Cryptography
Review for Final
1. Examples of cryptosystems:
•Shift:P=C=K=Zn,eK(x) = x+K, dK(y) = y−K,|P| =
|C| =|K| =n
•Affine:P=C=Zn,K={(a, b)|a, b ∈Zngcd(a, n) = 1},eK(x) =
ax +b, dK(y) = a−1(y−b), |P| =|C| =n,|K| =φ(n)n(In
particular φ(26) = 12)
•Substitution:P=C=Zn,K- all permutations of Zn,eπ(x) =
π(x), dπ(y) = π−1(y), |P| =|C | =n,|K| =n!
•Vigenere:P=C=K= (Zn)m,eK(x) = x+K, dK(y) = y−K,
|P| =|C| =|K| =nm
•Hill:P=C= (Zn)m,K- set of m×minvertible matrices over Zn,
eK(x) = xK, dK(y) = yK −1,|P| =|C| =nm,|K| ≤ nm2(when n
is prime, Qm−1
i=0 (pm−pi))
2. Friedman’s Test:
•The index of coincidence Ic(x) = Pfi(fi−1)
n(n−1) ≈Pp2
iand how to use
it to attack Vigenere Cipher.
•Mg=Ppifi+g
n′,n′=n/m and how to use it to guess a key in
Vignenere Cipher.
3. Basic Probability:
•Conditional probability.
•Bayes’ Theorem.
4. Perfect Secrecy:
•Check if a cryptosystem has perfect secrecy (compute conditional
probabilities).
•Shift, Affine (check directly that they have perfect secrecy).
•Characterization of cryptosystems with perfect secrecy (when |P | =
|C| =|K|) (with a proof).
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