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Application of Information Theory, Lecture 11 Pseudo-Entropy and Pseudorandom Generators Iftach Haitner Tel Aviv University. January 6, 2015 Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 1 / 23 Part I

  1. Encryption schemes Definition 1 A pair of algorithms ( E , D ) is (perfectly correct) encryption scheme, if for any k ∈ { 0 , 1 } n and m ∈ { 0 , 1 } ℓ , it holds that D ( k , E ( k , m )) = m ◮ What security should we ask from such scheme? ◮ Perfect secrecy: E K ( m ) ≡ E K ( m ′ ) , for any m , m ′ ∈ { 0 , 1 } ℓ and K ∼ { 0 , 1 } n , letting E k ( x ) := E ( k , x ) . ◮ Theorem (Shannon): Perfect secrecy implies n ≥ ℓ . ◮ Is is bad? Is it optimal? ◮ Proof : Let M ∼ { 0 , 1 } n . ◮ Perfect secrecy = ⇒ H ( M , E K ( M )) = H ( M , E K ( 0 ℓ )) ⇒ H ( M | E K ( M )) = H ( M , E K ( M )) − H ( E K ( M )) = H ( M | E K ( 0 ℓ )) = n = ◮ ◮ Perfect correctness = ⇒ H ( M | E K ( M ) , K ) = 0 = ⇒ H ( M | E K ( M )) ≤ H ( M , K | E K ( M )) ≤ H ( K | E K ( M )) + 0 ≤ H ( K ) ◮ = ⇒ n ≤ ℓ . ◮ ◮ Statistical security? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 3 / 23

  2. Encryption schemes Definition 1 A pair of algorithms ( E , D ) is (perfectly correct) encryption scheme, if for any k ∈ { 0 , 1 } n and m ∈ { 0 , 1 } ℓ , it holds that D ( k , E ( k , m )) = m ◮ What security should we ask from such scheme? ◮ Perfect secrecy: E K ( m ) ≡ E K ( m ′ ) , for any m , m ′ ∈ { 0 , 1 } ℓ and K ∼ { 0 , 1 } n , letting E k ( x ) := E ( k , x ) . ◮ Theorem (Shannon): Perfect secrecy implies n ≥ ℓ . ◮ Is is bad? Is it optimal? ◮ Proof : Let M ∼ { 0 , 1 } n . ◮ Perfect secrecy = ⇒ H ( M , E K ( M )) = H ( M , E K ( 0 ℓ )) ⇒ H ( M | E K ( M )) = H ( M , E K ( M )) − H ( E K ( M )) = H ( M | E K ( 0 ℓ )) = n = ◮ ◮ Perfect correctness = ⇒ H ( M | E K ( M ) , K ) = 0 = ⇒ H ( M | E K ( M )) ≤ H ( M , K | E K ( M )) ≤ H ( K | E K ( M )) + 0 ≤ H ( K ) ◮ = ⇒ n ≤ ℓ . ◮ ◮ Statistical security? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 3 / 23

  3. Encryption schemes Definition 1 A pair of algorithms ( E , D ) is (perfectly correct) encryption scheme, if for any k ∈ { 0 , 1 } n and m ∈ { 0 , 1 } ℓ , it holds that D ( k , E ( k , m )) = m ◮ What security should we ask from such scheme? ◮ Perfect secrecy: E K ( m ) ≡ E K ( m ′ ) , for any m , m ′ ∈ { 0 , 1 } ℓ and K ∼ { 0 , 1 } n , letting E k ( x ) := E ( k , x ) . ◮ Theorem (Shannon): Perfect secrecy implies n ≥ ℓ . ◮ Is is bad? Is it optimal? ◮ Proof : Let M ∼ { 0 , 1 } n . ◮ Perfect secrecy = ⇒ H ( M , E K ( M )) = H ( M , E K ( 0 ℓ )) ⇒ H ( M | E K ( M )) = H ( M , E K ( M )) − H ( E K ( M )) = H ( M | E K ( 0 ℓ )) = n = ◮ ◮ Perfect correctness = ⇒ H ( M | E K ( M ) , K ) = 0 = ⇒ H ( M | E K ( M )) ≤ H ( M , K | E K ( M )) ≤ H ( K | E K ( M )) + 0 ≤ H ( K ) ◮ = ⇒ n ≤ ℓ . ◮ ◮ Statistical security? HW. Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 3 / 23

