An Overview of Homomorphic Encryption

Transcription

1 An Overview of Homomorphic Encryption Alexander Lange Department of Computer Science Rochester Institute of Technology Rochester, NY May 9, 2011 Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

2 Outline 1 Algebraic Homomorphisms Group & Ring Homomorphism 2 Application to Cryptography Example: RSA 3 History Data Banks Blind Signatures 4 Additive Homomorphisms ElGamal Paillier 5 Applications E-Voting Private Information Retrieval 6 Fully Homorphic Encryption Overview Craig Gentry Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

3 Algebraic Homomorphisms Group & Ring Homomorphism Algebraic Homomorphisms Definition (Group Homomorphism) Let (G, ) and (H, ) be groups. The map ϕ : G H is a homomorphism if ϕ(x y) = ϕ(x) ϕ(y) x, y G Definition (Ring Homomorphism) Let R and S be rings with addition and multiplication. The map ϕ : R S is a homomorphism if 1 ϕ is a group homomorphism on the additive groups (R, +) and (S, +) 2 ϕ(xy) = ϕ(x)ϕ(y) x, y R Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

4 Application to Cryptography Application to Cryptography A homomorphic encryption function allows for the manipulation of encrypted data with out the seemingly inherent loss of the encryption. Applications E-Cash E-Voting Private information retrieval Cloud computing A fully homomorphic encryption function (two operations) has been an open problem in cryptography for 30+ years. The first ever system was proposed by Craig Gentry in However, encryption systems that respect one operation have been utilized for decades. Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

5 Application to Cryptography Example: RSA Example: The RSA Cryptosystem Definition (RSA) Let n = pq where p and q are primes. Pick a and b such that ab 1 (mod φ(n)). n and b are public while p, q and a are private. e K (x) = x b mod n d K (y) = y a mod n The Homomorphism: Suppose x 1 and x 2 are plaintexts. Then, e K (x 1 )e K (x 2 ) = x b 1 x b 2 mod n = (x 1x 2 ) b mod n = e K (x 1 x 2 ) Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

6 History Data Banks History On Data Banks and Privacy Homomorphisms Rivest, Adleman and Dertouzos, 1978 Introduced idea of Privacy Homomorphisms...it appears likely that there exist encryption functions which permit encrypted data to be operated on without preliminary decryption. Encrypted data of loan company What is the size of the average loan? How many loans over $5,000? Introduced four possible encryption functions (RSA was one of them) Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

7 History Blind Signatures History Blind signatures for untraceable payments David Chaum, 1982 Calls for payment system with: Anonymity of payment Proof of payment Analogy to secure voting Place vote in a carbon envelope The signer can then sign the envelope, consequently signing the vote with out ever knowing what the vote is Although no mention of a private homomorphism, the paper helps introduce the need for secure voting as well as the relationship between e-cash and e-voting Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

8 Additive Homomorphisms ElGamal ElGamal Cryptosystem Definition (ElGamal) Let p be a prime and pick α Z p such that α is a generator of Z p. Pick a and β such that β α a (mod p). p, α and β are public; a is private. Let r Z p 1 be a secret random number. Then, e K (x, r) = (α r mod p, xβ r mod p) The Homomorphism: Let x 1 and x 2 be plaintexts. Then, e K (x 1, r 1 )e K (x 2, r 2 ) = ( α r 1 mod p, x 1 β r 1 mod p )( α r 2 mod p, x 2 β r 2 mod p ) = ( α r 1 α r 2 mod p, x 1 β r 1 x 2 β r 2 mod p ) = ( α r 1+r 2 mod p, (x 1 x 2 )β r 1+r 2 mod p ) = e K (x 1 x 2, r 1 + r 2 ) Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

