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

IIT Bombay

OLE extension 


from 


OT extension

joint work with 


Guru Vamsi Policharla 
 Rajeev Raghunath 
 Parjanya Vyas

Manoj Prabhakaran

New Results for OLE over

GF(2n)

• Random OLE over

: Alice gets (a, t ) & Bob gets (b, u) s.t. a+b = tu GF(2n)

New Results for OLE over

GF(2n)

• Random OLE over

: Alice gets (a, t ) & Bob gets (b, u) s.t. a+b = tu GF(2n)

• O(n) string OTs

OLE over GF(2n)

perfect security

New Results for OLE over

GF(2n)

• Random OLE over

: Alice gets (a, t ) & Bob gets (b, u) s.t. a+b = tu GF(2n)

• Optimal: Ω(n) string OTs necessary (no matuer how long the strings are)

O(n) string OTs OLE over GF(2n)

perfect security

New Results for OLE over

GF(2n)

• Random OLE over

: Alice gets (a, t ) & Bob gets (b, u) s.t. a+b = tu GF(2n)

• Optimal: Ω(n) string OTs necessary (no matuer how long the strings are)
• Gives OLE Extension

O(n) string OTs OLE over GF(2n)

perfect security

New Results for OLE over

GF(2n)

• Random OLE over

: Alice gets (a, t ) & Bob gets (b, u) s.t. a+b = tu GF(2n)

• Optimal: Ω(n) string OTs necessary (no matuer how long the strings are)
• Gives OLE Extension
• A few OLEs → a few string OTs → many string OTs → many OLEs

O(n) string OTs OLE over GF(2n)

perfect security

• New Results for OLE over

GF(2n)

• Random OLE over

: Alice gets (a, t ) & Bob gets (b, u) s.t. a+b = tu GF(2n)

• Optimal: Ω(n) string OTs necessary (no matuer how long the strings are)
• Gives OLE Extension
• A few OLEs → a few string OTs → many string OTs → many OLEs

O(n) string OTs OLE over GF(2n)

perfect security

OLE over and GF(2n)

ℤ4

• A bijection from

× to : GF(2n) GF(2n)

ℤn

4

OLE over and GF(2n)

ℤ4

• A bijection from

× to : GF(2n) GF(2n)

ℤn

4

φ(a,t) = f (a ) + g (t )

f, g : → f (x) = 2[√x ] g (x) + g (y) - g (x+y) = f (xy) GF(2n)

ℤn

4

a+b = tu ⇔ φ(a,t) + φ(b,u) ∊ S where S = { g(x) | x ∊ } ⊆ GF(2n)

ℤn

4

Z2

4 labels

(0, 0) (0, 1) (1, 0) (3, 3) (0, 2) (0, 3) (1, 2) (3, 1) (2, 2) (2, 3) (3, 2) (1, 1) (2, 0) (2, 1) (3, 0) (1, 3) (t, a) (00, 00) (01, 00) (10, 00) (11, 00) (00, 01) (01, 01) (10, 01) (11, 01) (00, 10) (01, 10) (10, 10) (11, 10) (00, 11) (01, 11) (10, 11) (11, 11) (u, b) (00, 00) (01, 00) (10, 00) (11, 00) (00, 01) (01, 01) (10, 01) (11, 01) (00, 10) (01, 10) (10, 10) (11, 10) (00, 11) (01, 11) (10, 11) (11, 11) Z2

4 labels

(0, 0) (0, 1) (1, 0) (3, 3) (0, 2) (0, 3) (1, 2) (3, 1) (2, 2) (2, 3) (3, 2) (1, 1) (2, 0) (2, 1) (3, 0) (1, 3)

α ~ β ⇔ α + β ∊ S

Z2

4 labels

(0, 0) (0, 1) (1, 0) (3, 3) (0, 2) (0, 3) (1, 2) (3, 1) (2, 2) (2, 3) (3, 2) (1, 1) (2, 0) (2, 1) (3, 0) (1, 3) (t, a) (00, 00) (01, 00) (10, 00) (11, 00) (00, 01) (01, 01) (10, 01) (11, 01) (00, 10) (01, 10) (10, 10) (11, 10) (00, 11) (01, 11) (10, 11) (11, 11) (u, b) (00, 00) (01, 00) (10, 00) (11, 00) (00, 01) (01, 01) (10, 01) (11, 01) (00, 10) (01, 10) (10, 10) (11, 10) (00, 11) (01, 11) (10, 11) (11, 11) Z2

4 labels

(0, 0) (0, 1) (1, 0) (3, 3) (0, 2) (0, 3) (1, 2) (3, 1) (2, 2) (2, 3) (3, 2) (1, 1) (2, 0) (2, 1) (3, 0) (1, 3)

S α ~ β ⇔ α + β ∊ S

• New Results for OLE over

GF(2n)

• Optimal: Ω(n) string OTs necessary (no matuer how long the strings are)
• Gives OLE Extension
• A few OLEs → a few string OTs → many string OTs → many OLEs

O(n) string OTs OLE over GF(2n)

perfect security

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 (a1, a2), Bob gets (b1, b2) s.t. a1 + b1 + a2 + b2 = 0
• Generic 2-round protocols for random self-reduction, self-testing etc.

Group Correlations: This and more (Coming soon on eprint)