Pseudorandomness of Ring-LWE for Any Ring and Modulus Chris Peikert University of Michigan Oded Regev Noah Stephens-Davidowitz (to appear, STOC’17) 10 March 2017 1 / 14
Lattice-Based Cryptography p d o m x g = y N = = ⇒ p m e mod N · q e ( g a , g b ) (Images courtesy xkcd.org) 2 / 14
Lattice-Based Cryptography = ⇒ (Images courtesy xkcd.org) 2 / 14
Lattice-Based Cryptography = ⇒ Main Attractions ◮ Efficient: linear, embarrassingly parallel operations (Images courtesy xkcd.org) 2 / 14
Lattice-Based Cryptography = ⇒ Main Attractions ◮ Efficient: linear, embarrassingly parallel operations ◮ Resists quantum attacks (so far) (Images courtesy xkcd.org) 2 / 14
Lattice-Based Cryptography = ⇒ Main Attractions ◮ Efficient: linear, embarrassingly parallel operations ◮ Resists quantum attacks (so far) ◮ Security from worst-case assumptions (Images courtesy xkcd.org) 2 / 14
Lattice-Based Cryptography = ⇒ Main Attractions ◮ Efficient: linear, embarrassingly parallel operations ◮ Resists quantum attacks (so far) ◮ Security from worst-case assumptions ◮ Solutions to ‘holy grail’ problems in crypto: FHE and related (Images courtesy xkcd.org) 2 / 14
Learning With Errors [Regev’05] ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α 3 / 14
Learning With Errors [Regev’05] ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α ◮ Search: find secret s ∈ Z n q given many ‘noisy inner products’ a 1 ← Z n , b 1 ≈ � a 1 , s � ∈ Z q q a 2 ← Z n , b 2 ≈ � a 2 , s � ∈ Z q q . . . 3 / 14
Learning With Errors [Regev’05] ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α ◮ Search: find secret s ∈ Z n q given many ‘noisy inner products’ a 1 ← Z n , b 1 = � a 1 , s � + e 1 ∈ Z q q a 2 ← Z n , b 2 = � a 2 , s � + e 2 ∈ Z q q . . . width αq 3 / 14
Learning With Errors [Regev’05] ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α ◮ Search: find secret s ∈ Z n q given many ‘noisy inner products’ a 1 ← Z n , b 1 = � a 1 , s � + e 1 ∈ Z q q a 2 ← Z n , b 2 = � a 2 , s � + e 2 ∈ Z q q . . . width αq ◮ Decision: distinguish ( a i , b i ) from uniform ( a i , b i ) 3 / 14
Learning With Errors [Regev’05] ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α ◮ Search: find secret s ∈ Z n q given many ‘noisy inner products’ a 1 ← Z n , b 1 = � a 1 , s � + e 1 ∈ Z q q a 2 ← Z n , b 2 = � a 2 , s � + e 2 ∈ Z q q . . . width αq ◮ Decision: distinguish ( a i , b i ) from uniform ( a i , b i ) LWE is Hard and Versatile worst case ( n/α ) -SIVP on ≤ search-LWE ≤ decision-LWE ≤ much crypto n -dim lattices (quantum [R’05]) [BFKL’93,R’05,. . . ] 3 / 14
Learning With Errors [Regev’05] ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α ◮ Search: find secret s ∈ Z n q given many ‘noisy inner products’ a 1 ← Z n , b 1 = � a 1 , s � + e 1 ∈ Z q q a 2 ← Z n , b 2 = � a 2 , s � + e 2 ∈ Z q q . . . width αq ◮ Decision: distinguish ( a i , b i ) from uniform ( a i , b i ) LWE is Hard and Versatile worst case ( n/α ) -SIVP on ≤ search-LWE ≤ decision-LWE ≤ much crypto n -dim lattices (quantum [R’05]) [BFKL’93,R’05,. . . ] ◮ Classically , GapSVP ≤ search-LWE (worse params) [P’09,BLPRS’13] 3 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : ⋆ Any prime q = poly ( n ) [R’05] 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : ⋆ Any prime q = poly ( n ) [R’05] ⋆ Any “somewhat smooth” q = p 1 · · · p t (large enough primes p i ) [P’09] 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : ⋆ Any prime q = poly ( n ) [R’05] ⋆ Any “somewhat smooth” q = p 1 · · · p t (large enough primes p i ) [P’09] ⋆ Any q = p e for large enough prime p [ACPS’09] 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : ⋆ Any prime q = poly ( n ) [R’05] ⋆ Any “somewhat smooth” q = p 1 · · · p t (large enough primes p i ) [P’09] ⋆ Any q = p e for large enough prime p [ACPS’09] ⋆ Any q = p e with uniform error mod p i [MM’11] 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : ⋆ Any prime q = poly ( n ) [R’05] ⋆ Any “somewhat smooth” q = p 1 · · · p t (large enough primes p i ) [P’09] ⋆ Any q = p e for large enough prime p [ACPS’09] ⋆ Any q = p e with uniform error mod p i [MM’11] ⋆ Any q = p e — but increases α [MP’12] 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : ⋆ Any prime q = poly ( n ) [R’05] ⋆ Any “somewhat smooth” q = p 1 · · · p t (large enough primes p i ) [P’09] ⋆ Any q = p e for large enough prime p [ACPS’09] ⋆ Any q = p e with uniform error mod p i [MM’11] ⋆ Any q = p e — but increases α [MP’12] ⋆ Any q via “mod-switching” — but increases α [P’09,BV’11,BLPRS’13] 4 / 14
LWE Hardness and Parameters ◮ Parameters: dimension n , integer modulus q , error ‘rate’ α Worst case SIVP ≤ Search-LWE ◮ One reduction for best known parameters: any q ≥ √ n/α [R’05] Search-LWE ≤ Decision-LWE ◮ Messy. Many incomparable reductions for different forms of q : ⋆ Any prime q = poly ( n ) [R’05] ⋆ Any “somewhat smooth” q = p 1 · · · p t (large enough primes p i ) [P’09] ⋆ Any q = p e for large enough prime p [ACPS’09] ⋆ Any q = p e with uniform error mod p i [MM’11] ⋆ Any q = p e — but increases α [MP’12] ⋆ Any q via “mod-switching” — but increases α [P’09,BV’11,BLPRS’13] ◮ Increasing q, α yields a weaker ultimate hardness guarantee. 4 / 14
LWE is Efficient (Sort Of) ◮ Getting one pseudorandom scalar requires an n -dim inner . . product mod q . � � · · · a i · · · s + e = b ∈ Z q . . . 5 / 14
LWE is Efficient (Sort Of) ◮ Getting one pseudorandom scalar requires an n -dim inner . . product mod q . � � · · · a i · · · s + e = b ∈ Z q ◮ Can amortize each a i over many . . secrets s j , but still ˜ O ( n ) work . per scalar output. 5 / 14
LWE is Efficient (Sort Of) ◮ Getting one pseudorandom scalar requires an n -dim inner . . product mod q . � � · · · a i · · · s + e = b ∈ Z q ◮ Can amortize each a i over many . . secrets s j , but still ˜ O ( n ) work . per scalar output. ◮ Cryptosystems have rather large keys: Ω( n 2 log 2 q ) bits: . . . . . . pk = , Ω( n ) A b . . . . . . � �� � n 5 / 14
Wishful Thinking. . . . . . . ◮ Get n pseudorandom scalars . . . . . . . . from just one cheap product ∈ Z n a i ⋆ s + e i = b i q operation? . . . . . . . . . . . . 6 / 14
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