9 Additive Homomorphisms ElGamal ElGamal The Problem: This homomorphism is multiplicative E-cash and e-voting would benefit from an additive homomorphism One solution: Modify ElGamal Put the plaintext in the exponent If we modify ElGamal so that e K (x, r) = (α r mod p, α x β r mod p) Then the homomorphism is e K (x 1, r 1 )e K (x 2, r 2 ) = ( α r 1 mod p, α x 1 β r 1 mod p )( α r 2 mod p, α x 2 β r 2 mod p ) = ( α r 1+r 2 mod p, α x 1+x 2 β r 1+r 2 mod p ) = e K (x 1 + x 2, r 1 + r 2 ) Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

10 Additive Homomorphisms ElGamal The problem with this modification is that d K = α x, introducing the discrete logarithm problem into the decryption. For large enough texts, this becomes impractical. We would like another cryptosystem which takes advantage of this additive property of exponentiation, but does so with out extra decryption time. Solution: the Paillier Cryptosystem Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

11 Additive Homomorphisms Paillier Paillier Cryptosystem Introduced by Pascal Paillier in Public-Key Cryptosystems Based on Composite Degree Residuosity Classes, 1999 Probabilistic, asymmetric algorithm Decisional composite residuosity assumption Given composite n and integer z, it is hard to determine if y exists such that z y n (mod n 2 ) Homomorphic and self-blinding Extended by Damgård and Jurik in 2001 modulo n 2 = modulo n s+1 Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

12 Additive Homomorphisms Paillier Paillier Cryptosystem Definition Pick two large primes p and q and let n = pq. Let λ denote the Carmichael function, that is, λ(n) = lcm(p 1, q 1). Pick random g Z n such that L(g λ 2 mod n 2 ) is invertible modulo n (where L(u) = u 1 n ). n and g are public; p and q (or λ) are private. For plaintext x and resulting ciphertext y, select a random r Z n. Then, e K (x, r) = g m r n mod n 2 d K (y) = L(y λ mod n 2 ) L(g λ mod n 2 ) mod n Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

13 Applications E-Voting Paillier Example: E-Voting Suppose Alice, Bob and Oscar are running in an election. Only 6 people voted in the election, and the results are tabulated below. Vote Oscar Bob Alice 1! = 1 2! = 4 3! = 4 4! = 1 5! = 16 6! = 1 Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

14 Applications E-Voting Paillier Example: E-Voting Let p = 5 and q = 7. Then n = 35, n 2 = 1225 and λ = 12. g is chosen to be 141. For the first vote x 1 = 1, r is randomly chosen as 4. Then, e K (x 1, r 1 ) = e K (1, 4) = = = 359 mod 1225 All votes, r values and resulting encryptions are shown below x r e K (x, r) Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

15 Applications E-Voting Paillier Example: E-Voting In order to sum the votes, we multiply the encrypted data modulo n 2 : mod 1225 = 983 We then decrypt: L(y λ mod n 2 ) = L( mod 1225) = 36 1 = 1 35 L(g λ mod n 2 ) = L( mod 1225) = = d K (y) = ( L(y λ mod n 2 ) )( L(g λ mod n 2 ) ) 1 mod n = mod 35 = 27 We convert 27 to ( ) for the final results. Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

16 Applications Private Information Retrieval Another Application: Private Information Retrieval Idea first introduced by Chor, Goldreich, Kushilevitz and Sudan in 1997 The problem: How can the user access an item from a database with out the database knowing which item it is? (Private Information Retrieval) How can the user do this with out knowing about any other item of the database? (Symmetric Private Information Retrieval) The additive homomorphic properties of Paillier allow for the indexing and filtering of an encrypted database Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

17 Applications Private Information Retrieval Some PIR Protocols Stern Protocol - Uses a simple homomorphic scheme and a linear indexing technique Chang Protocol - Expands on Stern by allowing the indexing to take place on a hyper cube. Uses the Paillier Cryptosystem Lipmaa Protocol - Expands on Chang by using Damgård-Jurik system - removes the limit set on the plaintext due to Paillier Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

