Lattice-based (Post Quantum) Cryptography
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1 Lattice-based (Post Quantum) Cryptography Divesh Aggarwal Center of Quantum Technologies, Singapore February 8, 28 Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 1 / 30
2 Lattices A lattice is a set of points L = {a 1 v a nv n a i integers}. for some linearly independent vectors v 1,..., v n R n. We call v 1,..., v n a basis of L, and n the dimension of the lattice. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 2 / 30
3 Basis is Not Unique Good Basis: v 1, v 2 Bad Basis: v 1, v 2 Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 3 / 30
4 Hard Problems SVP: Given a lattice, find the shortest non-zero vector (of length λ 1 ). Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 4 / 30
5 Hard Problems SVP: Given a lattice, find the shortest non-zero vector (of length λ 1 ). ApproxSVP γ : Given a lattice basis, find a vector of length γ λ 1. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 4 / 30
6 Hard Problems SVP: Given a lattice, find the shortest non-zero vector (of length λ 1 ). ApproxSVP γ : Given a lattice basis, find a vector of length γ λ 1. GapSVP γ : Given a basis and d, decide whether λ 1 d or λ 1 > γ d. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 4 / 30
7 Hard Problems SVP: Given a lattice, find the shortest non-zero vector (of length λ 1 ). ApproxSVP γ : Given a lattice basis, find a vector of length γ λ 1. GapSVP γ : Given a basis and d, decide whether λ 1 d or λ 1 > γ d. There are also other hard problems like CVP, SIVP Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 4 / 30
8 GapSVP γ - Algorithms and Complexity γ 2 (log n)1 ɛ n n O(1) (1 + ɛ) n NP Hard Problem co-np Cryptography P [Ajt98,..., HR07] [GG98, AR05] [Ajtai96,...] [LLL82, Sch87] Fastest known algorithms for γ = r n/r run in 2 O(r) poly(n) time, i.e., For γ = poly(n), the running time is exponential in n. In particular, fastest known algorithm for γ close to 1 runs in time 2 n+o(n) [ADRS15,AS18] Under the Gap exponential time hypothesis, no algorithm can solve GapSVP γ for a constant γ 1 in time better than 2 cn for some constant c [AS17]. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 5 / 30
9 Lattices and Cryptography The first applications included attacking knapsack-based cryptosystems [LagOdl85] and variants of RSA [Has85,Cop]. Lattices began to be used to create cryptography starting with a breakthrough work of Ajtai[Ajt96]. Cryptography based on lattices has many advantages compared with traditional cryptography like RSA: It has strong, mathematically proven, security. It is believed to be resistant to quantum computers. In some cases, it is much faster. It can do more, e.g., fully homomorphic encryption, which is one of the most important cryptographic primitives. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 6 / 30
10 Lattice-based Crypto Public-key Encryption [Reg05,KTX07,PKW08] CCA-Secure PKE [PW08,Pei09]. Identity-based Encryption [GPV08] Oblivious Transfer [PVW08] Circular Secure Encryption [ACPS09] Hierarchical Identity-based Encryption [Gen09,CHKP09,ABB09]. Fully Homomorphic Encryption [Gen09,BV11,Bra12]. And more... Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 7 / 30
11 Talk Outline GGH Public Key Encryption Scheme LWE Problem and Applications Efficiency from Rings Recent Implementations. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 8 / 30
12 GGH Public Key Encryption Scheme Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, 28 9 / 30
13 Public key encryption Suppose Alice wants to send a message m privately to Bob over a public channel. Key Generation: Bob generates a pair (pk, sk). Encryption: Alice sends c = Enc(pk, m) to Bob. Decryption: Bob obtains the message m = Dec(sk, m). Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
14 Public key encryption Suppose Alice wants to send a message m privately to Bob over a public channel. Key Generation: Bob generates a pair (pk, sk). Encryption: Alice sends c = Enc(pk, m) to Bob. Decryption: Bob obtains the message m = Dec(sk, m). Security: An eavesdropper shouldn t learn anything about the message given the public-key and the ciphertext. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
15 Digital Signatures Suppose you wish to digitally sign the message m. Key Generation: The algorithm generates a pair (pk, sk). Sign: A tag for the message is computed using the signing algorithm t = Sign(sk, m). Verify: The signature can be verified using the public key Verify(pk, t, m ) {True, False}. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
16 Digital Signatures Suppose you wish to digitally sign the message m. Key Generation: The algorithm generates a pair (pk, sk). Sign: A tag for the message is computed using the signing algorithm t = Sign(sk, m). Verify: The signature can be verified using the public key Verify(pk, t, m ) {True, False}. Security: An eavesdropper shouldn t be able to forge a signature given a valid signature and the public key. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
17 Reducing a vector modulo a basis u mod B = u B B 1 t If u = α 1 u α n u n, then u mod B = (α 1 α 1 ) u (α n α n ) u n. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
18 Bad Basis u mod B is likely a long vector. Note that u u mod B is a lattice vector close to u. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