  4. Encryption schemes Definition 1 A pair of algorithms ( E , D ) is (perfectly correct) encryption scheme, if for any k ∈ { 0 , 1 } n and m ∈ { 0 , 1 } ℓ , it holds that D ( k , E ( k , m )) = m ◮ What security should we ask from such scheme? ◮ Perfect secrecy: E K ( m ) ≡ E K ( m ′ ) , for any m , m ′ ∈ { 0 , 1 } ℓ and K ∼ { 0 , 1 } n , letting E k ( x ) := E ( k , x ) . ◮ Theorem (Shannon): Perfect secrecy implies n ≥ ℓ . ◮ Is is bad? Is it optimal? ◮ Proof : Let M ∼ { 0 , 1 } n . ◮ Perfect secrecy = ⇒ H ( M , E K ( M )) = H ( M , E K ( 0 ℓ )) ⇒ H ( M | E K ( M )) = H ( M , E K ( M )) − H ( E K ( M )) = H ( M | E K ( 0 ℓ )) = n = ◮ ◮ Perfect correctness = ⇒ H ( M | E K ( M ) , K ) = 0 = ⇒ H ( M | E K ( M )) ≤ H ( M , K | E K ( M )) ≤ H ( K | E K ( M )) + 0 ≤ H ( K ) ◮ = ⇒ n ≤ ℓ . ◮ ◮ Statistical security? HW. Computational security? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 3 / 23

  5. Part II Statistical Vs. Computational distance Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 4 / 23

  6. Distributions and statistical distance Let P and Q be two distributions over a finite set U . Their statistical distance (also known as, variation distance) is defined as SD ( P , Q ) := 1 � |P ( x ) − Q ( x ) | = max S⊆U ( P ( S ) − Q ( S )) 2 x ∈U We will only consider finite distributions. Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 5 / 23

  7. Distributions and statistical distance Let P and Q be two distributions over a finite set U . Their statistical distance (also known as, variation distance) is defined as SD ( P , Q ) := 1 � |P ( x ) − Q ( x ) | = max S⊆U ( P ( S ) − Q ( S )) 2 x ∈U We will only consider finite distributions. Claim 2 For any pair of (finite) distributions P and Q , it holds that D { ∆ D ( P , Q ) := Pr SD ( P , Q ) = max x ←P [ D ( x ) = 1 ] − Pr x ←Q [ D ( x ) = 1 ] } , where D is any algorithm. Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 5 / 23

  8. Distributions and statistical distance Let P and Q be two distributions over a finite set U . Their statistical distance (also known as, variation distance) is defined as SD ( P , Q ) := 1 � |P ( x ) − Q ( x ) | = max S⊆U ( P ( S ) − Q ( S )) 2 x ∈U We will only consider finite distributions. Claim 2 For any pair of (finite) distributions P and Q , it holds that D { ∆ D ( P , Q ) := Pr SD ( P , Q ) = max x ←P [ D ( x ) = 1 ] − Pr x ←Q [ D ( x ) = 1 ] } , where D is any algorithm. Let P , Q , R be finite distributions, then Triangle inequality: SD ( P , R ) ≤ SD ( P , Q ) + SD ( Q , R ) Repeated sampling: SD ( P 2 = ( P , P ) , Q 2 = ( Q , Q )) ≤ 2 · SD ( P , Q ) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 5 / 23

  9. Section 1 Computational Indistinguishability Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 6 / 23

  10. Computational indistinguishability Definition 3 (computational indistinguishability) P and Q are ( s , ε ) -indistinguishable, if ∆ D P , Q ≤ ε , for any s -size D. Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 7 / 23

  11. Computational indistinguishability Definition 3 (computational indistinguishability) P and Q are ( s , ε ) -indistinguishable, if ∆ D P , Q ≤ ε , for any s -size D. ◮ Adversaries are circuits (possibly randomized) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 7 / 23

  12. Computational indistinguishability Definition 3 (computational indistinguishability) P and Q are ( s , ε ) -indistinguishable, if ∆ D P , Q ≤ ε , for any s -size D. ◮ Adversaries are circuits (possibly randomized) ◮ ( ∞ , ε ) -indistinguishable is equivalent to statistical distance ε Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 7 / 23

