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Lattice-based Proxy Re-encryption PKC 2014 , 26.03.14 Elena Kirshanova Horst Grtz Institute for IT Security Ruhr University Bochum Outline 1 Definition of PRE and Security Model 2 Previous constructions and our contribution 3 One-way functions

  1. Lattice-based Proxy Re-encryption PKC 2014 , 26.03.14 Elena Kirshanova Horst Görtz Institute for IT Security Ruhr University Bochum

  2. Outline 1 Definition of PRE and Security Model 2 Previous constructions and our contribution 3 One-way functions on lattices 4 Extended G-trapdoor and Re-Encryption

  3. The informal definition of a Proxy Re-Encyption PRE based on lattices|PKC 2014|26.03.14 3/23

  4. The informal definition of a Proxy Re-Encyption PRE based on lattices|PKC 2014|26.03.14 3/23

  5. The informal definition of a Proxy Re-Encyption PRE based on lattices|PKC 2014|26.03.14 3/23

  6. The informal definition of a Proxy Re-Encyption PRE based on lattices|PKC 2014|26.03.14 3/23

  7. The formal Definition Definition 1 (Proxy Re-Encryption) A unidirectional Proxy Re-Encryption (PRE) is a tuple of algorithms: ◮ ( pk , sk ) ← KeyGen ( 1 n ) ◮ c pk ← Enc ( pk , m ) ◮ m ← Dec ( sk , c ) PRE based on lattices|PKC 2014|26.03.14 4/23

  8. The formal Definition Definition 1 (Proxy Re-Encryption) A unidirectional Proxy Re-Encryption (PRE) is a tuple of algorithms: ◮ ( pk , sk ) ← KeyGen ( 1 n ) ◮ c pk ← Enc ( pk , m ) ◮ m ← Dec ( sk , c ) ◮ rk pk → pk ′ ← ReKeyGen ( pk , sk , pk ′ ) PRE based on lattices|PKC 2014|26.03.14 4/23

  9. The formal Definition Definition 1 (Proxy Re-Encryption) A unidirectional Proxy Re-Encryption (PRE) is a tuple of algorithms: ◮ ( pk , sk ) ← KeyGen ( 1 n ) ◮ c pk ← Enc ( pk , m ) ◮ m ← Dec ( sk , c ) ◮ rk pk → pk ′ ← ReKeyGen ( pk , sk , pk ′ ) ◮ c ′ ← ReEnc ( rk pk → pk ′ , c pk ) PRE based on lattices|PKC 2014|26.03.14 4/23

  10. PRE-CCA1 Security (simplified) PRE CCA1 A A , Π ( n ) pk ∗ ( pk , pk ′ ) rk pk → pk ′ ( Dec ( c ) , pk ) · · · m 0 , m 1 m 0 , m 1 ∈ M b ← { 0 , 1 } c ∗ = Enc ( pk ∗ , m b ) c ∗ If b = b ′ output 1 b ′ ∈ { 0 , 1 } b ′ else output 0

  11. PRE-CCA1 Security (simplified) PRE CCA1 A A , Π ( n ) pk ∗ ( pk , pk ′ ) rk pk → pk ′ ( Dec ( c ) , pk ) · · · m 0 , m 1 m 0 , m 1 ∈ M b ← { 0 , 1 } c ∗ = Enc ( pk ∗ , m b ) c ∗ If b = b ′ output 1 b ′ ∈ { 0 , 1 } b ′ else output 0

  12. PRE-CCA1 Security (simplified) PRE CCA1 A A , Π ( n ) pk ∗ ( pk , pk ′ ) rk pk → pk ′ ( Dec ( c ) , pk ) · · · m 0 , m 1 m 0 , m 1 ∈ M b ← { 0 , 1 } c ∗ = Enc ( pk ∗ , m b ) c ∗ If b = b ′ output 1 b ′ ∈ { 0 , 1 } b ′ else output 0

  13. PRE-CCA1 Security (simplified) PRE CCA1 A A , Π ( n ) pk ∗ ( pk , pk ′ ) rk pk → pk ′ ( Dec ( c ) , pk ) · · · m 0 , m 1 m 0 , m 1 ∈ M b ← { 0 , 1 } c ∗ = Enc ( pk ∗ , m b ) c ∗ If b = b ′ output 1 b ′ ∈ { 0 , 1 } b ′ else output 0

  14. PRE-CCA1 Security (simplified) PRE CCA1 A A , Π ( n ) pk ∗ ( pk , pk ′ ) rk pk → pk ′ ( Dec ( c ) , pk ) · · · m 0 , m 1 m 0 , m 1 ∈ M b ← { 0 , 1 } c ∗ = Enc ( pk ∗ , m b ) c ∗ If b = b ′ output 1 b ′ ∈ { 0 , 1 } b ′ else output 0