19 Good Basis u mod B is a short vector. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
20 Good Basis u mod B is a short vector. Note that u u mod B is a lattice vector close to u. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
21 The encryption scheme Map the message m to a vector close to the origin. Encryption: c = m mod B pk (public basis) Processed plaintext Ciphertext Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
22 The encryption scheme Map the message m to a vector close to the origin. Encryption: c = m mod B pk (public basis) Decryption: m = c mod B sk (secret basis) Processed plaintext Ciphertext Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
23 More on the GGH Scheme There is a dual digital signature scheme based on the same principle. Map the message m to a random vector in space. Signature: t = m m mod B sk (secret(good) basis) resulting in a lattice vector close to m. Verification: Use the public (bad) basis to check whether t is a lattice vector and that t m is short. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
24 More on the GGH Scheme There is a dual digital signature scheme based on the same principle. Map the message m to a random vector in space. Signature: t = m m mod B sk (secret(good) basis) resulting in a lattice vector close to m. Verification: Use the public (bad) basis to check whether t is a lattice vector and that t m is short. Nguyen in 1999 pointed out a flaw in the GGH scheme. He showed that every ciphertext reveals information about the plaintext. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
25 More on the GGH Scheme There is a dual digital signature scheme based on the same principle. Map the message m to a random vector in space. Signature: t = m m mod B sk (secret(good) basis) resulting in a lattice vector close to m. Verification: Use the public (bad) basis to check whether t is a lattice vector and that t m is short. Nguyen in 1999 pointed out a flaw in the GGH scheme. He showed that every ciphertext reveals information about the plaintext. The principle of GGH, however, has been used in several follow-up schemes. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
26 More on the GGH Scheme There is a dual digital signature scheme based on the same principle. Map the message m to a random vector in space. Signature: t = m m mod B sk (secret(good) basis) resulting in a lattice vector close to m. Verification: Use the public (bad) basis to check whether t is a lattice vector and that t m is short. Nguyen in 1999 pointed out a flaw in the GGH scheme. He showed that every ciphertext reveals information about the plaintext. The principle of GGH, however, has been used in several follow-up schemes. In fact, the first fully homomorphic encryption scheme candidate by Gentry [2009] was based on the same principle. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
27 Talk Outline GGH Public Key Encryption Scheme LWE Problem and Applications Efficiency from Rings Recent Implementations. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
28 LWE Problem and Applications Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
29 Learning with Errors Problem [Regev05] Parameters: Dimension n, modulus q = poly(n), error distribution Search: Find uniformly random secret s Z n q given A 11 A 12 A A 1n A = A 21 A 22 A A 2n and b = A s + e, A m1 A m2 A m3... A mn where A Z m n q is chosen uniformly at random and e Z m q is some short noise distribution. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
30 Learning with Errors Problem [Regev05] Parameters: Dimension n, modulus q = poly(n), error distribution Search: Find uniformly random secret s Z n q given A 11 A 12 A A 1n A = A 21 A 22 A A 2n and b = A s + e, A m1 A m2 A m3... A mn where A Z m n q is chosen uniformly at random and e Z m q is some short noise distribution. Decision: Distinguish (A, b) from (A, u) where u is uniform in Z m q. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
31 Learning with Errors Problem [Regev05] Parameters: Dimension n, modulus q = poly(n), error distribution Search: Find uniformly random secret s Z n q given A 11 A 12 A A 1n A = A 21 A 22 A A 2n and b = A s + e, A m1 A m2 A m3... A mn where A Z m n q is chosen uniformly at random and e Z m q is some short noise distribution. Decision: Distinguish (A, b) from (A, u) where u is uniform in Z m q. Worst case to Average Case: Break Crypto = Decision LWE = Search LWE = GapSVP poly(n) Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
32 A PKE scheme from decision LWE hardness Suppose Alice wants to send a bit µ {0, 1} privately to Bob over a public channel. Key Generation: Bob chooses q, m, n and A Z m n q, s Z n q are chosen uniformly at random and e Z m q is chosen from the LWE noise distribution. Then pk = A, b = A s + e, sk = s. Encryption: Alice chooses uniform r {0, 1} m and sends c = (r A, r b + µ q 2 ). Decryption: Bob on receiving the ciphertext (C 1, c 2 ) checks whether c 2 C 1 s is closer to 0 or q and hence deciphers the 2 message bit µ. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
33 A PKE scheme from decision LWE hardness Suppose Alice wants to send a bit µ {0, 1} privately to Bob over a public channel. Key Generation: Bob chooses q, m, n and A Z m n q, s Z n q are chosen uniformly at random and e Z m q is chosen from the LWE noise distribution. Then pk = A, b = A s + e, sk = s. Encryption: Alice chooses uniform r {0, 1} m and sends c = (r A, r b + µ q 2 ). Decryption: Bob on receiving the ciphertext (C 1, c 2 ) checks whether c 2 C 1 s is closer to 0 or q and hence deciphers the 2 message bit µ. Security is almost immediate from the Decision LWE Hardness assumption. The ciphertext looks random given the public key. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