  13. Computational indistinguishability Definition 3 (computational indistinguishability) P and Q are ( s , ε ) -indistinguishable, if ∆ D P , Q ≤ ε , for any s -size D. ◮ Adversaries are circuits (possibly randomized) ◮ ( ∞ , ε ) -indistinguishable is equivalent to statistical distance ε ◮ We sometimes think of s = n ω ( 1 ) and ε = 1 / s , where n is the “security parameter” Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 7 / 23

  14. Computational indistinguishability Definition 3 (computational indistinguishability) P and Q are ( s , ε ) -indistinguishable, if ∆ D P , Q ≤ ε , for any s -size D. ◮ Adversaries are circuits (possibly randomized) ◮ ( ∞ , ε ) -indistinguishable is equivalent to statistical distance ε ◮ We sometimes think of s = n ω ( 1 ) and ε = 1 / s , where n is the “security parameter” ◮ Can it be different from the statistical case? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 7 / 23

  15. Computational indistinguishability Definition 3 (computational indistinguishability) P and Q are ( s , ε ) -indistinguishable, if ∆ D P , Q ≤ ε , for any s -size D. ◮ Adversaries are circuits (possibly randomized) ◮ ( ∞ , ε ) -indistinguishable is equivalent to statistical distance ε ◮ We sometimes think of s = n ω ( 1 ) and ε = 1 / s , where n is the “security parameter” ◮ Can it be different from the statistical case? ◮ Unless said otherwise, distributions are over { 0 , 1 } n Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 7 / 23

  16. Repeated sampling Question 4 Assume P and Q are ( s , ε ) -indistinguishable, what about P 2 and Q 2 ? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 8 / 23

  17. Repeated sampling Question 4 Assume P and Q are ( s , ε ) -indistinguishable, what about P 2 and Q 2 ? ◮ Let D be an s ′ -size algorithm with ∆ D ( P 2 , Q 2 ) = ε ′ Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 8 / 23

  18. Repeated sampling Question 4 Assume P and Q are ( s , ε ) -indistinguishable, what about P 2 and Q 2 ? ◮ Let D be an s ′ -size algorithm with ∆ D ( P 2 , Q 2 ) = ε ′ Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 8 / 23

  19. Repeated sampling Question 4 Assume P and Q are ( s , ε ) -indistinguishable, what about P 2 and Q 2 ? ◮ Let D be an s ′ -size algorithm with ∆ D ( P 2 , Q 2 ) = ε ′ ε ′ = x ←P 2 [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ] Pr = ( Pr x ←P 2 [ D ( x ) = 1 ] − x ← ( P , Q ) [ D ( x ) = 1 ]) Pr + ( x ← ( P , Q ) [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ]) Pr Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 8 / 23

  20. Repeated sampling Question 4 Assume P and Q are ( s , ε ) -indistinguishable, what about P 2 and Q 2 ? ◮ Let D be an s ′ -size algorithm with ∆ D ( P 2 , Q 2 ) = ε ′ ε ′ = x ←P 2 [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ] Pr = ( Pr x ←P 2 [ D ( x ) = 1 ] − x ← ( P , Q ) [ D ( x ) = 1 ]) Pr + ( x ← ( P , Q ) [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ]) Pr = ∆ D ( P 2 , ( P , Q )) + ∆ D (( P , Q ) , Q 2 ) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 8 / 23

  21. Repeated sampling Question 4 Assume P and Q are ( s , ε ) -indistinguishable, what about P 2 and Q 2 ? ◮ Let D be an s ′ -size algorithm with ∆ D ( P 2 , Q 2 ) = ε ′ ε ′ = x ←P 2 [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ] Pr = ( Pr x ←P 2 [ D ( x ) = 1 ] − x ← ( P , Q ) [ D ( x ) = 1 ]) Pr + ( x ← ( P , Q ) [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ]) Pr = ∆ D ( P 2 , ( P , Q )) + ∆ D (( P , Q ) , Q 2 ) ◮ So either ∆ D ( P 2 , ( P , Q )) ≥ ε ′ / 2, or ∆ D (( P , Q ) , Q 2 ) ≥ ε ′ / 2 Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 8 / 23

