# Notes, Chapter 2

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2 Principles of cryptography (KL1.1, KL1.2, KL1.4, KL2) -------------------------------------------------------- These considerations and similar ones applying to other historical ciphers have led to the principles for the design of cryptosystems. Before stating these we describe somewhat more formally what a (private-key) cryptosystem is. A cryptosystem comprises three algorithms: - A key generation algorithm Gen which outputs a key according to some format an probability distribution. - An encryption algorithm Enc which given plaintext m and key k produces a ciphertext Enc_k(m) - A decryption algorithm Dec which given ciphertext c and key k produces a plaintext Dec_k(m). The set of all keys possibly put out by the key generator Gen is called keyspace (K). For every key k produced by Gen it must hold that Dec_k(Enc_k(m)) = m. There is also a finite set of possible messages M and ciphertexts C. Clearly, Gen must be randomized, otherwise it would always return the same key. Sometimes Enc is also randomized, i.e. ciphertexts may contain some random information and do not depend functionally on plaintext and key. One of the most important design principles is Kerckhoff's rule (19th century): Security of the cryptosystem should rely on secrecy of the key not on secrecy of the method. In other words, the cryptosystem should remain secure if everything about the three algorithms involved is publicly known. "No security through obscurity". In addition to this, the cryptosystem should withstand some or all of the following kinds of attack: Ciphertext only attack: an eavesdropper observes ciphertext and aims at determining the underlying plaintexts. Known plaintext attack (KPA): This is when the eavesdropper knows some ciphertext-plaintext pairs for one and the same key and tries to use this information to then decrypt additional ciphertexts (again w.r.t. the same key). Chosen plaintext attack (CPA): Here the adversary may prepare plaintexts of its choice and request the corresponding ciphertexts. Chosen ciphertext attack (CCA): Here the adversary can even request the decryption of ciphertexts of its choice. Modern cryptography, say since WW2, holds the following additional principles: Principle 1: Formulation of exact definitions Which notion of security is one aiming at? No adversary should be able to know the key ... But what if they can decrypt ciphertexts without knowing the key? No adversary should be able to decrypt any ciphertext unless they know the key. But what if they can find out some bits of the ciphertext, e.g. every second letter, or a block of letters somewhere in the middle? Principle 2: Reliance on precise assumptions Typically, security of a cryptosystem can be guaranteed only by making assumptions on the computational power of an adversary (with unlimited resources they could always try out all plaintexts) and also on the computational difficulty of solving certain algorithmic problems. Desirable as it would be to have unconditional security; if one has to work with assumptions one should at least state them precisely. Sometimes, security can only be proven for idealized versions of cryptosystems that use components not known to exist but which are believed to be "approximated" by existing components. Again, it is then important to clearly state what these idealized components or versions are. Principle 3: Rigorous proofs of security Once the desired notion of security and the assumptions are clearly defined one should then establish a rigorous mathematical proof of conditional security. "Given that Assumption X is true, Construction Y is secure according to the given definition." As an example, we now look at perfect security in the sense of Shannon and see how Vernam's cipher (patented in 1919) implements it. In order to get around the fact that an adversary can always try out all possible plaintexts but that this kind of attack "doesn't count" Claude Shannon suggested to use probabilities: if one out of two messages is selected at random and the adversary gets to see the ciphertexts only it has a chance of no more than 50% of determining the plaintext correctly from the ciphertext. For cryptosystem Pi and adversary A we thus define the following Eavesdropping indistinguishability experiment PrivK^eav_{A,Pi}: 1. The adversary produces two messages m0, m1 from the message space M. 2. Gen is used to generate a (random) key k and a bit b <- {0,1} is chosen uniformly at random (50/50). The (or rather a) ciphertext c = Enc_k(m_b) is then given to A. 3. A answers by outputting a bit b'. 