Homomorphic Encryption
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Homomorphic encryption allows computations to be carried out on encrypted data without decrypting it first. This summary discusses Craig Gentry's scheme for fully homomorphic encryption based on ideal lattices. The scheme works by encrypting bits as ciphertexts with small noise that grows with computations. A bootstrapping procedure called re-crypt reduces the noise to keep ciphertexts decryptable. While promising for applications like cloud computing, the scheme has high computational costs that scale poorly with security level. Current research aims to make homomorphic encryption more efficient and practical.
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![Craig Gentry scheme (integers)
Key: odd integer p > 2N
Encryption algorithm: given a bit b -> E(b) = c = b +
2x + kp, where x is in [-n/2,n/2] and k is an integer
chosen from some range
Decryption algorithm: b = (c mod p) mod 2, where (c
mod p) is the noise and belongs to [-n,n]
Decryption works if b + 2x [-N,N] [-p/2,p/2]](https://image.slidesharecdn.com/homomorphic-130320052226-phpapp01/85/Homomorphic-Encryption-9-320.jpg)
![Craig Gentry scheme (integers)
Graig Gentry schemes homomorphic assumptions
Addiction: c1 + c2 = b1+ b2 + 2(x1+x2) + (k1+k2)p =
b1 xor b2 + 2x + kp
Decryption works if (b1+2x1) + (b2+2x2) is in [-
N,N]
Multiplication: c1*c2 = b1*b2 + 2(b1x2 + b2x1 +
2x1x2) + kp = b1*b2 + 2x + kp
Decryption works if (b1+2x1) * (b2+2x2) is in [-
N,N]](https://image.slidesharecdn.com/homomorphic-130320052226-phpapp01/85/Homomorphic-Encryption-10-320.jpg)
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Homomorphic Encryption
- 1. Homomorphic Encryption Craig Gentry scheme
- 2. Why homomorphic encryption? Proposed by Rivest, Adleman and Dertouzos Confidentiality problems Ability to compute over ciphertext instead of plaintext One could use information without knowing the content of that information Privacy garanteed
- 3. Homomorphic Encryption Crypto Magic 5 * 6 = CT(5) * CT(6) -> D ( k, E(k,5) * E(k,6) ) = 5 * 6 Homomorphic Assumption Partially homomorphic/fully homomorphic
- 4. Homomorphic Encryption Partially homomorphic schemes RSA: CT(x)*CT(y) = (xe mod M) * (ye mod M) = xeye mod M = (xy)e mod M = CT(x*y), where e is the exponent key and M the modulus p=61; q=53; N=3233; 陸(N)=60*52=3120; e=17; d=2753;
- 5. Homomorphic Encryption Partially homomorphic schemes RSA: Obtain 5*6 performing RSA(5)*RSA(6) RSA(5) = 517 (mod 3233) = 3086; RSA(6) = 617 (mod 3233) = 824; 3068*824 = 2542864; RSA-1(2542864) = 25428642753 (mod 3233) = 30; 5*6 = 30;
- 6. Homomorphic Encryption Fully homomorphic schemes Craig Gentry scheme Based on ideal lattices Zaryab Khan scheme Based on perfectly colorblind function
- 7. Craig Gentry scheme Suppose a scheme with a noise parameter attached to each CT; Encryption algorithm outputs a CT with a small noise parameter (say less than n); Decryption algorithm only works if noise is less than some parameter N >> n; To compute E(a+b) / E(a*b), include noise; This gives a somewhat homomorphic scheme.