  22. Repeated sampling Question 4 Assume P and Q are ( s , ε ) -indistinguishable, what about P 2 and Q 2 ? ◮ Let D be an s ′ -size algorithm with ∆ D ( P 2 , Q 2 ) = ε ′ ε ′ = x ←P 2 [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ] Pr = ( Pr x ←P 2 [ D ( x ) = 1 ] − x ← ( P , Q ) [ D ( x ) = 1 ]) Pr + ( x ← ( P , Q ) [ D ( x ) = 1 ] − Pr x ←Q 2 [ D ( x ) = 1 ]) Pr = ∆ D ( P 2 , ( P , Q )) + ∆ D (( P , Q ) , Q 2 ) ◮ So either ∆ D ( P 2 , ( P , Q )) ≥ ε ′ / 2, or ∆ D (( P , Q ) , Q 2 ) ≥ ε ′ / 2 ◮ Hence, ε ′ < 2 ε implies s ′ ≥ s − n . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 8 / 23

  23. Repeated sampling cont. What about P k and Q k ? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  24. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  25. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  26. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  27. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  28. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ ◮ ε ′ = Pr � D ( H k ) = 1 � � D ( H 0 ) = 1 � − Pr . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  29. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ ◮ ε ′ = Pr � D ( H k ) = 1 � � D ( H 0 ) = 1 � − Pr . ◮ ε ′ = � � D ( H i ) = 1 � � D ( H i − 1 ) = 1 � i ∈ [ k ] Pr − Pr Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  30. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ ◮ ε ′ = Pr � D ( H k ) = 1 � � D ( H 0 ) = 1 � − Pr . ◮ ε ′ = � � D ( H i ) = 1 � � D ( H i − 1 ) = 1 � i ∈ [ k ] Pr − Pr Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  31. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ ◮ ε ′ = Pr � D ( H k ) = 1 � � D ( H 0 ) = 1 � − Pr . ◮ ε ′ = � � D ( H i ) = 1 � � D ( H i − 1 ) = 1 � i ∈ [ k ] ∆ D ( H i , H i − 1 ) i ∈ [ k ] Pr − Pr = � Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  32. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ ◮ ε ′ = Pr � D ( H k ) = 1 � � D ( H 0 ) = 1 � − Pr . ◮ ε ′ = � � D ( H i ) = 1 � � D ( H i − 1 ) = 1 � i ∈ [ k ] ∆ D ( H i , H i − 1 ) i ∈ [ k ] Pr − Pr = � ⇒ ∃ i ∈ [ k ] with ∆ D ( H i , H i − 1 ) ≥ ε ′ / k . = ◮ Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  33. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ ◮ ε ′ = Pr � D ( H k ) = 1 � � D ( H 0 ) = 1 � − Pr . ◮ ε ′ = � � D ( H i ) = 1 � � D ( H i − 1 ) = 1 � i ∈ [ k ] ∆ D ( H i , H i − 1 ) i ∈ [ k ] Pr − Pr = � ⇒ ∃ i ∈ [ k ] with ∆ D ( H i , H i − 1 ) ≥ ε ′ / k . = ◮ ◮ Thus, ε ′ ≤ k ε implies s ′ > s − kn Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  34. Repeated sampling cont. What about P k and Q k ? Claim 5 Assume P and Q are ( s , ε ) -indistinguishable, then P k and Q k are ( s − kn , k ε ) -indistinguishable. Proof : ? ◮ For i ∈ { 0 , . . . , k } , let H i = ( P 1 , . . . , P i , Q i + 1 , . . . , Q k ) , where the P i ’s are iid ∼ P and the Q i ’s are iid ∼ Q . (hybrids) ◮ Let D be a s ′ -size algorithm with ∆ D ( P k , Q k ) = ε ′ ◮ ε ′ = Pr � D ( H k ) = 1 � � D ( H 0 ) = 1 � − Pr . ◮ ε ′ = � � D ( H i ) = 1 � � D ( H i − 1 ) = 1 � i ∈ [ k ] ∆ D ( H i , H i − 1 ) i ∈ [ k ] Pr − Pr = � ⇒ ∃ i ∈ [ k ] with ∆ D ( H i , H i − 1 ) ≥ ε ′ / k . = ◮ ◮ Thus, ε ′ ≤ k ε implies s ′ > s − kn ◮ When considering bounded time algorithms, things behaves very differently! Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 9 / 23

