PostQuantum Cryptography #4


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1 PostQuantum Cryptography #4 Prof. Claude Crépeau McGill University 185
2 ( 186
3 Attack scenarios Ciphertextonly attack: This is the most basic type of attack and refers to the scenario where the adversary just observes a ciphertext (or multiple ciphertexts) and attempts to determine the underlying plaintext (or plaintexts). m? cwill you marry me? 187
4 cwill you marry me? Attack scenarios Knownplaintext attack: The adversary learns one or more pairs of plaintexts/ciphertexts encrypted under the same key. The aim is to determine the plaintext that was encrypted in some other ciphertext. m m? c Will you marry me? 188
5 Attack scenarios Chosenplaintext attack: The adversary has the ability to obtain the encryption of plaintexts of its choice. It then attempts to determine the plaintext that was encrypted in some other ciphertext. m? m cwill you marry me? c Will you marry me? 189
6 Attack scenarios Chosenciphertext attack: The adversary is even given the capability to obtain the decryption of ciphertexts of its choice. The adversary s aim, once again, is to determine the plaintext that was encrypted in some other ciphertext. c cwill you marry me? m m? c Will you marry me? 190
7 What is secure encryption? Answer 1 an encryption scheme is secure if no adversary can find the secret key when given a ciphertext. 191
8 secure encryption. Answer 2 an encryption scheme is secure if no adversary can find the plaintext that corresponds to the ciphertext. 192
9 secure encryption. Answer 3 an encryption scheme is secure if no adversary can determine any character of the plaintext that corresponds to the ciphertext. 193
10 secure encryption. Answer 4 an encryption scheme is secure if no adversary can derive any meaningful information about the plaintext from the ciphertext. Definitions of security should suffice for all potential applications. 194
11 secure encryption. The Final Answer an encryption scheme is secure if no adversary can compute any function of the plaintext from the ciphertext. 195
12 Perfect Secrecy DEFINITION 2.1 An encryption scheme (Gen, Enc, Dec) over a message space M is perfectly secret if for every probability distribution over M, every message m M, and every ciphertext c C for which Pr[C = c] > 0 : Pr[M = m C = c] = Pr[M = m]. 196
13 An equivalent formulation LEMMA 2.2 An encryption scheme (Gen, Enc, Dec) over a message space M is perfectly secret if and only if for every probability distribution over M, every message m M, and every ciphertext c C : Pr[C = c M = m] = Pr[C = c]. 197
14 Perfect indistinguishability LEMMA 2.3 An encryption scheme (Gen, Enc, Dec) over a message space M is perfectly secret if and only if for every probability distribution over M, every m0, m1 M, and every c C : Pr[ C = c M = m0 ] = Pr[ C = c M = m1 ]. 198
15 Adversarial indistinguishability. 199
16 Adversarial indistinguishability. This other definition is based on an experiment involving an adversary A, and formalizes A s inability to distinguish the encryption of one plaintext from the encryption of another; we thus call it adversarial indistinguishability. 199
17 Adversarial indistinguishability. This other definition is based on an experiment involving an adversary A, and formalizes A s inability to distinguish the encryption of one plaintext from the encryption of another; we thus call it adversarial indistinguishability. This definition will serve as our starting point when we introduce the notion of computational security in the next chapter. 199
18 Adversarial indistinguishability. 200
19 Adversarial indistinguishability. The experiment is defined for any encryption scheme Π = (Gen, Enc, Dec) over message space M and for any adversary A. 200
20 Adversarial indistinguishability. The experiment is defined for any encryption scheme Π = (Gen, Enc, Dec) over message space M and for any adversary A. We let PrivK ea A, v denote an execution of the Π experiment for a given Π and A. The experiment is defined as follows: 200
21 PrivK e A a, v Π A 201
22 PrivK e A a, v Π m0, m1 M A 201
23 PrivK e A a, v Π k Gen m0, m1 M A 201