- 8. Craig Gentry scheme Now suppose a new algorithm RECRYPT, such that: Input: E(a), with noise N < N Output: E(a), with noise N Somewhat homomorphic -> fully homomorphic! Apply RECRYPT to E(a) and E(b) to ensure that the noise in E(a*b) or E(a+b) is smaller than N Bootstrappable
- 9. Craig Gentry scheme (integers) Key: odd integer p > 2N Encryption algorithm: given a bit b -> E(b) = c = b + 2x + kp, where x is in [-n/2,n/2] and k is an integer chosen from some range Decryption algorithm: b = (c mod p) mod 2, where (c mod p) is the noise and belongs to [-n,n] Decryption works if b + 2x [-N,N] [-p/2,p/2]
- 10. Craig Gentry scheme (integers) Graig Gentry schemes homomorphic assumptions Addiction: c1 + c2 = b1+ b2 + 2(x1+x2) + (k1+k2)p = b1 xor b2 + 2x + kp Decryption works if (b1+2x1) + (b2+2x2) is in [- N,N] Multiplication: c1*c2 = b1*b2 + 2(b1x2 + b2x1 + 2x1x2) + kp = b1*b2 + 2x + kp Decryption works if (b1+2x1) * (b2+2x2) is in [- N,N]
- 11. Craig Gentry scheme (integers) Addition example: 4+4 CT(100): 22 21 21 CT(1) = 1 + 2*3 + 5*3 = 22 +22 21 21 CT(0) = 0 + 2*3 + 5*3 = 21 44 42 42 CT(0) = 0 + 2*3 + 5*3 = 21 D(44 42 42): D(44) = 44 mod 3 = 2 D(42) = 42 mod 3 = 0 1000 = 8 = 4+4 D(42) = 42 mod 3 = 0
- 12. Craig Gentry scheme (integers) Multiplication example: 4*4 CT(100): CT(1) = 1 + 2*3 + 5*3 = 22 22 21 21 CT(0) = 0 + 2*3 + 5*3 = 21 22 21 21 CT(0) = 0 + 2*3 + 5*3 = 21 484 924 1365 882 441 D(484 924 1365 882 441): D(484) = 484 mod 3 = 1 D(924) = 924 mod 3 = 0 D(1365) = 1365 mod 3 = 0 10000 = 16 = 4*4 D(882) = 882 mod 3 = 0 D(441) = 441 mod 3 = 0
- 13. Craig Gentry scheme (ideal lattices) Replace integers by ideal lattices Ideal lattices have many representations or bases Bases: Good: good to decrypt, bad to encrypt Bad: bad to decrypt, good to encrypt Public key scheme, where good bases are private keys and bad bases are public keys
- 14. Cryptography over lattices L = 龍(B) = {Bc : c Zk}, B Rnk, where the k columns of the basis are linearly independent NP-hard problems over lattices: SVP (shortest vector problem): given a basis for lattice L of size n, find the shortest nonzero vector v L s.t. ||v|| = 了(L); CVP (closest vector problem): given a basis for lattice L of size n and a vector t Rn, find a nonzero vector v L s.t. ||t-v|| 粒;
- 15. Cryptography over lattices NP-hard problems over lattices: SIVP (shortest independent vector problem): like the SVP, except the output are linearly independent vectors v1, , v2 L of length at most 了(L); BDDP (bounded distance decoding problem): same as CVP but with the promise that there is a unique solution.
- 16. Craig Gentry scheme Why inefficient? CT size and computation time increase sharply as the security level increases; 2k security -> CT size and computation time are high-degree polynomials in k; Efforts are being made to reduce the computational requirements of Craig Gentry construction
- 17. Homomorphic Encryption Nowadays: Craig Gentry presented a working implementation of the fully homomorphic system, including the bootstrapping function Exists a practical application of homomorphic encryption to a hybrid wireless network Perform statistical tests over encrypted data such as temperature, humidity, etc. There are also some practical implementations of simplifications of this scheme over databases
- 18. Problems solved Cloud security Problems related to personal records like medical records Work with information stored in databases Querys to search engines
- 19. My Project Design an API and include it on a Web Service that will work over CLOUD platforms The API should provide homomorphic encryption functions to be used Create a prototype that will work under the constructed API
- 20. QUESTIONS?