34 Talk Outline GGH Public Key Encryption Scheme LWE Problem and Applications Efficiency from Rings Recent Implementations. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
35 Efficiency from Rings. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
36 How efficient is LWE? Getting one pseudorandom b i Z q requires an n-dimensional inner product modulo q. Cryptosystems have larger keys of size larger than n 2 log q. Wishful thinking: (a 1, a 2,..., a n) (s 1, s 2,..., s n) + (e 1, e 2,..., e n) = (b 1, b 2,..., b n). Get n pseudorandom elements in Z q from one inner product. Replace every n 2 length key by a key of length n. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
37 How efficient is LWE? Getting one pseudorandom b i Z q requires an n-dimensional inner product modulo q. Cryptosystems have larger keys of size larger than n 2 log q. Wishful thinking: (a 1, a 2,..., a n) (s 1, s 2,..., s n) + (e 1, e 2,..., e n) = (b 1, b 2,..., b n). Get n pseudorandom elements in Z q from one inner product. Replace every n 2 length key by a key of length n. Question: How to define the operation? Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
38 How efficient is LWE? Getting one pseudorandom b i Z q requires an n-dimensional inner product modulo q. Cryptosystems have larger keys of size larger than n 2 log q. Wishful thinking: (a 1, a 2,..., a n) (s 1, s 2,..., s n) + (e 1, e 2,..., e n) = (b 1, b 2,..., b n). Get n pseudorandom elements in Z q from one inner product. Replace every n 2 length key by a key of length n. Question: How to define the operation? With small error, co-ordinate wise multiplication is insecure. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
39 LWE over Rings Let R = Z[X]/(X n + 1), and R q = R/qR. Operations in R q are very efficient using algorithms similar to FFT. Search: Find secret s R q given a 1 R q, b 1 = a 1 s + e 1 a 2 R q, b 2 = a 2 s + e 2 Decision: Distinguish (a i, b i ) from (a i, u i ) where u i is uniform in R q. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
40 LWE over Rings Let R = Z[X]/(X n + 1), and R q = R/qR. Operations in R q are very efficient using algorithms similar to FFT. Search: Find secret s R q given a 1 R q, b 1 = a 1 s + e 1 a 2 R q, b 2 = a 2 s + e 2 Decision: Distinguish (a i, b i ) from (a i, u i ) where u i is uniform in R q. Worst case to Average Case [LPR10, PRS17] Decision R-LWE = Search R-LWE = ApproxSVP poly(n) on ideal lattices Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
41 Complexity of SVP on Ideal Lattices We know that if we can solve R-LWE, then we can also solve SVP on ideal lattices. The other direction is unknown. There has been some recent progress[cdpr16, BS16, K16, CDW16] giving an efficient quantum algorithm for 2 O( n log n) approximation of SVP on Ideal Lattices. This does not say anything about the hardness of Ring-LWE. There is more algebraic structure in Ring-LWE that can possibly lead to quantum attacks, but so far there has been little success. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
42 Complexity of SVP on Ideal Lattices We know that if we can solve R-LWE, then we can also solve SVP on ideal lattices. The other direction is unknown. There has been some recent progress[cdpr16, BS16, K16, CDW16] giving an efficient quantum algorithm for 2 O( n log n) approximation of SVP on Ideal Lattices. This does not say anything about the hardness of Ring-LWE. There is more algebraic structure in Ring-LWE that can possibly lead to quantum attacks, but so far there has been little success. Ring-LWE based crypto is more efficient, but perhaps less secure. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
43 Talk Outline GGH Public Key Encryption Scheme LWE Problem and Applications Efficiency from Rings Recent Implementations. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
44 Recent Implementations. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
45 Key Exchange [BCNS15] Ring-LWE based key exchange. NewHope [ADPS 15]: Optimized Ring-LWE key exchange with λ = 200 bit quantum security. Comparable to or even faster than current ECDH with 128-bit classical security. Google has experimentally deployed NewHope + ECDH. Frodo [BCDMNNRS 16]: Plain-LWE key exchange with some optimizations. Conjectures 128-bit quantum security. About 10 times slower than NewHope, but almost as fast as ECDH (and much faster than RSA). NTRU EES743EP1 [WEJ13]. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
46 Other Implementations BLISS: [DDLL 13] An efficient digital signature scheme based on [Lyu09,Lyu12] DILITHIUM: [DLLSSS17] Another efficient digital signature based on [GLP12] that eliminates some of the vulnerabilities of BLISS. HELib: [HaleviShoup] Implementation of Fully Homomorphic Encryption. Λ λ: [CroPei 16] A high level framework aimed at advanced lattice cryptosystems. Lots of ongoing work including many proposals of new post-quantum crypto schemes submitted to NIST. Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
47 Questions? Divesh Aggarwal (CQT) Lattice-based Cryptography February 8, / 30
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