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Paper by: Craig Gentry Presented By: Daniel Henneberger

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What is homomorphic encryption?

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Computations on ciphertext which predictably modifies the plaintext Operate on messages while they are encrypted Data can be securely processed in unsecure environments ◦ Cloud Computing ◦ Databases ◦ Voting machines

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Keygen Encrypt Decrypt Evaluate

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1978 – Privacy Homomorphism US government pumps millions in it

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Additive ◦ E(m1) + E(m2) = E(m1+m2) Multiplicative ◦ E(m1) * E(m2) = E(m1*m2) Why just Add and Mul? ◦ Can evaluate any function ◦ Turing complete over a ring

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Somewhat Homomorphic ◦ You can do only do some functions ◦ RSA Fully Homomorphic ◦ You can do all functions Leveled Fully Homomorphic ◦ Keysize can grow with depth of the function Bootstrappable ◦ Can evaluate its own decryption circuit

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Craig Gentry Stanford University and IBM Watson 2009

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Before this paper, it was unknown if fully homomorphic encryption could exist First feasible result Holy grail of encryption 17 results on YouTube!

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Ideal lattices are a form of difficult to compute mathematical problems Similar to: ◦ Integer Factorization ◦ Discrete logarithm problem ◦ Elliptic curves over finite fields (Elliptical curve) Closest vector problem Learning with errors Unbreakable with quantum computing ◦ Uses arbitrary approximations

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“Recipe”: 1. Take two linearly independent vectors in R 2. 2. Close them for addition and for multiplication by an integer scalar. Each point corresponds to a vector in the lattice etc....

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A cyclic lattice is ‘ideal’ (ring-based) NTRU – Asymmetric key cryptosystem that uses ring-based lattices Low circuit complexity Very fast Allows additive and multiplicative homomorphism

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Lots of math involved with this: ◦ Cyclotomic Polynomials Too much for this class time

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Evaluate(pk,C, Encrypt(pk,m1),..., Encrypt(pk,mt)) = Encrypt(pk,C(m1,..., mt)) Steps ◦ Create a general bootstrapping result ◦ Initial construction using ideal lattices ◦ Squash the decryption circuit to permit bootstrapping

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Find a Public key scheme that is homomorphic for shallow circuits and uses ideal lattices ◦ NTRUEncrypt Ciphertext has a form of an ideal lattice + offset Use a cyclic ring of keys ◦ Hard to do ◦ Large key size (GB)

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Evaluate its own decryption circuit Provides ability to recrypt plaintext Must be allowed to recrypt augmented versions to provide mathematical operations

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Allows ‘unlimited’ additions ◦ Recrypt algorithm Greater multiplicative depth ◦ log log (N) - log log (n-1) ◦ Still bad

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Can only evaluate in logarithmic depth ◦ Ciphertext grows ◦ Noise increases Addition- circuits can be corrected (recrypting) Multiplication- noise grows quickly Not yet practical ◦ Client must begin the decryption process to be bootstrappable ◦ Solution is approximate ◦ >1 day to compute 1 message

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PollyCracker Fully Homomorphic Encryption over the Integers Fully Homomorphic Encryption over the Binary Polynomials

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Many people have created new variants Implementations All slow Finding shortcuts AES-128 – Completed June 15 th 2012 ◦ Computed with 256GB of ram (still limiting factor) ◦ 24 Xeon cores ◦ Took 5 days per operation

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