FULLY HOMOMORPHIC ENCRYPTION: CURRENT STATE OF THE ...

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Our Plaintext Space for AES � We make our plaintext “slots” are isomorphic to F 2 8. � More accurately, to F 2 d where 8 divides d, and d is the degree of the polynomials f i(x) that divide Ф N(x). � Each slot holds a byte of the AES state.
The Inversion Step of AES � For m 2 F 2 8, we have that m -1 = m 254 . � We can homomorphically compute encryptions of m, m 2 , m 4 , …, m 128 “for free” (Frobenius aut.) � m 254 = m 128 ×m 64 ×m 32 ×m 16 ×m 8 ×m 4 ×m 2 . � A product of 7 things � Computable by a circuit with 3 levels (binary tree). � Each AES round actually costs us about 6 levels. � Other steps are not completely “free”. � L = 60 for 10 round AES.
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 and 10: How To Construct a SWHE Scheme? Mos
Page 11 and 12: Polly Cracker Cryptanalysis [see AF
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: AES (Advanced Encryption Standard)
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
Page 67 and 68: Relinearizing and “Key Switching
Page 69 and 70: Basic PERMUTEs (Rotations) Array of
Page 71 and 72: n-PERMUTE(π) from n-ADD, n-MULT, a

FULLY HOMOMORPHIC ENCRYPTION: CURRENT STATE OF THE ...

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