24 PrivK e A a, v Π k Gen b { 0, 1 } m0, m1 M A 201
25 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M A 201
26 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A 201
27 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A b 201
28 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A b b 201
29 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A b b b = b? 201
30 Adversarial indistinguishability. 202
31 Adversarial indistinguishability. PrivK e A a, v Π : 202
32 Adversarial indistinguishability. PrivK e A a, v Π : 1. Adversary A outputs a pair of messages m0, m1 M. 202
33 Adversarial indistinguishability. PrivK ea A, v : Π 1. Adversary A outputs a pair of messages m0, m1 M. 2. A random key k is generated by running Gen, and a random bit b { 0, 1 } is chosen (by some imaginary entity that is running the experiment with A.) A ciphertext c Enck(mb) is computed and given to A. 202
34 Adversarial indistinguishability. PrivK ea A, v : Π 1. Adversary A outputs a pair of messages m0, m1 M. 2. A random key k is generated by running Gen, and a random bit b { 0, 1 } is chosen (by some imaginary entity that is running the experiment with A.) A ciphertext c Enck(mb) is computed and given to A. 3. A outputs a bit b. 202
35 Adversarial indistinguishability. PrivK ea A, v : Π 1. Adversary A outputs a pair of messages m0, m1 M. 2. A random key k is generated by running Gen, and a random bit b { 0, 1 } is chosen (by some imaginary entity that is running the experiment with A.) A ciphertext c Enck(mb) is computed and given to A. 3. A outputs a bit b. 4. The output of the experiment is defined to be 1 if b = b, and 0 otherwise. 202
36 Adversarial indistinguishability. 203
37 Adversarial indistinguishability. We write PrivK e A a, v Π = 1 if the output is 1 and in this case we say that A succeeded. 203
38 Adversarial indistinguishability. We write PrivK ea A, v = 1 if the output is 1 and in Π this case we say that A succeeded. One should think of A as trying to guess the value of b that is chosen in the experiment, and A succeeds when its guess b is correct. 203
39 Adversarial indistinguishability. We write PrivK ea A, v = 1 if the output is 1 and in Π this case we say that A succeeded. One should think of A as trying to guess the value of b that is chosen in the experiment, and A succeeds when its guess b is correct. The alternate definition we now give states that an encryption scheme is perfectly secret if no adversary A can succeed with probability any better than 1 /2. 203
40 PrivK e A a, v Π A 204
41 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A 204
42 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A b 204
43 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A b b 204
44 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A b b Pr[ b = b ] = 1 /2 204
45 PrivK e A a, v Π k Gen b { 0, 1 } c Enck(mb) m0, m1 M c A perfectly secret b b Pr[ b = b ] = 1 /2 204
46 Adversarial indistinguishability. DEFINITION 2.4 An encryption scheme Π = (Gen, Enc, Dec) over a message space M is perfectly secret if for every adversary A it holds that Pr[ PrivK ea A, v = 1 ] = 1 Π /2. 205
47 Adversarial indistinguishability. PROPOSITION 2.5 Let (Gen, Enc, Dec) be an encryption scheme over a message space M. Then (Gen, Enc, Dec) is perfectly secret with respect to Definition 2.1 if and only if it is perfectly secret with respect to Definition
48 4 Equivalent Formulations DEFINITION 2.1 An encryption scheme (Gen, Enc, Dec) over a message space M is perfectly secret if for every probability distribution over M, every message m M, and every ciphertext c C for which Pr[C = c] > 0 : Pr[M = m C = c] = Pr[M = m]. LEMMA 2.3 An encryption scheme (Gen, Enc, Dec) over a message space M is perfectly secret if and only if for every probability distribution over M, every m0, m1 M, and every c C : Pr[ C = c M = m0 ] = Pr[ C = c M = m1 ]. LEMMA 2.2 An encryption scheme (Gen, Enc, Dec) over a message space M is perfectly secret if and only if for every probability distribution over M, every message m M, and every ciphertext c C : Pr[C = c M = m] = Pr[C = c]. DEFINITION 2.4 An encryption scheme Π = (Gen, Enc, Dec) over a message space M is perfectly secret if for every adversary A it holds that Pr[ PrivK e a v A, Π = 1 ] = 1 /2. 207