  15. PRE-CCA1 Security (simplified) PRE CCA1 A A , Π ( n ) pk ∗ ( pk , pk ′ ) rk pk → pk ′ ( Dec ( c ) , pk ) · · · m 0 , m 1 m 0 , m 1 ∈ M b ← { 0 , 1 } c ∗ = Enc ( pk ∗ , m b ) c ∗ If b = b ′ output 1 b ′ ∈ { 0 , 1 } b ′ else output 0

  16. PRE-CCA1 Security (simplified) PRE CCA1 A A , Π ( n ) pk ∗ ( pk , pk ′ ) rk pk → pk ′ ( Dec ( c ) , pk ) · · · m 0 , m 1 m 0 , m 1 ∈ M b ← { 0 , 1 } c ∗ = Enc ( pk ∗ , m b ) c ∗ If b = b ′ output 1 b ′ ∈ { 0 , 1 } b ′ else output 0 PRE based on lattices|PKC 2014|26.03.14 5/23

  17. Desired properties of PRE schemes ◮ Unidirectional (rk pk → pk ′ � = rk pk ′ → pk ) PRE based on lattices|PKC 2014|26.03.14 6/23

  18. Desired properties of PRE schemes ◮ Unidirectional (rk pk → pk ′ � = rk pk ′ → pk ) ◮ Non-interactive (ReKeyGen ( pk , sk , pk ′ ) ) PRE based on lattices|PKC 2014|26.03.14 6/23

  19. Desired properties of PRE schemes ◮ Unidirectional (rk pk → pk ′ � = rk pk ′ → pk ) ◮ Non-interactive (ReKeyGen ( pk , sk , pk ′ ) ) ◮ Collusion ‘safe’ PRE based on lattices|PKC 2014|26.03.14 6/23

  20. Desired properties of PRE schemes ◮ Unidirectional (rk pk → pk ′ � = rk pk ′ → pk ) ◮ Non-interactive (ReKeyGen ( pk , sk , pk ′ ) ) ◮ Collusion ‘safe’ ◮ Key optimal ◮ Non-transitive ◮ Proxy invisibility PRE based on lattices|PKC 2014|26.03.14 6/23

  21. Outline 1 Definition of PRE and Security Model 2 Previous constructions and our contribution 3 One-way functions on lattices 4 Extended G-trapdoor and Re-Encryption

  22. PRE overview Unidirectional Non-interactive Collusion-safe Assumption Security Model [BBS98] ✗ ✗ ✗ DDH IND-CPA PRE based on lattices|PKC 2014|26.03.14 8/23

  23. PRE overview Unidirectional Non-interactive Collusion-safe Assumption Security Model [BBS98] ✗ ✗ ✗ DDH IND-CPA [AFGH06] ✓ ✓ ✓ eDBDH IND-CPA PRE based on lattices|PKC 2014|26.03.14 8/23

  24. PRE overview Unidirectional Non-interactive Collusion-safe Assumption Security Model [BBS98] ✗ ✗ ✗ DDH IND-CPA [AFGH06] ✓ ✓ ✓ eDBDH IND-CPA [CH07] ✗ ✗ ✗ DBDH IND-CCA PRE based on lattices|PKC 2014|26.03.14 8/23

  25. PRE overview Unidirectional Non-interactive Collusion-safe Assumption Security Model [BBS98] ✗ ✗ ✗ DDH IND-CPA [AFGH06] ✓ ✓ ✓ eDBDH IND-CPA [CH07] ✗ ✗ ✗ DBDH IND-CCA [Xag10] ✗ ✗ ✗ LWE IND-CPA PRE based on lattices|PKC 2014|26.03.14 8/23

  26. PRE overview Unidirectional Non-interactive Collusion-safe Assumption Security Model [BBS98] ✗ ✗ ✗ DDH IND-CPA [AFGH06] ✓ ✓ ✓ eDBDH IND-CPA [CH07] ✗ ✗ ✗ DBDH IND-CCA [Xag10] ✗ ✗ ✗ LWE IND-CPA This work ✓ ✓ ✓ LWE IND-CCA1 PRE based on lattices|PKC 2014|26.03.14 8/23

  27. Main result Theorem 2 Our unidirectional Proxy Re-Encryption scheme is IND-CCA1-secure assuming the hardness of decision-LWE. PRE based on lattices|PKC 2014|26.03.14 9/23

