Talk announcement please consider attending!


 Richard Bridges
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1 Talk announcement please consider attending! Where: Maurer School of Law, Room 335 When: Thursday, Feb 5, 12PM 1:30PM Speaker: Rafael Pass, Associate Professor, Cornell University, Topic: Reasoning Cryptographically About Knowledge ABSTRACT: Cryptographic notions of knowledge consider the knowledge obtained, or possessed, by computationallybounded agents under adversarial conditions. In this talk, we will survey some recent cryptographicallyinspired approaches for reasoning about agents' knowledge in the context of mechanism design, coding theory, voting theory and weather forecasting.
2 Optional lab for Assignment 1 solutions: Date: TOMORROW (Tuesday, Feb 3) Time: 1PM 2PM Place: LH 101 (Assignment 2 is due one week from Wednesday!)
3 a1q3a Many students did not follow my hint! Let E be the event that m has property P and let F N be the event that A(m) outputs N consecutive zeros Question asks you to solve for Pr E F N A common (but incorrect) response: Pr[E] = ε (true) Pr F 0 E = δ (true) Pr F 0 E = 1 δ (true) Hence, Pr F 0 = ε δ (true) Hence, Pr F 0 = ε (1 δ) (not true!!! ) Hence, Pr F N = ε (1 δ) N (not true!!! ) Hence Pr[E F N ] = (1 ε 1 δ N ) (not true!!!) What is this probability???
4 a1q3c Another ubiquitous mistake (nobody lost marks, though). What is wrong with the following? 1 2 N /99 N hence, N = 24 consecutive 0s suffice. Log Log N 1 / N N N N 23.24
5 Last Wednesday s lecture: Pseudorandom functions and permutations Birthday bounds Block ciphers Modes of operation (maybe?) Today s lecture: Modes of operation Message authentication codes
6 Review: PRF formal definition Let Func[n] be the set of all function from {0,1} n to 0,1 n Def: A keyed function F: {0,1} {0,1} {0,1} is a pseudorandom function (PRF) iff, for every PPT distinguisher D, there exists a negligible function ε: N R + such that Pr D F(k, ) 1 n = 1 k R 0,1 n Pr D f 1 n = 1 f R Func[n] ε(n)
7 Review: PRF indistinguishability game 1 n 1 N Game 0: Challenger k R 0,1 n D given oracle access to F(k, ) Distinguisher D F(k, ) 1 n 1 N b {0,1} Game 1: 1 n 1 N Challenger f R Func[n] D given oracle access to f Distinguisher D f( ) 1 n 1 N b {0,1} Distinguisher s wins if it outputs b = 0 in Game 0 or b = 1 in Game 1
8 Review: PRP formal definition Let Perm[n] denote the set of all permutations on {0,1} n A keyed permutation is efficient if there is a PPT algorithm to compute its inverse. Def: A keyed permutation Π: {0,1} {0,1} {0,1} is a pseudorandom permutation (PRP) iff it is efficient and, for every PPT distinguisher D, there exists a negligible function ε: N R + such that Pr D Π(k, ) 1 n = 1 k R 0,1 n Pr D π( ) 1 n = 1 π R Perm[n] ε(n)
9 Review: Strong PRPs Def: A keyed bijection Π: {0,1} {0,1} {0,1} is a strong pseudorandom permutation (SPRP) iff it is efficient and, for every PPT distinguisher D, there exists a negligible function ε: N R + such that Pr D Π k,,π 1 (k, ) 1 n = 1 k R 0,1 n Pr D π,π 1 ( ) 1 n = 1 π R Perm[n] ε(n) NOTE: Distinguisher gets oracle access both to the permutation and to the inverse permutation!
10 Fixedlength encryption from strong PRPs Gen(1 n ) outputs k R 0,1 n Enc k (m) outputs c Π k, m Dec k (c) outputs m Π 1 k, c Is this secure? Goals: YES, but only if the key is used to encrypt a single fixedlength plaintext Security for multiple encryptions variablelength encryption with short ciphertexts
11 I538: Introduction to Cryptography Ryan Henry Computationally secret encryption Modes of operation
12 Electronic Code Book (ECB) m m 1 m 2 m l m 1 m 2 k k k m 3 Π k Π k Π k c 1 c 2 c l c c 1 c 2 c l Split m into k bit blocks and apply PRP Π k to each block individually Does ECB mode offer INDCPA security? NO! Indistinguishable encryptions in the presence of an eavesdropper? NO!