49 3.2 Defining Computationally Secure Encryption DEFINITION 3.7 A privatekey encryption scheme is a tuple of probabilistic polynomialtime algorithms (Gen, Enc, Dec) such that: 1/3. The keygeneration algorithm Gen takes as input the security parameter 1 n and outputs a key k; we write this as k Gen(1 n ) (thus emphasizing the fact that Gen is a randomized algorithm). We will assume without loss of generality that any key k Gen(1 n ) satisfies k n. 208
50 Defining Computationally Secure Encryption DEFINITION 3.7 A privatekey encryption scheme is a tuple of probabilistic polynomialtime algorithms (Gen, Enc, Dec) such that: 2/3. The encryption algorithm Enc takes as input a key k and a plaintext message m {0,1}, and outputs a ciphertext c. Since Enc may be randomized, we write c Enck(m). 209
51 Defining Computationally Secure Encryption DEFINITION 3.7 A privatekey encryption scheme is a tuple of probabilistic polynomialtime algorithms (Gen, Enc, Dec) such that: 3/3. The decryption algorithm Dec takes as input a key k and a ciphertext c, and outputs a message m. We assume that Dec is deterministic, and so write this as m Deck(c). 210
52 Defining Computationally Secure Encryption It is required that for every n, every key k output by Gen(1 n ), and every m {0,1}, it holds that Deck(Enck(m)) = m. If (Gen, Enc, Dec) is such that for k output by Gen(1 n ), algorithm Enck is only defined for m {0,1} (n), then we say that (Gen, Enc, Dec) is a fixedlength privatekey encryption scheme for messages of length (n). 211
53 Indistinguishability in the presence of an eavesdropper An experiment is defined for any privatekey encryption scheme Π = (Gen, Enc, Dec), any PPT adversary A and any value n for the security parameter. The eavesdropping indistinguishability experiment PrivK e A a, v Π(n) : 212
54 PrivK e A a, v Π 1 n A 213
55 PrivK e A a, v Π 1 n m0, m1 M A 213
56 PrivK e A a, v Π 1 n k Gen(1 n ) m0, m1 M A 213
57 PrivK e A a, v Π 1 n k Gen(1 n ) b { 0, 1 } m0, m1 M A 213
58 PrivK e A a, v Π 1 n k Gen(1 n ) b { 0, 1 } c Enck(mb) m0, m1 M A 213
59 PrivK e A a, v Π 1 n k Gen(1 n ) b { 0, 1 } c Enck(mb) m0, m1 M c A 213
60 PrivK e A a, v Π 1 n k Gen(1 n ) b { 0, 1 } c Enck(mb) m0, m1 M c A b 213
61 PrivK e A a, v Π 1 n k Gen(1 n ) b { 0, 1 } c Enck(mb) m0, m1 M c A b b 213
62 PrivK e A a, v Π 1 n k Gen(1 n ) b { 0, 1 } c Enck(mb) m0, m1 M c A b b Pr[ b = b ] ½ + negl(n) 213
63 PrivK e A a, v Π 1 n k Gen(1 n ) b { 0, 1 } c Enck(mb) m0, m1 M c A computationally secret b b Pr[ b = b ] ½ + negl(n) 213
64 PrivK e A a, v Π(n) 1. The adversary A is given input 1 n, and outputs a pair of messages m0, m1 of the same length. 2. A key k is generated by running Gen(1 n ), and a random bit b {0,1} is chosen. A (challenge) ciphertext c Enck(mb) is computed and given to A. 3. A outputs a bit b. 4. The output of the experiment is defined to be 1 if b = b, and 0 otherwise. (If PrivK e A a, v Π(n) = 1, we say that A succeeded.) 214
65 PrivK e A a, v Π(n) If Π is a fixedlength scheme for messages of length (n), the previous experiment is modified by requiring m0, m1 {0,1} (n). 215
66 Defining Computationally Secure Encryption DEFINITION 3.8 A privatekey encryption scheme Π = (Gen, Enc, Dec) has indistinguishable encryptions in the presence of an eavesdropper if for all PPT adversaries A there exists a negligible function negl such that Pr[ PrivK e A a, v Π(n) = 1 ] ½ + negl(n), where the probability is taken over the random coins used by A, as well as the random coins used in the experiment (for choosing the key, the random bit b, and any random coins used in the encryption process). 216
67 3.2.2* Properties of the Definition DEFINITION 3.12 A privatekey encryption scheme (Gen, Enc, Dec) is semantically secure in the presence of an eavesdropper if for every PPT algorithm A there exists a PPT algorithm A such that for all efficientlysampleable distributions X = (X1,...) and all polynomialtime computable functions f and h, there exists a negligible function negl s.t. Pr[ A(1 n, Enck(m), h(m)) = f(m) ] Pr[ A (1 n, h(m)) = f(m) ] negl(n), where m is chosen according to distribution Xn, and the probabilities are taken over the choice of m and the key k, and any random coins used by A, A, and the encryption process. 217