  28. Outline 1 Definition of PRE and Security Model 2 Previous constructions and our contribution 3 One-way functions on lattices 4 Extended G-trapdoor and Re-Encryption

  29. Lattice definition ◮ Lattice Λ of dimension m is a discrete additive subgroup of Z m . b 1 b 2 PRE based on lattices|PKC 2014|26.03.14 11/23

  30. Lattice definition ◮ Lattice Λ of dimension m is a discrete additive subgroup of Z m . b 1 b 2 ◮ Basis B = { b 1 , . . . , b k } : Λ( B ) = { Bz : z ∈ Z k } . PRE based on lattices|PKC 2014|26.03.14 11/23

  31. Gaussians on Lattices v ← D Λ , s ⇔ v ∝ ρ s ( x ) = exp ( − π � x � 2 s 2 ) PRE based on lattices|PKC 2014|26.03.14 12/23

  32. One-way functions from lattices � � ◮ Public ∈ Z n × m A , q = poly ( n ) , m ≈ n log q q PRE based on lattices|PKC 2014|26.03.14 13/23

  33. One-way functions from lattices � � ◮ Public ∈ Z n × m A , q = poly ( n ) , m ≈ n log q q SIS LWE g A ( s , e ) = s t A + e t mod q ∈ Z m u := f A ( x ) = Ax mod q ∈ Z n q q PRE based on lattices|PKC 2014|26.03.14 13/23

  34. One-way functions from lattices � � ◮ Public ∈ Z n × m A , q = poly ( n ) , m ≈ n log q q SIS LWE g A ( s , e ) = s t A + e t mod q ∈ Z m u := f A ( x ) = Ax mod q ∈ Z n q q : sample x ′ ← D Λ u , s f − 1 g − 1 : find the unique s A A s.t. Ax ′ = u (or e ) PRE based on lattices|PKC 2014|26.03.14 13/23

  35. G-trapdoor [PM12] and a short R ← Z ¯ nk × nk define ◮ For a uniform A 0 ∈ Z n × ¯ m q � I � − R A = [ A 0 | G ] = [ A 0 | G − A 0 R ] I for some G with easy f − 1 and g − 1 G . G PRE based on lattices|PKC 2014|26.03.14 14/23

  36. G-trapdoor [PM12] and a short R ← Z ¯ nk × nk define ◮ For a uniform A 0 ∈ Z n × ¯ m q � I � − R A = [ A 0 | G ] = [ A 0 | G − A 0 R ] I for some G with easy f − 1 and g − 1 G . G ◮ [ A 0 | A 0 R ] is uniform by the leftover hash lemma, so is A . PRE based on lattices|PKC 2014|26.03.14 14/23

  37. G-trapdoor [PM12] and a short R ← Z ¯ nk × nk define ◮ For a uniform A 0 ∈ Z n × ¯ m q � I � − R A = [ A 0 | G ] = [ A 0 | G − A 0 R ] I for some G with easy f − 1 and g − 1 G . G ◮ [ A 0 | A 0 R ] is uniform by the leftover hash lemma, so is A . � R � ◮ A · = G I PRE based on lattices|PKC 2014|26.03.14 14/23

  38. Outline 1 Definition of PRE and Security Model 2 Previous constructions and our contribution 3 One-way functions on lattices 4 Extended G-trapdoor and Re-Encryption

  39. Extended G-trapdoor ◮ Idea: generate multiple R -transformations trapdoor for g A � �� � A = [ A 0 | G − A 0 R 1 | G − A 0 R 2 ] � �� � trapdoor for f A ◮ R 1 allows to sample short vectors (i.e. generate rk) ◮ R 2 allows to invert s t A + e t (i.e. decrypt) PRE based on lattices|PKC 2014|26.03.14 16/23

  40. Encryption ◮ pk = [ A 0 | G − A 0 R 1 | G − A 0 R 2 ] ∈ Z n × m , sk := [ R 1 | R 2 ] q PRE based on lattices|PKC 2014|26.03.14 17/23

  41. Encryption ◮ pk = [ A 0 | G − A 0 R 1 | G − A 0 R 2 ] ∈ Z n × m , sk := [ R 1 | R 2 ] q ◮ Enc ( mes , pk ) : c 1 = s t · pk + e t 1 mod q , c 2 = s t · A aux + e t 2 + enc ( mes ) mod q , $ $ and enc ( mes ) := mes · ⌊ q − Z n − Z n × nk for s ← q , e 1 , e 2 ← D s , A aux ← 2 ⌋ . q PRE based on lattices|PKC 2014|26.03.14 17/23

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