13 Insecurity of ECB mode (b, 1 n ) 0,1 1 N 1 n 1 N Challenger m 0, m 1 M, m 0 = m 1 k Gen 1 n c Enc k (m b ) Attacker (A) b {0,1} Choose m, m {0,1} n one block each Set m 0 m m and m 1 m m two blocks each Output 1 if c has form c 1 c 1 and output 0 if c has form c 1 c 2 Advantage is Adv eav A = 1 2
14 Insecurity of ECB mode (in picture form) plaintext (bitmap) ECB mode ciphertext CBC mode ciphertext
15 Cipher Block Chaining (CBC) m m 1 m 2 m l m 1 m 2 k k k m l Π k Π k Π k IV c 1 c 2 c l c IV, c 1 c 2 c l Choose IV R {0,1} n and XOR first block with IV prior to encryption XOR each subsequent plaintext block with ciphertext of preceding block Reveal IV along with ciphertext
16 Output Feedback (OFB) m m 1 m 2 m l k k k F k F k F k IV m 1 c 1 m 2 c 2 m l c l c (IV, c 1 c 2 c l ) Choose IV R {0,1} n and XOR first block with F k (IV) i To encrypt ith plaintext block, XOR by F k Reveal IV along with ciphertext IV
17 Counter mode (CTR) m m 1 m 2 m l k IV+1 k IV+2 k IV+l F k F k F k m 1 m 2 m l c 1 c 2 c l c (IV, c 1 c 2 c l ) Essentially a blockbyblock stream cipher
18 Comparing modes of operation Electronic Code Book (ECB) mode Pros: simple, parallelizable, nonmalleable Cons: no CPA security, no secrecy in the presence of an eavesdropper Cipher Block Chaining (CBC) mode Pros: CPAsecure, parallelizable decryption Cons: encryption is inherently sequential, malleable Output Feedback (OFB) mode Pros: CPAsecure Cons: encryption and decryption both inherently sequential, malleable Counter (CTR) mode Pros: CPAsecure, fully parallelizable Cons: malleable
19 I538: Introduction to Cryptography Ryan Henry Message authentication codes Motivation, definitions, and constructions
20 Secrecy vs Authenticity So far we have only worried about message secrecy Secrecy / authenticity! OTP, stream cipher, CTR and OFB mode Flip bit of CT flip corresponding bit of PT ECB mode Flip bit of CT randomize block of PT Add/remove/permute CT blocks add/remove/permute PT blocks CBC mode Are authenticity attacks possible? YES! Is this detectable?
21 Malleability of CBC mode c (IV, c 1 c 2 c 3 ) c 1 c 2 k k k c 3 IV What integrity attacks are possible? Π 1 k Π 1 k Π 1 k m 1 m 2 m 3 Flip bit in IV flip corresponding bit in m 1 Truncate c 3 from CT truncate m 3 from PT Flip bit in c i randomize m i and flip corresponding bit in m i+1 m m 1 m 2 m 3
22 Noncryptographic integrity checks Cyclic Redundancy Check (CRC) Detects random errors due to noise channel Not intended to detect intentional errors introduced by an Attacker Straw man proposal: Alice computes c Enc k (m) using CTR mode block cipher Alice computes t CRC(c) Ciphertext is c = c t What can possibly go wrong? c is malleable attacker can modify c and then update t to match!
23 Noncryptographic integrity checks A candidate fix: Compute t CRC m Ciphertext is c Enc k (m t) What can possibly go wrong now? CRC is linear!! CRC x y z = CRC x CRC y CRC(z) Integrity in the presence of attackers requires a secret key
24 Message authentication codes (MACs) Def: A message authentication code (MAC) is a triple of algorithms (Gen, MAC, Ver) Gen: 1 N K is a randomized algorithm MAC: K M T is a (usually) deterministic algorithm Ver: K M T {0,1} is a deterministic algorithm Often write MAC k (m) instead of MAC k, m, Ver k (m, t) instead of Ver k, m, t M is the message space K is the key space T is the tag space set of possible messages set of possible keys set of possible tags
25 Correctness Intuitively: the property of being able to verify (given the right key) Def: A MAC (Gen, MAC, Ver) is correct if k K and m M, Pr[ Ver k (m, MAC k (m)) = 1 ] = 1
26 MACForgery game 1 n 1 N MAC Oracle m 1 M Attacker 1 n 1 N k Gen(1 n ) t 1 MAC k (m 1 ) m 2 M t 2 MAC k (m 2 ) A MAC(k, ) m q M t q MAC k (m q ) (m, t) M T Attackers wins if (m, t) {(m 1, t 1 ),..., (m q, t q )} and Ver k m, t = 1
27 Existential unforgeability Intuitively: the property of being secure against malicious tampering A s success probability in the MACForgery game denoted Adv MAC A Def: A MAC (Gen, MAC, Ver) is existentially unforgeable under adaptive chosen message attacks if, for every PPT adversary A, there exists a negligible function ε: N R + such that Adv MAC A ε n.