68 A 218
69 1 n A 218
70 k Gen(1 n ) 1 n A 218
71 k Gen(1 n ) 1 n c Enck(m) A 218
72 k Gen(1 n ) 1 n h(m) c Enck(m) A 218
73 k Gen(1 n ) 1 n c Enck(m) h(m) c A 218
74 k Gen(1 n ) 1 n c Enck(m) h(m) c A z 218
75 k Gen(1 n ) 1 n c Enck(m) h(m) c A z 218
76 k Gen(1 n ) 1 n c Enck(m) h(m) c A z A 218
77 k Gen(1 n ) 1 n c Enck(m) h(m) c A z 1 n A 218
78 k Gen(1 n ) 1 n c Enck(m) h(m) c A z 1 n h(m) A 218
79 k Gen(1 n ) 1 n c Enck(m) h(m) c A z 1 n h(m) z A 218
80 k Gen(1 n ) 1 n c Enck(m) h(m) c A Pr[z = f(m)] Pr[z = f(m)] negl(n), z 1 n h(m) z A 218
81 Semantic Security THEOREM 3.13 A privatekey encryption scheme has indistinguishable encryptions in the presence of an eavesdropper if and only if it is semantically secure in the presence of an eavesdropper. Shafi Goldwasser Silvio Micali 219
82 ) 220
83 PostQuantum Cryptography Finite Fields based cryptography Codes Multivariate Polynomials Integers based cryptography Approximate Integer GCD Lattices 221
84 Lattice based cryptography x 3b1+2b2 b2 0 b1 222
85 Lattices Given nlinearly independent vectors b 1,...,b n R n, the lattice they generate is the set of vectors L(b 1,...,b n ) = i n =1 x i b i :x i Z. The vectors b 1,...,b n are known as a basis of the lattice. 223
86 Lattices x 3b1+2b2 b2 0 b1 224
87 Integer Lattices Given nlinearly independent vectors b 1,...,b n Z n, the lattice they generate is the set of vectors L(b 1,...,b n ) = i n =1 x i b i :x i Z. The vectors b 1,...,b n are known as a basis of the lattice. 225
88 Lattices x b1+b2 b2 0 b1 226
89 Closest Vector Problem Given a basis b 1,...,b n R n, and a vector t R n find the closest vector in the lattice L(b 1,...,b n ) (x 1,...,x n ) Z n : d(t, i n =1 x i b i ) is minimal. d(u,v) is Euclidean distance i n =1 (u i v i ) 2 227
90 CVP t b2 0 b1 Analoguous to correcting errors in codes 228
91 CVP t b2 0 b1 Analoguous to correcting errors in codes 229
92 Shortest Vector Problem Given a basis b 1,...,b n R n find the shortest vector in the lattice L(b 1,...,b n ) (x 1,...,x n ) Z n \0 : d(0, i n =1 x i b i ) is minimal. d(u,v) is Euclidean distance i n =1 (u i v i ) 2 230
93 SVP shortest b2 b1 0 shortest Analoguous to finding min distance in code 231
94 GGH 232
95 GGH The GGH cryptosystem, proposed by Goldreich, Goldwasser, and Halevi is essentially a lattice analogue of the McEliece/Niederreiter cryptosystem 232
96 GGH The GGH cryptosystem, proposed by Goldreich, Goldwasser, and Halevi is essentially a lattice analogue of the McEliece/Niederreiter cryptosystem The private key is a good lattice basis B. 232
97 GGH The GGH cryptosystem, proposed by Goldreich, Goldwasser, and Halevi is essentially a lattice analogue of the McEliece/Niederreiter cryptosystem The private key is a good lattice basis B. Typically, a good basis consists of short, almost orthogonal vectors. 232
98 GGH The GGH cryptosystem, proposed by Goldreich, Goldwasser, and Halevi is essentially a lattice analogue of the McEliece/Niederreiter cryptosystem The private key is a good lattice basis B. Typically, a good basis consists of short, almost orthogonal vectors. Algorithmically, good bases allow to efficiently solve certain instances of the closest vector problem in L(B), e.g., instances where the target is very close to the lattice. 232
99 GGH/HNF 233
100 GGH/HNF The public key H is a bad basis for the same lattice L(H) = L(B). 233
101 GGH/HNF The public key H is a bad basis for the same lattice L(H) = L(B). Micciancio proposed to use the Hermite Normal Form (HNF) of B. This normal form gives a lower triangular basis for L(B). 233
102 GGH/HNF The public key H is a bad basis for the same lattice L(H) = L(B). Micciancio proposed to use the Hermite Normal Form (HNF) of B. This normal form gives a lower triangular basis for L(B). Notice that any attack on the HNF public key can be easily adapted to work with any other basis B of L(B) by first computing H from B. 233
103 GGH/HNF 234
104 GGH/HNF The encryption process consists of adding a short noise vector r (somehow encoding the message to be encrypted) to a properly chosen lattice point v. 234
