: e_k(x1, x2, ..., xm) = (x1, x2, ..., xm) + (k1, k2, k3, ..., km) % 26 and d_k(y1, y2, ..., ym) = (y1, y2, ..., ym) - (k1, k2, k3, ..., km) % 26

  • example: thiscryptosystemisnotsecure and k = cipher encrypt to vpxzgiaxivwpubttmjpwizitwzt