  35. Part III Pseudorandom Generators Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 10 / 23

  36. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  37. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  38. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  39. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Definition 7 (pseudorandom generators (PRGs)) A poly-time computable function g : { 0 , 1 } n �→ { 0 , 1 } ℓ ( n ) is a ( s , ε ) -pseudorandom generator, if for any n ∈ N Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  40. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Definition 7 (pseudorandom generators (PRGs)) A poly-time computable function g : { 0 , 1 } n �→ { 0 , 1 } ℓ ( n ) is a ( s , ε ) -pseudorandom generator, if for any n ∈ N ◮ g is length extending (i.e., ℓ ( n ) > n ) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  41. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Definition 7 (pseudorandom generators (PRGs)) A poly-time computable function g : { 0 , 1 } n �→ { 0 , 1 } ℓ ( n ) is a ( s , ε ) -pseudorandom generator, if for any n ∈ N ◮ g is length extending (i.e., ℓ ( n ) > n ) ◮ g ( U n ) is ( s ( n ) , ε ( n )) -pseudorandom Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  42. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Definition 7 (pseudorandom generators (PRGs)) A poly-time computable function g : { 0 , 1 } n �→ { 0 , 1 } ℓ ( n ) is a ( s , ε ) -pseudorandom generator, if for any n ∈ N ◮ g is length extending (i.e., ℓ ( n ) > n ) ◮ g ( U n ) is ( s ( n ) , ε ( n )) -pseudorandom Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  43. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Definition 7 (pseudorandom generators (PRGs)) A poly-time computable function g : { 0 , 1 } n �→ { 0 , 1 } ℓ ( n ) is a ( s , ε ) -pseudorandom generator, if for any n ∈ N ◮ g is length extending (i.e., ℓ ( n ) > n ) ◮ g ( U n ) is ( s ( n ) , ε ( n )) -pseudorandom ◮ We omit the “security parameter", i.e., n , when its value is clear from the context Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  44. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Definition 7 (pseudorandom generators (PRGs)) A poly-time computable function g : { 0 , 1 } n �→ { 0 , 1 } ℓ ( n ) is a ( s , ε ) -pseudorandom generator, if for any n ∈ N ◮ g is length extending (i.e., ℓ ( n ) > n ) ◮ g ( U n ) is ( s ( n ) , ε ( n )) -pseudorandom ◮ We omit the “security parameter", i.e., n , when its value is clear from the context ◮ Do such generators exist? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  45. Pseudorandom generator Definition 6 (pseudorandom distributions) A distribution P over { 0 , 1 } n is ( s , ε ) -pseudorandom, if it is ( s , ε ) -indistinguishable from U n . ◮ Do such distributions exit for interesting ( s , ε ) Definition 7 (pseudorandom generators (PRGs)) A poly-time computable function g : { 0 , 1 } n �→ { 0 , 1 } ℓ ( n ) is a ( s , ε ) -pseudorandom generator, if for any n ∈ N ◮ g is length extending (i.e., ℓ ( n ) > n ) ◮ g ( U n ) is ( s ( n ) , ε ( n )) -pseudorandom ◮ We omit the “security parameter", i.e., n , when its value is clear from the context ◮ Do such generators exist? ◮ Applications? Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 11 / 23

  46. Section 2 Pseudorandom generators (PRGs) from One-Way Permutations (OWPs) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 12 / 23

