OLE extension from OT extension Manoj Prabhakaran joint work with - PowerPoint PPT Presentation

OLE extension 
 from 
 OT extension Manoj Prabhakaran joint work with 
 Guru Vamsi Policharla 
 Rajeev Raghunath 
 Parjanya Vyas IIT Bombay

New Results for OLE over GF ( 2 n ) • Random OLE over GF ( 2 n ) : Alice gets ( a, t ) & Bob gets ( b , u ) s.t. a + b = tu

New Results for OLE over GF ( 2 n ) • Random OLE over GF ( 2 n ) : Alice gets ( a, t ) & Bob gets ( b , u ) s.t. a + b = tu • O( n ) string OTs perfect security OLE over GF ( 2 n )

New Results for OLE over GF ( 2 n ) • Random OLE over GF ( 2 n ) : Alice gets ( a, t ) & Bob gets ( b , u ) s.t. a + b = tu • O( n ) string OTs perfect security OLE over GF ( 2 n ) • Optimal : Ω ( n ) string OTs necessary (no ma tu er how long the strings are)

New Results for OLE over GF ( 2 n ) • Random OLE over GF ( 2 n ) : Alice gets ( a, t ) & Bob gets ( b , u ) s.t. a + b = tu • O( n ) string OTs perfect security OLE over GF ( 2 n ) • Optimal : Ω ( n ) string OTs necessary (no ma tu er how long the strings are) • Gives OLE Extension

New Results for OLE over GF ( 2 n ) • Random OLE over GF ( 2 n ) : Alice gets ( a, t ) & Bob gets ( b , u ) s.t. a + b = tu • O( n ) string OTs perfect security OLE over GF ( 2 n ) • Optimal : Ω ( n ) string OTs necessary (no ma tu er how long the strings are) • Gives OLE Extension • A few OLEs → a few string OTs → many string OTs → many OLEs

������������ New Results for OLE over GF ( 2 n ) • Random OLE over GF ( 2 n ) : Alice gets ( a, t ) & Bob gets ( b , u ) s.t. a + b = tu • O( n ) string OTs perfect security OLE over GF ( 2 n ) • Optimal : Ω ( n ) string OTs necessary (no ma tu er how long the strings are) • Gives OLE Extension • A few OLEs → a few string OTs → many string OTs → many OLEs

OLE over GF ( 2 n ) and ℤ 4 • A bijection from GF ( 2 n ) GF ( 2 n ) ℤ n × to : 4

OLE over GF ( 2 n ) and ℤ 4 • A bijection from GF ( 2 n ) GF ( 2 n ) ℤ n × to : 4 φ ( a,t ) = f ( a ) + g ( t ) → GF ( 2 n ) ℤ n f, g : 4 f ( x ) = 2[ √ x ] g ( x ) + g ( y ) - g ( x + y ) = f ( xy ) a + b = tu ⇔ φ ( a , t ) + φ ( b , u ) ∊ S where S = { g ( x ) | x ∊ GF ( 2 n ) ℤ n } ⊆ 4

Z 2 Z 2 ( u, b ) 4 labels 4 labels ( t, a ) α ~ β ⇔ α + β ∊ S (0 , 0) (00 , 00) (00 , 00) (0 , 0) (0 , 1) (01 , 00) (01 , 00) (0 , 1) (1 , 0) (10 , 00) (10 , 00) (1 , 0) (3 , 3) (11 , 00) (11 , 00) (3 , 3) (0 , 2) (00 , 01) (00 , 01) (0 , 2) (0 , 3) (01 , 01) (01 , 01) (0 , 3) (1 , 2) (10 , 01) (10 , 01) (1 , 2) (3 , 1) (11 , 01) (11 , 01) (3 , 1) (2 , 2) (00 , 10) (00 , 10) (2 , 2) (2 , 3) (01 , 10) (01 , 10) (2 , 3) (3 , 2) (10 , 10) (10 , 10) (3 , 2) (1 , 1) (11 , 10) (11 , 10) (1 , 1) (2 , 0) (00 , 11) (00 , 11) (2 , 0) (2 , 1) (01 , 11) (01 , 11) (2 , 1) (3 , 0) (10 , 11) (10 , 11) (3 , 0) (1 , 3) (11 , 11) (11 , 11) (1 , 3)

Z 2 Z 2 ( u, b ) 4 labels 4 labels ( t, a ) α ~ β ⇔ α + β ∊ S (0 , 0) (00 , 00) (00 , 00) (0 , 0) (0 , 1) (01 , 00) (01 , 00) (0 , 1) (1 , 0) (10 , 00) (10 , 00) (1 , 0) S (3 , 3) (11 , 00) (11 , 00) (3 , 3) (0 , 2) (00 , 01) (00 , 01) (0 , 2) (0 , 3) (01 , 01) (01 , 01) (0 , 3) (1 , 2) (10 , 01) (10 , 01) (1 , 2) (3 , 1) (11 , 01) (11 , 01) (3 , 1) (2 , 2) (00 , 10) (00 , 10) (2 , 2) (2 , 3) (01 , 10) (01 , 10) (2 , 3) (3 , 2) (10 , 10) (10 , 10) (3 , 2) (1 , 1) (11 , 10) (11 , 10) (1 , 1) (2 , 0) (00 , 11) (00 , 11) (2 , 0) (2 , 1) (01 , 11) (01 , 11) (2 , 1) (3 , 0) (10 , 11) (10 , 11) (3 , 0) (1 , 3) (11 , 11) (11 , 11) (1 , 3)

������������ New Results for OLE over GF ( 2 n ) • O( n ) string OTs perfect security OLE over GF ( 2 n ) • Optimal : Ω ( n ) string OTs necessary (no ma tu er how long the strings are) • Gives OLE Extension • A few OLEs → a few string OTs → many string OTs → many OLEs Group Correlations: This and more (Coming soon on eprint)

Bi- affine Correlations • E.g., Bilinear correlations (like OT, OLE, vector OLE, Beaver’s triples, …) • E.g., Alice gets ( a 1 , a 2 ) , Bob gets ( b 1 , b 2 ) s.t. a 1 + b 1 + a 2 + b 2 = 0 • Generic 2-round protocols for random self-reduction, self-testing etc. Group Correlations: This and more (Coming soon on eprint)