105 GGH/HNF The encryption process consists of adding a short noise vector r (somehow encoding the message to be encrypted) to a properly chosen lattice point v. It was proposed to select the vector v such that all the coordinates of (r + v) are reduced modulo the corresponding element along the diagonal of the HNF public basis H. 234
106 GGH/HNF The encryption process consists of adding a short noise vector r (somehow encoding the message to be encrypted) to a properly chosen lattice point v. It was proposed to select the vector v such that all the coordinates of (r + v) are reduced modulo the corresponding element along the diagonal of the HNF public basis H. The resulting vector is denoted r mod H, and it provably makes cryptanalysis hardest because r mod H can be efficiently computed from any vector of the form (r + v) with v L(B). 234
107 GGH/HNF 235
108 GGH/HNF The decryption problem corresponds to finding the lattice point v closest to the target ciphertext c = (r mod H) = v+r, and the error vector r = c v. 235
109 GGH/HNF The decryption problem corresponds to finding the lattice point v closest to the target ciphertext c = (r mod H) = v+r, and the error vector r = c v. The correctness of the GGH/HNF cryptosystem rests on the fact that the error vector r is short enough so that the lattice point v can be recovered from the ciphertext v+r using the private basis B, e.g., by using Babai s rounding procedure, which gives v = B[B 1 (v + r)] where [x] stands for the nearest integer to x 235
110 236
111 qary Lattices Given nlinearly independent vectors b 1,...,b n Z n, the qary lattice they generate is the set of vectors L(b 1,...,b n,q 1,...,q n ) = i n =1 x i b i mod q:x i Z where each vector q i is of the form (0,...,0,q,0,...,0) 237
112 qary Lattices mod q x 3b1+2b2 b2 0 b1 238
113 qary Lattices 239
114 qary Lattices Structure very similar to linear codes 239
115 qary Lattices Structure very similar to linear codes We define two types of qary lattices from a matrix A Z nxm q q (A)={y Z m q : y = A T s mod q, s Z qn } q(a)={y Z m q : Ay = 0 mod q} 239
116 Learning With Errors LWE uses a discrete normal distribution   with mean 0 and standard deviation q / 2π defined as [ ] mod q 240
117 Learning With Errors LWE uses a discrete normal distribution   with mean 0 and standard deviation q / 2π defined as [ ] mod q q/2 +q/2 241
118 Learning With Errors A generalization of Learning Parity with Noise where q=2 and Bernouilli errors. 242
119 Learning With Errors A generalization of Learning Parity with Noise where q=2 and Bernouilli errors. LWE is parametrized by n and q=poly(n) 242
120 Learning With Errors A generalization of Learning Parity with Noise where q=2 and Bernouilli errors. LWE is parametrized by n and q=poly(n) A: Z q mxn, a uniform public matrix 242
121 Learning With Errors A generalization of Learning Parity with Noise where q=2 and Bernouilli errors. LWE is parametrized by n and q=poly(n) A: Z q mxn, a uniform public matrix S: Z qn, a uniform secret (trapdoor) vector 242
122 Learning With Errors A generalization of Learning Parity with Noise where q=2 and Bernouilli errors. LWE is parametrized by n and q=poly(n) A: Z q mxn, a uniform public matrix S: Z qn, a uniform secret (trapdoor) vector E: Z qm, a secret vector where each entry has distribution   with s.t. q n (reductions & there is an exp(( q) 2 )time attack) 242
123 Learning With Errors A generalization of Learning Parity with Noise where q=2 and Bernouilli errors. LWE is parametrized by n and q=poly(n) A: Z q mxn, a uniform public matrix S: Z qn, a uniform secret (trapdoor) vector E: Z qm, a secret vector where each entry has distribution   with s.t. q n (reductions & there is an exp(( q) 2 )time attack) (search)lwe: Given A and P=AS+E find S. 242
124 Learning With Errors 243
125 Learning With Errors DecisionLWE is made of 243
126 Learning With Errors DecisionLWE is made of A: Z q mxn, a uniform public matrix 243
127 Learning With Errors DecisionLWE is made of A: Z q mxn, a uniform public matrix S: Z qn, a uniform secret (trapdoor) vector 243