  47. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  48. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  49. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  50. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . ◮ Let δ = Pr [ D ( U n + 1 ) = 1 ] (hence, Pr [ D ( g ( U n )) = 1 ] = δ + ε ′ ) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  51. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . ◮ Let δ = Pr [ D ( U n + 1 ) = 1 ] (hence, Pr [ D ( g ( U n )) = 1 ] = δ + ε ′ ) ◮ Compute δ = Pr [ D ( f ( U n ) , U 1 ) = 1 ] Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  52. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . ◮ Let δ = Pr [ D ( U n + 1 ) = 1 ] (hence, Pr [ D ( g ( U n )) = 1 ] = δ + ε ′ ) ◮ Compute δ = Pr [ D ( f ( U n ) , U 1 ) = 1 ] Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  53. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . ◮ Let δ = Pr [ D ( U n + 1 ) = 1 ] (hence, Pr [ D ( g ( U n )) = 1 ] = δ + ε ′ ) ◮ Compute δ = Pr [ D ( f ( U n ) , U 1 ) = 1 ] ( f is a permuation) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  54. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . ◮ Let δ = Pr [ D ( U n + 1 ) = 1 ] (hence, Pr [ D ( g ( U n )) = 1 ] = δ + ε ′ ) ◮ Compute δ = Pr [ D ( f ( U n ) , U 1 ) = 1 ] ( f is a permuation) = Pr [ U 1 = b ( U n )] · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] + Pr [ U 1 = b ( U n )] · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  55. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . ◮ Let δ = Pr [ D ( U n + 1 ) = 1 ] (hence, Pr [ D ( g ( U n )) = 1 ] = δ + ε ′ ) ◮ Compute δ = Pr [ D ( f ( U n ) , U 1 ) = 1 ] ( f is a permuation) = Pr [ U 1 = b ( U n )] · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] + Pr [ U 1 = b ( U n )] · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] = 1 2 ( δ + ε ′ ) + 1 2 · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  56. OWP to PRG Claim 8 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a poly-time permutation and let b : { 0 , 1 } n �→ { 0 , 1 } be a poly-time ( s , ε ) -hardcore predicate of f , then g ( x ) = ( f ( x ) , b ( x )) is a ( s − O ( n ) , ε ) -PRG. ◮ Hence, OWP = ⇒ PRG ◮ Proof : Let D be an s ′ -size algorithm with ∆ D ( g ( U n ) , U n + 1 ) = ε ′ , we will show ∃ ( s ′ + O ( n )) -size P with Pr [ P ( f ( U n )) = b ( U n )] = 1 2 + ε ′ . ◮ Let δ = Pr [ D ( U n + 1 ) = 1 ] (hence, Pr [ D ( g ( U n )) = 1 ] = δ + ε ′ ) ◮ Compute δ = Pr [ D ( f ( U n ) , U 1 ) = 1 ] ( f is a permuation) = Pr [ U 1 = b ( U n )] · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] + Pr [ U 1 = b ( U n )] · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] = 1 2 ( δ + ε ′ ) + 1 2 · Pr [ D ( f ( U n ) , U 1 ) = 1 | U 1 = b ( U n )] . � � ◮ Hence, Pr = δ − ε ′ D ( f ( U n ) , b ( U n )) = 1 Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 13 / 23

  57. OWP to PRG cont. ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ + ε ′ ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ − ε ′ Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 14 / 23

  58. OWP to PRG cont. ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ + ε ′ ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ − ε ′ Algorithm 9 ( P ) Input: y ∈ { 0 , 1 } n 1. Flip a random coin c ← { 0 , 1 } . 2. If D ( y , c ) = 1 output c , otherwise, output c . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 14 / 23

  59. OWP to PRG cont. ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ + ε ′ ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ − ε ′ Algorithm 9 ( P ) Input: y ∈ { 0 , 1 } n 1. Flip a random coin c ← { 0 , 1 } . 2. If D ( y , c ) = 1 output c , otherwise, output c . ◮ It follows that Pr [ P ( f ( U n )) = b ( U n )] = Pr [ c = b ( U n )] · Pr [ D ( f ( U n ) , c ) = 1 | c = b ( U n )] + Pr [ c = b ( U n )] · Pr [ D ( f ( U n ) , c ) = 0 | c = b ( U n )] Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 14 / 23

  60. OWP to PRG cont. ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ + ε ′ ◮ Pr [ D ( f ( U n ) , b ( U n )) = 1 ] = δ − ε ′ Algorithm 9 ( P ) Input: y ∈ { 0 , 1 } n 1. Flip a random coin c ← { 0 , 1 } . 2. If D ( y , c ) = 1 output c , otherwise, output c . ◮ It follows that Pr [ P ( f ( U n )) = b ( U n )] = Pr [ c = b ( U n )] · Pr [ D ( f ( U n ) , c ) = 1 | c = b ( U n )] + Pr [ c = b ( U n )] · Pr [ D ( f ( U n ) , c ) = 0 | c = b ( U n )] = 1 2 · ( δ + ε ′ ) + 1 2 ( 1 − δ + ε ′ ) = 1 2 + ε ′ . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 14 / 23

