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Introduction to cryptographyChristophe RitzenthalerNovember 19, 20071 Private key cryptographyLet us introduce some terminology. The information that Alice wants to send to Bob willbe called plaintext (Message clair). She will cipher or encrypt (chiﬀrer) the plaintext toobtain a ciphertext (texte chiﬀr´e ou cryptogramme) with the help of a key. Bob whoknows the key will then decipher (d´echiﬀrer) the ciphertext, whereas Oscar, who does not,can try to decrypt (d´ecrypter) it, in order to obtain the plaintext.Deﬁnition 1.1. A (secret key) cryptosystem (cryptosyst`eme) is a three-tuple (P,C,K)where the following conditions are satisﬁed :1. P is a ﬁnite set of possible plaintexts;2. C is a ﬁnite set of possible ciphertexts;3. K is a ﬁnite set of possible keys called keyspace (espace des cl´es);4. For each K ∈K, there is an encryption rule (fonction de chiﬀrement) E :P →CKand a decryptionrule (fonction de d´echiﬀrement)D :C →P suchthatD ◦E =K K KId.Remark 1. We see already that the function E must be injective. Moreover if P =C, EK Kand D are bijective : they are permutations, inverse of each other.KAlice and Bob will employ the following protocol to use a speciﬁc cryptosystem. Firstthey choose a random key K ∈ K. This is done when they are in the same place and notbeing observed by Oscar, or, alternatively, when they do have access to a secure channel. Atalatertime, suppose AlicewantstocommunicateamessagetoBoboveraninsecurechannel.We suppose that this ...

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