18 Fully Homorphic Encryption Overview Fully Homomorphic Encryption Up until now, the homomorphic systems described have been partially homomorphic They preserve the structures of multiplication or division, but cannot do both If a fully homomorphic encryption was implemented, then any arbitrary computation could be performed on a ciphertext, preserving the encryption as if the computation was performed on the plaintext The additive and multiplicative preservation of a Ring Homomorphism modulo 2 directly correspond to the XOR and AND operations of a circuit Applications: Private queries on search engines - The search engine would be able to return encrypted data with out every decrypting the query Cloud Computing - Storing encrypted data on the cloud is seemingly useless; no manipulation of the data can be obtained with out allowing the cloud access and/or decrypting the data off the cloud Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

19 Fully Homorphic Encryption Craig Gentry Craig Gentry In 2009, Craig Gentry proposed the first fully homomorphic encryption scheme in his PhD thesis A Fully Homomorphic Encryption Scheme Centers around a function which introduces a certain level of noise into the encryption Each operation on the ciphertext results in compounding noise Resolved with the bootstrapability of the encryption Each re-encryption cuts down the noise Analogy to Alice s jewelry shop Involves operations on Ideal Lattices Allows for less complex circuit implementation Correspond to the structure of Rings Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

20 Fully Homorphic Encryption Craig Gentry A Simple Example Over the Integers A Somewhat Homomorphic Scheme KEYGEN ɛ : Output a random odd integer p For bit m {0, 1}, let a random m = m mod 2 (ie. m is even if m = 0, odd if m = 1). Pick a random q. Then ENCRYPT ɛ (m, p) = c = m + pq. m is the noise associated with the plaintext. Let c = c mod p where c ( p/2, p/2). Then DECRYPT ɛ (p, c) = c mod 2. c is considered to be the noise associated with the ciphertext (ie. the shortest distance to a multiple of p) The Homomorphism: (Multiplication) Let m 1, m 2 {0, 1}. Then e(m 1, p)e(m 2, p) = (m 1 + pq 1 )(m 2 + pq 2 ) = d(c) = (m 1 + pq 1 )(m 2 + pq 2 ) mod p mod 2 = m 1 m 2 mod 2 = m 1 m 2 Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

21 Fully Homorphic Encryption Craig Gentry The Problem The compounding noise (m 1 m 2 in the example) results in loss of homomorphic property after a certain number of operations The bootstrapping of the algorithm allows for this noise to be reduced, allowing for no limit in operations However, the combination of the noise production followed by the noise reduction makes the scheme completely impractical Complexity grows as more and more operations are performed (inherent limitation of the algorithm) Gentry has stated that in order to perform one search on Google using this encryption, the amount of computations needed would increase by a trillion More schemes have been introduced to try and decrease this complexity, but all rely on the same Despite this impracticality, Gentry s discovery is an amazing breakthrough in cryptography and proves that (at least theoretical) fully homomorphic encryption schemes exist Alexander Lange (RIT) Homomorphic Encryption May 9, / 22

22 Fully Homorphic Encryption Craig Gentry Questions? Sources R. L. Rivest, L. Adleman, and M. L. Dertouzos. On data banks and privacy homomorphisms. Foundations of Secure Computation, Craig Gentry. Computing Arbitrary Functions of Encrypted Data. Association for Computing Machinery, Pascal Paillier. Public-Key Cryptosystems Based on Composite Degree Residuosity Classes. EUROCRYPT Laura Lincoln. Symmetric Private Information Retrieval via Additive Homomorphic Probabilistic Encryption. RIT, Department of Computer Science, David Chaum. Blind signatures for untraceable payments. Advances in Cryptology - Crypto 82, Springer-Verlag 1983 Paillier Cryptosystem Interactive Simulator. Andreas Steffen. HSR Hochschule fur Technik Rapperswil Steve Weis. Verifying Elections with Cryptography. Google Tech Talks: Theory and Practive of Cyptography. December Youtube. Alexander Lange (RIT) Homomorphic Encryption May 9, / 22