  61. Part IV PRG from Regular OWF Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 15 / 23

  62. Computational notions of entropy Definition 10 X has ( s , ε ) -pseudoentropy at least k , if ∃ rv Y with H ( Y ) ≥ k and ∆ D ( X , Y ) ≤ ε for any s -size D. ( s , ε ) -pseudo min/Reiny -entropy are analogously defined. Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 16 / 23

  63. Computational notions of entropy Definition 10 X has ( s , ε ) -pseudoentropy at least k , if ∃ rv Y with H ( Y ) ≥ k and ∆ D ( X , Y ) ≤ ε for any s -size D. ( s , ε ) -pseudo min/Reiny -entropy are analogously defined. ◮ Example Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 16 / 23

  64. Computational notions of entropy Definition 10 X has ( s , ε ) -pseudoentropy at least k , if ∃ rv Y with H ( Y ) ≥ k and ∆ D ( X , Y ) ≤ ε for any s -size D. ( s , ε ) -pseudo min/Reiny -entropy are analogously defined. ◮ Example ◮ Repeated sampling Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 16 / 23

  65. Computational notions of entropy Definition 10 X has ( s , ε ) -pseudoentropy at least k , if ∃ rv Y with H ( Y ) ≥ k and ∆ D ( X , Y ) ≤ ε for any s -size D. ( s , ε ) -pseudo min/Reiny -entropy are analogously defined. ◮ Example ◮ Repeated sampling ◮ Non-monotonicity Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 16 / 23

  66. Computational notions of entropy Definition 10 X has ( s , ε ) -pseudoentropy at least k , if ∃ rv Y with H ( Y ) ≥ k and ∆ D ( X , Y ) ≤ ε for any s -size D. ( s , ε ) -pseudo min/Reiny -entropy are analogously defined. ◮ Example ◮ Repeated sampling ◮ Non-monotonicity ◮ Ensembles Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 16 / 23

  67. Computational notions of entropy Definition 10 X has ( s , ε ) -pseudoentropy at least k , if ∃ rv Y with H ( Y ) ≥ k and ∆ D ( X , Y ) ≤ ε for any s -size D. ( s , ε ) -pseudo min/Reiny -entropy are analogously defined. ◮ Example ◮ Repeated sampling ◮ Non-monotonicity ◮ Ensembles ◮ In the following we will simply write ( s , ε ) -entropy, etc Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 16 / 23

  68. High entropy OWF from regular OWF Claim 11 Let f : { 0 , 1 } n �→ { 0 , 1 } n be a 2 k -regular ( s , ε ) -one-way, let H = { h : { 0 , 1 } n �→ { 0 , 1 } k + 2 } be 2-universal family, and let g ( h , x ) = ( f ( x ) , h , h ( x )) . Then 1. H 2 ( g ( U n , H )) ≥ 2 n − 1 2 , for H ← H . 2. g is (Θ( s ε 2 ) , 2 ε ) -one-way. ◮ k and m and H are parameterized by of n ◮ We assume log |H| = n and s ≥ n Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 17 / 23

  69. g has high Renyi entropy Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 18 / 23

  70. g has high Renyi entropy w , w ′ ←{ 0 , 1 } n ×H [ g ( w ) = g ( w ′ )] CP ( g ( U n , H )) := Pr h , h ′ ←H [ h = h ′ ] · ( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ f ( x ) = f ( x ′ )] = Pr Pr h ←H ;( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ h ( x ) = h ( x ′ ) | f ( x ) = f ( x ′ )] · Pr Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 18 / 23

  71. g has high Renyi entropy w , w ′ ←{ 0 , 1 } n ×H [ g ( w ) = g ( w ′ )] CP ( g ( U n , H )) := Pr h , h ′ ←H [ h = h ′ ] · ( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ f ( x ) = f ( x ′ )] = Pr Pr h ←H ;( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ h ( x ) = h ( x ′ ) | f ( x ) = f ( x ′ )] · Pr = CP ( H ) · CP ( f ( U n )) · ( 2 − k + ( 1 − 2 − k ) · 2 − k − 2 ) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 18 / 23