128 Learning With Errors DecisionLWE is made of A: Z q mxn, a uniform public matrix S: Z qn, a uniform secret (trapdoor) vector E: Z qm, a secret vector where each entry has distribution
129 Learning With Errors DecisionLWE is made of A: Z q mxn, a uniform public matrix S: Z qn, a uniform secret (trapdoor) vector E: Z qm, a secret vector where each entry has distribution  . Decision LWE : Given either A and P=AS+E or A,P for unfiorm P, identify which is the case. 243
130 Learning With Errors DecisionLWE is made of A: Z q mxn, a uniform public matrix S: Z qn, a uniform secret (trapdoor) vector E: Z qm, a secret vector where each entry has distribution  . Decision LWE : Given either A and P=AS+E or A,P for unfiorm P, identify which is the case. Equivalent to the search problem. 243
131 LWE hardness GapSVP SIVP searchlwe decisionlwe crypto 244
132 LWE hardness Quantum!!! GapSVP SIVP searchlwe decisionlwe crypto 244
133 LWE based cryptography 245
134 LWE based cryptography Private key: S: Z qn, E: Z q m sampled using
135 LWE based cryptography Private key: S: Z qn, E: Z q m sampled using   Public Key: A: Z q mxn, P=AS+E 245
136 LWE based cryptography Private key: S: Z qn, E: Z q m sampled using   Public Key: A: Z q mxn, P=AS+E Input message: b: {0,1} 245
137 LWE based cryptography Private key: S: Z qn, E: Z q m sampled using   Public Key: A: Z q mxn, P=AS+E Input message: b: {0,1} Enc AP (v) := (A T a,p T a+bq/2) where a: {0,1} m 245
138 LWE based cryptography Private key: S: Z qn, E: Z q m sampled using   Public Key: A: Z q mxn, P=AS+E Input message: b: {0,1} Enc AP (v) := (A T a,p T a+bq/2) where a: {0,1} m Dec S (u,c) := 1 (0) iff cs T u is closer to q/2 (0) cs T u = P T a+bq/2s T A T a = P T a+bq/2p T a+ea = bq/2+ea 245
139 LWE based cryptography 246
140 LWE based cryptography In the first part, one shows that distinguishing between public keys (A,P) as generated by the cryptosystem and pairs chosen uniformly at random from Z q mxn Z q m implies a solution to the LWE problem with parameters n,m,q,
141 LWE based cryptography In the first part, one shows that distinguishing between public keys (A,P) as generated by the cryptosystem and pairs chosen uniformly at random from Z q mxn Z q m implies a solution to the LWE problem with parameters n,m,q,  . The second part consists of showing that if one tries to encrypt with a public key (A,P) chosen at random, then with very high probability, the result carries essentially no statistical information about the encrypted message. (m > n log q) 246
142 LWE based cryptography In the first part, one shows that distinguishing between public keys (A,P) as generated by the cryptosystem and pairs chosen uniformly at random from Z q mxn Z q m implies a solution to the LWE problem with parameters n,m,q,  . The second part consists of showing that if one tries to encrypt with a public key (A,P) chosen at random, then with very high probability, the result carries essentially no statistical information about the encrypted message. (m > n log q) Together, these two parts establish the security of the cryptosystem (under chosen plaintext attacks). 246
143 LWE2 based cryptography 247
144 LWE2 based cryptography Private key: S,E: Z q n both sampled using  , 247
145 LWE2 based cryptography Private key: S,E: Z q n both sampled using  , Public Key: A: Z q nxn, P=AS+E 247
146 LWE2 based cryptography Private key: S,E: Z q n both sampled using  , Public Key: A: Z q nxn, P=AS+E Input message: b: {0,1} 247
147 LWE2 based cryptography Private key: S,E: Z q n both sampled using  , Public Key: A: Z q nxn, P=AS+E Input message: b: {0,1} Enc AP (v) := (A T a+x,p T a+bq/2+e ), a,x,e : Z q n using
148 LWE2 based cryptography Private key: S,E: Z q n both sampled using  , Public Key: A: Z q nxn, P=AS+E Input message: b: {0,1} Enc AP (v) := (A T a+x,p T a+bq/2+e ), a,x,e : Z q n using   Dec S (u,c) := 1 (0) iff cs T u is closer to q/2 (0) cs T u = P T a+bq/2+e S T A T as T x = P T a+bq/2+e P T a+eas T x = bq/2+ea+e S T x 247
149 LWE based cryptography 8 7 feb Peikert
150 Lattice based cryptography 249
151 PostQuantum Cryptography Prof. Claude Crépeau McGill University 250
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