28 Fixedlength MAC from PRFs Gen(1 n ) outputs k R 0,1 n MAC k (m) outputs t F k, m Ver k (m, t) outputs 1 if F k, m = t and 0 otherwise Thm: If F: {0,1} n {0,1} n {0,1} n is a PRF, then (Gen, MAC, Ver) as defined above is existentially unforgeable under adaptive chosen message attacks.
29 PRF fixedlength MAC reduction PRF distinguisher (D) 1 n, O D gets oracle access either to MAC k ( ) or to a PRF F k, In both cases, k R {0,1} n oracle D has advantage Adv PRF D = Adv MAC A! b 1 n t 1 O(m 1 ) t q O(m q ) MAC attacker (A) m 1 m q Adv CPA A = ε(n) (m, t) /
30 Replay attacks MACs guarantee the authenticity of messages in most cases An important case where they do not is replay attacks Common ways to prevent replay: sequence numbers and timestamps
31 Naïve CBCMAC Let F k k {0,1} be any PRF family Gen(1 n ) outputs k R 0,1 n MAC k m does the following: split m into nbit blocks m 1,..., m l initialize t 0 = 0 n compute t i = F k (t i 1 m i ) for i = 1,..., l output the tag t = t l Ver k (m, t) outputs 1 if t = MAC k (m) and 0 otherwise
32 Naïve CBCMAC m m 1 m 2 m 3 m 1 m 2 k k k m 3 F k F k F k 0 n t Is naïve CBCMAC existentially unforgeable under adaptive chosen message attacks? NO! (but why?)
33 Attacking naïve CBCMAC 1 n 1 N MAC Oracle Attacker 1 n 1 N k Gen(1 n ) A MAC(k, ) m M t MAC k (m) Indeed: t 1 F k m = t so that t 2 F k m t t = t (m m t, t) M T
34 CBCMAC Fix 1: Prepend the length Let F k k {0,1} be any PRF family Gen(1 n ) outputs k R 0,1 n MAC k m does the following: split m into nbit blocks m 1,..., m l initialize t 0 F k (l) pad l to n bits compute t i F k (t i 1 m i ) for i = 1,..., l output the tag t F k (t l )? Ver k (m, t) outputs 1 if t = MAC k (m) and 0 otherwise
35 CBCMAC Fix 2: nested CBCMAC (NMAC) Let F k k {0,1} be any PRF family Gen(1 n ) outputs k = (k 1, k 2 ) R 0,1 n {0,1} n MAC k m does the following: split m into nbit blocks m 1,..., m l initialize t 0 0 n compute t i F k1 (t i 1 m i ) for i = 1,..., l output the tag t F k2 (t l )? Naïve CBCMAC on the tag using second key Ver k (m, t) outputs 1 if t = MAC k (m) and 0 otherwise
36 CBCMAC Fix 3: Use a lengthspecific key Let F k k {0,1} be any PRF family Gen(1 n ) outputs k R 0,1 n MAC k m does the following: split m into nbit blocks m 1,..., m l compute a key k l F k (l) initialize t 0 0 n compute t i F kl (t i 1 m i ) for i = 1,..., l output the tag t t l derive lengthspecific key from length + master key? Ver k (m, t) outputs 1 if t = MAC k (m) and 0 otherwise
37 CBCMAC vs CBC mode encryption CBC mode uses random IV Otherwise, cannot be INDCPA secure! CBCMAC uses a fixed IV Otherwise, cannot be existentially unforgeable against chosen message attacks! CBC mode outputs every block of ciphertext Otherwise, cannot be correct! CBCMAC only outputs last block Otherwise, cannot be existentially unforgeable against chosen message attacks!
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