FULLY HOMOMORPHIC ENCRYPTION: CURRENT STATE OF THE ...

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“Polly Cracker”: An Early Attempt at SWHE [Fellows-Koblitz ‘93] Main Idea Encryptions of 0 evaluate to 0 at the secret key. �� KeyGen: Secret = some point s = (s (s1, 1, …,s …,sn) 2 Z n n) 2 Z n q . Public key: Polynomials {ai(x1,…,xn)} s.t. ai(s)=0 mod q. � Encrypt: From {ai}, generate a random polynomial b(x) such that b(s) = 0 mod q. For m in {0,1}, ciphertext is: c(x) = m + b(x) mod q. � Decrypt: Evaluate ciphertext at secret: c(s)=m mod q. � ADD and MULT: Output sum or product of ciphertexts.
Polly Cracker Cryptanalysis [see AFFP11] � An Attack: � Collect lots of encryptions {c i} of 0. � If the challenge ciphertext also encrypts 0, it will likely be in linear span of the given encryptions of 0. � Use Gaussian elimination (linear algebra).
Page 1 and 2: FULLY HOMOMORPHIC ENCRYPTION: CURRE
Page 3 and 4: Homomorphic Encryption “Somewhat
Page 5 and 6: Fully Homomorphic Encryption: Curre
Page 7 and 8: Somewhat Homomorphic Encryption wit
Page 9: How To Construct a SWHE Scheme? Mos
Page 13 and 14: Noisy Polly Cracker: A Framework fo
Page 15 and 16: SWHE with Integers [vDGHV10] Main I
Page 17 and 18: Encrypting More than Bits � So fa
Page 19 and 20: SWHE with Integers (Mod-15) Divisib
Page 21 and 22: Cyclotomic Ring R = Z[x]/Ф N(x)
Page 23 and 24: Security of SWHE over Cyclotomic Ri
Page 25 and 26: Managing Ciphertext Noise
Page 27 and 28: Better Noise Management? � Crazy
Page 29 and 30: How Does Modulus Switching Help? [B
Page 31 and 32: Where Are We? (The Good News) � W
Page 33 and 34: Packing Plaintexts into Ciphertexts
Page 35 and 36: Our Weird Cyclotomic Plaintext Spac
Page 37 and 38: SIMD (Single Instruction Multiple D
Page 39 and 40: SIMD (Single Instruction Multiple D
Page 41 and 42: A Taste of Cyclotomic Rings � Cho
Page 43 and 44: What Is Our Overhead Now? � A cip
Page 45 and 46: AES (Advanced Encryption Standard)
Page 47 and 48: The Inversion Step of AES � For m
Page 49 and 50: Parameter Sizes L (levels) N n = φ
Page 51 and 52: Bootstrapping
Page 53 and 54: Bootstrapping: What Is It? � So f
Page 55 and 56: Bootstrapping in [BGV12] � Set L
Page 57 and 58: Summary
Page 59 and 60: Next Steps
Page 61 and 62:
Open Problem: A Fast Refresh Proced
Page 63 and 64:
Homomorphic Encryption “Fully”
Page 65 and 66:
SWHE with Integers [vDGHV10] Divisi
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Relinearizing and “Key Switching
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Basic PERMUTEs (Rotations) Array of
Page 71 and 72:
n-PERMUTE(π) from n-ADD, n-MULT, a
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FULLY HOMOMORPHIC ENCRYPTION: CURRENT STATE OF THE ...

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