Overview of Cryptography

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1 18739A: Foundations of Security and Privacy Overview of Cryptography Anupam Datta CMU Fall

2 Is Cryptography A tremendous tool The basis for many security mechanisms Is not The solution to all security problems Reliable unless implemented properly Reliable unless used properly Something you should try to invent yourself unless you spend a lot of time becoming an expert you subject your design to outside review

3 Basic Cryptographic Concepts Encryption scheme: functions to encrypt, decrypt data key generation algorithm symmetric key vs. asymmetric (public) key Symmetric key: more efficient, generally key = key -1 Public key: publishing key does not reveal key -1 Signature scheme Functions to sign data, verify signature Hash function, MAC Map input to short hash; ideally, no collisions MAC (keyed hash) used for message integrity All of these primitives are used in SSL/TLS

4 Our Approach Analyze systems and protocols that use cryptographic primitives such as SSL, assuming that the primitives are themselves secure How is security of cryptographic primitives defined? Today s lecture How are primitives constructed that satisfy these definitions? : Applied Cryptography : Introduction to Theoretical Cryptography Will put up additional slides for optional reading

5 We will cover Symbolic model Modeling perfect cryptography always secure Abstraction enables mechanized analysis Complexity-theoretic model Security definitions given by games Security properties guaranteed with high probability (almost always) against probabilistic polynomial time adversaries

6 Symmetric encryption Can compute message m and key k from set S of messages implies can compute encryption {m} k from the same set S (1) S - m S -k S - {m} k Can compute encrypted message (cipher-text) {m} k and key k implies can recover message m (2) S - {m} k S - k S - m Formally, messages are terms of a term algebra

7 Asymmetric encryption Can compute message m and public key K implies can compute encryption {m} K (3) S - m S -K S - {m} K Can compute encrypted message (cipher-text) {m} K and inverse key K -1 implies can recover message m (4) S - {m} K S - K -1 S - m Each principal has a public-private encryption key pair <K, K -1 >

8 Digital signature Can compute message m and signing key s -1 implies can compute signature sig(s -1, m) (5) S - {m} S - s -1 S - sig(s -1, m) Given signature sig(s -1, m) and inverse key s, can verify signature Usually modeled using pattern matching

9 Hash function Can compute message m implies can compute hash h(m) (6) S - m S - h(m) Given message m and key k, can compute keyed hash h(k,m) (7) S - m S - k S - h(k,m) (7) Is a special case of (6) in the symbolic model

10 We will cover Symbolic model Modeling perfect cryptography always secure Abstraction enables mechanized analysis Complexity-theoretic model Security definitions given by games Security properties guaranteed with high probability (almost always) against probabilistic polynomial time adversaries

11 Digital signatures mi Sig(S -1,mi) Messages are bit-strings Signature scheme <K, S, V> C Sig(S -1,m) A Attacker is a PPT Turing Machine Attacker wins if m?mi UF-CMA security: PPT attackers A negligible function f n0 security parameters n = n0 Prob [m?mi A plays by the rules] <= f(n) Intuition: Signatures should be unforgeable with high probability

12 Keyed Hash (MAC) mi h(k,mi) Messages are bit-strings Attacker is a PPT Turing Machine C h(k,m) A Attacker wins if m?mi CMA security: PPT attackers A negligible function f n0 security parameters n = n0 Prob [m?mi A plays by the rules] <= f(n) Intuition: MACs should be unforgeable with high probability

13 Asymmetric encryption {m} K m Messages are bit-strings m0, m1 Attacker is a PPT Turing Machine C {mb} K {m} K A Attacker wins if d = b m d IND-CCA2 security: PPT attackers A negligible function f n0 security parameters n = n0 Prob [d = b A plays by the rules] <= ½ + f(n) Intuition: Encryption reveals no information about message

14 Hash functions Preimage resistant Second preimage resistant Collision resistant Given hash function h: X Y Find: x, x X s.t x? x and h(x) = h(x ) Should be difficult! PPT attackers A negligible function f n0 security parameters n = n0 Prob [A finds x, x X s.t x? x and h(x) = h(x )] <= f(n)

15 Putting it together: SSL/TLS N1, Version1 N2, Version2, sig(ca -1, S, Ks) C Sig(CA -1, C, Kc), Sig(Kc, handshake1), {secret} Ks, h(secret, handshake1, client ) S h(secret, handshake2, server )

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