Cryptography. Andreas Hülsing. 6 September 2016
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1 Cryptography Andreas Hülsing 6 September / 21
2 Announcements Homepage: http: // Lecture is recorded First row might be on recordings. Anything organizational: Ask Tanja on Thursday... 2 / 21
3 Setting: Alice and Bob want to chat. Alice Bob Eve 3 / 21
4 Security goals Secrecy, Integrity, Authenticity, Non-repudiation, (Privacy). 4 / 21
5 Security goals Secrecy, We focus on this today Integrity, Authenticity, Non-repudiation, (Privacy). 4 / 21
6 Setting: What about Eve? Alice Bob Eve 5 / 21
7 Attacker capabilities Passive: Listen. Active: Intercept & Manipulate. Change, add, drop content. 6 / 21
8 Encryption 7 / 21
9 Already the Greeks... 8 / 21
10 Later in Rome Kdoor Fubswr 9 / 21
11 Later in Rome Kdoor Fubswr Hallo Crypto Caesar cipher. Also known as ROT3. Key table : a b c d e f g h i j k l m d e f g h i j k l m n o p n o p q r s t u v w x y z q r s t u v w x y z a b c 9 / 21
12 Symmetric encryption Aka. secret key encryption. Examples: Caesar, Skytale,..., DES, AES. ONE (secret) key: Stick, rotation, bit string. 10 / 21
13 Symmetric encryption Aka. secret key encryption. Examples: Caesar, Skytale,..., DES, AES. ONE (secret) key: Stick, rotation, bit string. 10 / 21
14 Symmetric encryption Aka. secret key encryption. Examples: Caesar, Skytale,..., DES, AES. ONE (secret) key: Stick, rotation, bit string. 10 / 21
15 Semi-formal definition Symmetric Encryption Scheme A symmetric encryption scheme E = (Kg, Enc, Dec) consists of three PPT algorithms: Kg(1 n ): Key generation algorithm. Upon input of security parameter n in unary, outputs a secret key sk. Enc sk (m): Encryption algorithm. Upon input of a secret key sk and plaintext message m, outputs the encryption / ciphertext c of m under sk. Dec sk (c): Decryption algorithm. Upon input of a secret key sk and a ciphertext c, outputs the decryption m of c under sk. Such that: ( sk Kg(1 n ), m) : Dec sk (Enc sk (m)) = m (Completeness) Remark: Sometimes we use Enc(sk, m) Enc sk (m). 11 / 21
16 Semi-formal definition Symmetric Encryption Scheme A symmetric encryption scheme E = (Kg, Enc, Dec) consists of three PPT algorithms: Kg(1 n ): Key generation algorithm. Upon input of security parameter n in unary, outputs a secret key sk. Enc sk (m): Encryption algorithm. Upon input of a secret key sk and plaintext message m, outputs the encryption / ciphertext c of m under sk. Dec sk (c): Decryption algorithm. Upon input of a secret key sk and a ciphertext c, outputs the decryption m of c under sk. Such that: ( sk Kg(1 n ), m) : Dec sk (Enc sk (m)) = m (Completeness) Remark: Sometimes we use Enc(sk, m) Enc sk (m). 11 / 21
17 Security? Above definition only functional. What does it mean for an encryption scheme to be secure? Is Caesar cipher secure? Why not? Kdoor Fubswr Hallo Crypto 12 / 21
18 Semantic security Semantic security: Everything one can learn about a plaintext given its encryption, one can also learn without knowledge of the cipher text. Complicated formal definition. Hard to work with. Technical, equivalent notion: Indistinguishable ciphertexts. 13 / 21
19 Security Definitions Game-based: Adversary vs. Challenger 14 / 21
20 Indistinguishable ciphertexts (IND) m 1, m 2 sk, b Enc sk (m b ) b Adversary Challenger 15 / 21
21 Indistinguishable ciphertexts under choosen plaintext attacks (IND-CPA) Adversary might see more ciphertexts than the one she wants to learn more about. Attack against Enigma. To model worst-case, attacker is allowed to choose plaintexts and learn encryption of those. 16 / 21
22 Indistinguishable ciphertexts under choosen plaintext attacks (IND-CPA) m Enc sk (m) sk, b Adversary m 1, m 2 Enc sk (m b ) m Enc sk (m) b Challenger 17 / 21
23 Indistinguishable ciphertexts under choosen Ciphertext attacks (IND-CCA) Adversary might be able to learn decryptions of ciphertexts other than the target one. Users might leak plaintexts corresponding to ciphertexts the adversary saw. Practice: Often adversary only learns if a ciphertext is well-formed. Model: Additional access to decryption oracle. (Again, worst-case.) Oracle returns either Dec sk (c) or if c is no valid ciphertext. 18 / 21
24 Public-key encryption Symmetric encryption is very efficient, but how to share keys? Solution: Public-key / Asymmetric encryption. Key pair: Public encryption key pk and secret decryption key / private key sk. Public key can be published without requiring secrecy. Public key can be used to send encryption of (symmetric) secret key. 19 / 21
25 Semi-formal definition Asymmetric Encryption Scheme A symmetric encryption scheme E = (Kg, Enc, Dec) consists of three PPT algorithms: Kg(1 n ): Key generation algorithm. Upon input of security parameter n in unary, outputs a key pair (pk,sk). Enc pk (m): Encryption algorithm. Upon input of a public key pk and plaintext message m, outputs the encryption / ciphertext c of m under pk. Dec sk (c): Decryption algorithm. Upon input of a private key sk and a ciphertext c, outputs the decryption m of c under sk. Such that: ( (pk, sk) Kg(1 n ), m) : Dec sk (Enc pk (m)) = m (Completeness) 20 / 21
26 Part II: Break a toy public key encryption scheme. 21 / 21
27 Starting position p. 3
28 Selected nodes = private key p. 4
29 Perfect code we ll build one Each node is connected to exactly one selected node. Perfect code: there exists a selection of nodes so that each node is in the neighborhood of exactly one selected node (a selected node is in its own neighborhood.) p. 5
30 Additional edges To hide the structure of the selected nodes, further edges are included. These edges must not touch the selected nodes. This gives a perfect code proof it! p. 6
31 Public key All edges, no highlighting. p. 7
32 Encryption of m = = Partition 13, one share per node p. 8
33 Encryption of m = 13 For each node compute the sum of values at all nodes at distance at most 1, i.e. the value at the node itself plus all nodes directly connected to it = p. 9
34 Encrypted message For each node write the sum computed in the previous step next to it p. 10
35 Decryption Add values at points seleted as secret key = 13. Why does this work? p. 11
36 Overview A: 1. sheet: secret key (1), 2. sheet: public key (4) decryption (8) intermediate steps (1 3) B: 1. sheet: computations (5 6) 2.sheet: black numbers next to nodes (7) Why does this system work? Break the examples. Break this for graphs with 1000 nodes. p. 12
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