FULLY HOMOMORPHIC ENCRYPTION: CURRENT STATE OF THE ...

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A More Complicated SWHE Scheme… � Sadly, we proceed to a more complicated SWHE scheme… � Instead of Z (the integers), we will use a more complicated ring: the polynomial ring R = Z[x]/Ф Z[x]/ФN(x) (x) � Why? � Smaller parameters (better efficiency). � Pack many plaintexts into each ciphertext.
Cyclotomic Ring R = Z[x]/Ф N(x) � ФN(x) is the N-th cyclotomic polynomial � Largest irreducible divisor of xN-1. � The degree of ФN(x) is n = φ(N) < N. � φ is Euler totient function. E.g., for N = pq, φ(N) = (p-1)(q-1). � But we use much smaller N than RSA. Example: N = 68561, φ(N) = 62208. � We call n (=φ(N)) the “ring dimension”. � Properties of R � R consists of polynomials of degree n-1. � R has an additive identity (0) and multiplicative identity (1). � Addition: From a(x), b(x) 2 R, compute a(x)+b(x) in usual way. � Multiplication: From a(x), b(x) 2 R, compute c(x)=a(x)·b(x), and then reduce c(x) modulo Ф N(x). � R q = Z q[x]/Ф N(x) – i.e., elements of R with coefficients reduced mod q. a
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: SWHE with Integers (Mod-15) Divisib
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
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
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FULLY HOMOMORPHIC ENCRYPTION: CURRENT STATE OF THE ...

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