Information Security Theory vs. Reality
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1 Information Security Theory vs. Reality , Winter 2011 Lecture 14: More on vulnerability and exploits, Fully homomorphic encryption Eran Tromer Slides credit: Vinod Vaikuntanathan (U. Toronto) 1
2 2 More on vulnerability exploitation
3 Case study: sudo format string vulnerability Report: 3
4 Case study: sudo format string vulnerability (cont.) Sourcecode: 4
5 Case study: sudo format string vulnerability (cont.) Sourcecode diff: 5
6 Case study: sudo format string vulnerability (cont.) Report: 6
7 Case study: MS buffer overrun Report: 7
8 Case study: MS buffer overrun (cont.) Report: 8
9 Case study: MS buffer overrun Report: 9
10 Understanding binary patches: BinDiff 10
11 Understanding binary patches: BinDiff (cont.) 11
12 Metasploit Framework Framework for vulnerability exploitation and penetration testing Capabilities Library of exploit codes Library of payloads (shells, VNC) Victim fingerprinting Opcode database (instruction addresses for various software versions) Exploit encoding (avoiding special character, intrustion and intrusion detection systems) Modular architecture, many add-ons Powerful scriptable command-line interface Convenient GUI and web interfaces 12
13 Metasploit Framework (cont.) Book: Kennedy, O Gorman, Kearns, Aharoni, Metasplit: The Penetration Tester s Guide (2011 edition) Numerous on-line tutorials Example: 13
14 Metasploit Framework: back to MS Demo: 14
15 Meanwhile, in theory-land Fully Homomorphic Encryption 15
16 16 of 32 The goal Delegate processing of data without giving away access to it
17 17 of 32 Example 1: Private Search Delegate PROCESSING of data without giving away ACCESS to it You: Encrypt the query, send to Google (Google does not know the key, cannot see the query) Google: Encrypted query Encrypted results (You decrypt and recover the search results)
18 18 of 32 Example 2: Private Cloud Computing Delegate PROCESSING of data without giving away ACCESS to it Encrypt x Enc(x), P Enc(P(x)) (Input: x) (Program: P)
19 19 of 32 Fully Homomorphic Encryption Encrypted x, Program P Encrypted P(x) Definition: (KeyGen, Enc, Dec, Eval) (as in regular public/private-key encryption) Correctness of Eval: For every input x, program P If c = Enc(PK, x) and c = Eval (PK, c, P), then Dec (SK, c ) = P(x). Compactness: Length of c independent of size of P Security = Semantic Security [GM82]
20 x Fully Homomorphic Encryption [Rivest-Adleman-Dertouzos 78] Enc(x) Knows nothing of x. Function f Eval: f, Enc(x) Enc(f(x)) homomorphic evaluation 20 of 32
21 21 of 32 Fully Homomorphic Encryption First Defined: Privacy homomorphism [RAD 78] their motivation: searching encrypted data
22 22 of 32 Fully Homomorphic Encryption First Defined: Privacy homomorphism [RAD 78] their motivation: searching encrypted data Limited Variants: RSA & El Gamal: multiplicatively homomorphic GM & Paillier: additively homomorphic X c* = c 1 c 2 c n = (m 1 m 2 m n ) e mod N c 1 = m 1 e c 2 = m 2 e c n = m n e
23 Fully Homomorphic Encryption First Defined: Privacy homomorphism [RAD 78] their motivation: searching encrypted data Limited Variants: RSA & El Gamal: multiplicatively homomorphic GM & Paillier: additively homomorphic BGN 05 & GHV 10: quadratic formulas NON-COMPACT homomorphic encryption: 23 of 32 Based on Yao garbled circuits SYY 99 & MGH 08: c* grows exp. with degree/depth IP 07 works for branching programs
24 Fully Homomorphic Encryption First Defined: Privacy homomorphism [RAD 78] their motivation: searching encrypted data Big Breakthrough: [Gentry09] First Construction of Fully Homomorphic Encryption using algebraic number theory & ideal lattices Full course last semester Today: an alternative construction [DGHV 10]: using just integer addition and multiplication 24 of 32 easier to understand, implement and improve
25 25 of 32 Constructing fully-homomoprhic encryption assuming hardness of approximate GCD
26 A Roadmap 1. Secret-key Somewhat Homomorphic Encryption (under the approximate GCD assumption) (a simple transformation) 2. Public-key Somewhat Homomorphic Encryption (under the approximate GCD assumption) (borrows from Gentry s techniques) Public-key FULLY Homomorphic Encryption (under approx GCD + sparse subset sum)
27 Secret-key Homomorphic Encryption Secret key: a large n 2 -bit odd number p (sec. param = n) To Encrypt a bit b: pick a random large multiple of p, say q p pick a random small even number 2 r (q ~ n 5 bits) (r ~ n bits) Ciphertext c = q p+2 r+b noise To Decrypt a ciphertext c: c (mod p) = 2 r+b (mod p) = 2 r+b read off the least significant bit 27