4. The output of the experiment is defined 1 if b=b' and 0 otherwise. Definition (perfect secrecy): The encryption scheme Pi=(Gen,Enc,Dec) is perfectly secret if for all adversaries A it holds that Pr[PrivK^eav_{A,Pi} = 1] = 0.5 The following cryptosystems are not perfectly secure: A book cipher which uses a secret code book containing for each *word* another word to replace it. E,g. "bomb" gets enciphered as "cake" and "dollars" gets enciphered as "grams". Here the adversary could select m0="bomb bomb" and m1="dollar bomb". If the received ciphertext has the form c = x x then it will output b'=0 and if c = x y with x =/= y it goes for 1. The Vigenere cipher with key length uniformly chosen at random from {1,2} (and once the length is fixed keys also chosen at random). The adversary can choose m0 = aa and m1=ab. If the ciphertext has the form xx it goes for b'=0. Otherwise it returns b'=1. Now, if b=0 then there is a 1/2+1/2*1/26 chance that the ciphertext has the form xx (key length 1 or key length 2 with equal letters) and thus the adversary is right with probability 1/2+1/52 If b=1 then the adversary is right with probability 1-1/52 (the only way to be wrong is key length two and two key letters of distance 1). Thus, the total probability for A to be right is 1/2*(1/2+1/52) + 1/2*(1-1/52) = 3/4. Proposition: Fix a cryptosystem Pi and suppose that plaintexts and keys are drawn according to some, not necessarily uniform distribution. The sytem Pi is perfectly secret if and only if Pr[M=m | C=c] = Pr[M=m] for every fixed plaintext m and ciphertext c such that Pr[C=c] != 0. Proof (sketch): Suppose that Pr[M=m | C=c] != Pr[M=m] for some c and m. Since sum_m Pr[M=m | C=c] = sum_m Pr[M=m] = 1 (both are probability measures) there must exist m_0,m_1 (one of them m) such that Pr[M=m0 | C=c] < Pr[M=m0] and Pr[M=m1 | C=c] > Pr[M=m1]. By Bayes' theorem it follows that Pr[C=c|M=m0]<Pr[C=c] and Pr[C=c|M=m1]>Pr[C=c], thus in particular, Pr[C=c|M=m0] < Pr[C=c|M=m1]. Now, consider the adversary who presents these messages m_0, m_1 and if the ciphertext returned is c then outputs b'=1, i.e. assume that the plaintext was not m0 and outputs b'=0 otherwise. Its success probability is 1/2*(1-Pr[C=c|M=m0]) + 1/2*Pr[C=c|M=m1] = 1/2 + 1/2(Pr[C=c|M=m1] - Pr[C=c|M=m0]) which is strictly above 1/2. For the converse, one uses the fact that all the adversary knows is the ciphertext, but according to the assumed equality Pr[M=m | C=c] = Pr[M=m] this does not help. Q.E.D. Now, such perfectly secret cryptosystems exist indeed: for fixed n:N put M=C=K={0,1}^n and let Gen return k:K uniformly at random. The encryption function is Enc_k(m) = m+k and Dec_k(m) = m+k as well, where + denotes pointwise addition modulo 2 (XOR). For example, if n=5 and k=11010 and m=11100 then Enc_k(m)=00110. This system is known as Vernam's cipher or "one time pad" the point being that its secrecy relies on the key being chosen afresh at random every time one wants to use the scheme. It can also be seen as a special case of the Vigenere cipher with the key being as long as the entire text to be enciphered. Proposition: The Vernam cipher is perfectly secret. Proof: This is because for a given plaintext m the ciphertexts are uniformly distributed, i.e., their distributions are identical and independent for m0 and m1. As a result, the adversary cannot have an advantage. Q.E.D. The obvious disadvantage of the one-time pad is the key length, nevertheless it is sometimes used, e.g. purportedly during the Cold War for secure communication between the White House and the Kremlin. Unfortunately, this key length is the price of perfect secrecy: Theorem: If Pi is a perfectly secret cryptosystem with key space K and message space M then |K|>=|M|, i.e. there have to be at least as many possible keys as there are possible messages ("the key must be as long as the message"). Proof: Suppose for a contradiction that |K|<|M| and that ciphertext c:C occurs with nonzero probability. Let M(c) = {m | Dec_k(c)=m for some k:K}. We must have |M(c)|<=|K| since for each m:M(c) there is at least one k:K such that m=Dec_k(c) and the keys thus corresponding to distinct messages are distinct. If now |K|<|M| there must exist some m' not in M(c). But then Pr[M=m'|C=c] = 0 =/= Pr[M=m'], so the scheme is not perfectly secret. This result is due to Claude Shannon who gives in fact a more precise characterisation of perfect secrecy, see (KL2.4).

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