28 Secret-key Homomorphic Encryption How to Add and Multiply Encrypted Bits: Add/Mult two near-multiples of p gives a near-multiple of p. c 1 = q 1 p + (2 r 1 + b 1 ), c 2 = q 2 p + (2 r 2 + b 2 ) c 1 +c 2 = p (q 1 + q 2 ) + 2 (r 1 +r 2 ) + (b 1 +b 2 ) «p LSB = b 1 XOR b 2 c 1 c 2 = p (c 2 q 1 +c 1 q 2 -q 1 q 2 ) + 2 (r 1 r 2 +r 1 b 2 +r 2 b 1 ) + b 1 b 2 «p LSB = b 1 AND b 2 28
29 Problems Ciphertext grows with each operation Useless for many applications (cloud computing, searching encrypted ) Noise grows with each operation Consider c = qp+2r+b Enc(b) c (mod p) = r 2r+b lsb(r ) b 2r+b r 29 (q-1)p qp (q+1)p (q+2)p
30 Problems Ciphertext grows with each operation Useless for many applications (cloud computing, searching encrypted ) Noise grows with each operation Can perform limited number of hom. operations What we have: Somewhat Homomorphic Encryption 30
31 Public-key Homomorphic Encryption Secret key: an n 2 -bit odd number p Public key: [q 0 p+2r 0,q 1 p+2r 1,,q t p+2r t ] = (x 0,x 1,,x t ) t+1 encryptions of 0 Wlog, assume that x 0 is the largest of them To Decrypt a ciphertext c: c (mod p) = 2 r+b (mod p) = 2 r+b read off the least significant bit 31 Eval (as before)
32 Public-key Homomorphic Encryption Secret key: an n 2 -bit odd number p Public key: [q 0 p+2r 0,q 1 p+2r 1,,q t p+2r t ] = (x 0,x 1,,x t ) To Encrypt a bit b: pick random subset S c = xi + 2r + b (mod x 0 ) i S [1 t] 32 To Decrypt a ciphertext c: q c (mod p) = 2 r+b i r + (mod r p) = i 2 r+b c = p[ ] + 2[ ] + b (mod kx 0 (for x 0 ) a small k) i S read off the qleast i kq 0 significant r + bit r i kr 0 Eval (as(mult. before) of p) + ( small even noise) + b i S = p[ ] + 2[ ] + b i S i S
33 Public-key Ciphertext Homomorphic Size Reduction Encryption Secret key: an n 2 -bit odd number p Public key: [q 0 p+2r 0,q 1 p+2r 1,,q t p+2r t ] = (x 0,x 1,,x t ) To Encrypt a bit b: pick random subset S [1 t] Resulting ciphertext < x c = xi + 2r 0 + b (mod x 0 ) i S Underlying bit is the same (since x 0 has even noise) To Noise Decrypt does a ciphertext not increase c: by much (*) c (mod p) = 2 r+b (mod p) = 2 r+b read off the least significant bit 33 Eval: Reduce mod x 0 after each operation (*) additional tricks for mult
34 A Roadmap Secret-key Somewhat Homomorphic Encryption Public-key Somewhat Homomorphic Encryption Public-key FULLY Homomorphic Encryption
35 How Somewhat Homomorphic is this? Can evaluate (multi-variate) polynomials with m terms, and maximum degree d if d << n. 2 nd m 2 < p / 2 = 2 n / 2 or d ~ n f(x 1,, x t ) = x 1 x 2 x d + + x 2 x 5 x d-2 m terms Say, noise in Enc(x i ) < 2 n Final Noise ~ (2 n ) d + +(2 n ) d = m (2 n ) d 35
36 From Somewhat to Fully Theorem [Gentry 09]: Convert bootstrappable FHE. FHE = Can eval all fns. Somewhat Bootstrappable HE Augmented Decryption ckt. NAND Dec Dec 36 c 1 sk c 2 sk
37 Is our Scheme Bootstrappable? What functions can the scheme EVAL? (polynomials of degree < n) (?) Complexity of the (aug.) Decryption Circuit (degree ~ n 1.73 polynomial) Can be made bootstrappable Similar to Gentry 09 Caveat: Assume Hardness of Sparse Subset Sum 37
38 Security (of the somewhat homomorphic scheme) 38
39 The Approximate GCD Assumption Parameters of the Problem: Three numbers P,Q and R p q 1 p+r 1 p? (q 1 p+r 1,, q t p+r t ) q 1 [0 Q] Assumption: no PPT adversary r 1 [-R R] can guess the number p odd p [0 P] 39
40 p (q 1 p+r 1,, q t p+r t ) p? Assumption: no PPT adversary can guess the number p (proof = of security) Semantic Security [GM 82]: no PPT adversary can guess the bit b PK =(q 0 p+2r 0,{q i p+2r i }) Enc(b) =(qp+2r+b) 40
41 Progress in FHE Galactic Efficient [BV11a, BV11b, BGV11, GHS11, LTV11] asymptotically: nearly linear-time* algorithms practically: a few milliseconds for Enc, Dec [LNV11,GHS11] Strange assumptions Mild assumptions [BV11b, GH11, BGV11] Best Known [BGV11]: (leveled) FHE from worst-case hardness of n O(log n) -approx short vectors on lattices 41 *linear-time in the security parameter
42 42 sk 1, pk 1 Multi-key FHE x 1 Function f sk 2, pk 2 x 2
43 43 sk 1, pk 1 Multi-key FHE x 1 Dec y = Eval(f,c 1,c 2 ) Function f sk 2, pk 2 x 2 orrectness: Dec(sk 1,sk 2 y)=f(x 1,x 2 )
44 Fully homomorphic encryption: discussion Assumptions Mathematical Adversarial model Applicability Decryption? Keys? Alternative: multiparty computation When interaction is free What about integrity? Computationally-sound proofs, proof-carrying data 44
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