  72. g has high Renyi entropy w , w ′ ←{ 0 , 1 } n ×H [ g ( w ) = g ( w ′ )] CP ( g ( U n , H )) := Pr h , h ′ ←H [ h = h ′ ] · ( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ f ( x ) = f ( x ′ )] = Pr Pr h ←H ;( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ h ( x ) = h ( x ′ ) | f ( x ) = f ( x ′ )] · Pr = CP ( H ) · CP ( f ( U n )) · ( 2 − k + ( 1 − 2 − k ) · 2 − k − 2 ) ≤ CP ( H ) · CP ( f ( U n )) · 2 − k · 4 3 Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 18 / 23

  73. g has high Renyi entropy w , w ′ ←{ 0 , 1 } n ×H [ g ( w ) = g ( w ′ )] CP ( g ( U n , H )) := Pr h , h ′ ←H [ h = h ′ ] · ( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ f ( x ) = f ( x ′ )] = Pr Pr h ←H ;( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ h ( x ) = h ( x ′ ) | f ( x ) = f ( x ′ )] · Pr = CP ( H ) · CP ( f ( U n )) · ( 2 − k + ( 1 − 2 − k ) · 2 − k − 2 ) ≤ CP ( H ) · CP ( f ( U n )) · 2 − k · 4 3 = 2 − n · 2 − n · 4 3 . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 18 / 23

  74. g has high Renyi entropy w , w ′ ←{ 0 , 1 } n ×H [ g ( w ) = g ( w ′ )] CP ( g ( U n , H )) := Pr h , h ′ ←H [ h = h ′ ] · ( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ f ( x ) = f ( x ′ )] = Pr Pr h ←H ;( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ h ( x ) = h ( x ′ ) | f ( x ) = f ( x ′ )] · Pr = CP ( H ) · CP ( f ( U n )) · ( 2 − k + ( 1 − 2 − k ) · 2 − k − 2 ) ≤ CP ( H ) · CP ( f ( U n )) · 2 − k · 4 3 = 2 − n · 2 − n · 4 3 . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 18 / 23

  75. g has high Renyi entropy w , w ′ ←{ 0 , 1 } n ×H [ g ( w ) = g ( w ′ )] CP ( g ( U n , H )) := Pr h , h ′ ←H [ h = h ′ ] · ( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ f ( x ) = f ( x ′ )] = Pr Pr h ←H ;( x , x ′ ) ← ( { 0 , 1 } n ) 2 [ h ( x ) = h ( x ′ ) | f ( x ) = f ( x ′ )] · Pr = CP ( H ) · CP ( f ( U n )) · ( 2 − k + ( 1 − 2 − k ) · 2 − k − 2 ) ≤ CP ( H ) · CP ( f ( U n )) · 2 − k · 4 3 = 2 − n · 2 − n · 4 3 . Hence, H 2 ( g ( U n , H )) ≥ 2 n + log 3 4 ≥ 2 n − 1 2 . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 18 / 23

  76. g is one-way Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 19 / 23

  77. g is one-way Let A be an s ′ -size algorithm that inverts g w.p ε ′ and let ℓ = k − 2 log 1 � � . ε ′ Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 19 / 23

  78. g is one-way Let A be an s ′ -size algorithm that inverts g w.p ε ′ and let ℓ = k − 2 log 1 � � . ε ′ Consider the following inverter for f Algorithm 12 ( B ) Input: y ∈ { 0 , 1 } n . Return D ( y , h , z ) , for h ← H and z ← { 0 , 1 } ℓ . Algorithm 13 ( D ) Input: y ∈ { 0 , 1 } n , h ∈ H and z 1 ∈ { 0 , 1 } ℓ . For all z 2 ∈ { 0 , 1 } k + 2 − ℓ : 1. Let ( x , h ) = A ( y , h , z 1 ◦ z 2 ) . 2. If f ( x ) = y , return x . Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 19 / 23

  79. g is one-way Let A be an s ′ -size algorithm that inverts g w.p ε ′ and let ℓ = k − 2 log 1 � � . ε ′ Consider the following inverter for f Algorithm 12 ( B ) Input: y ∈ { 0 , 1 } n . Return D ( y , h , z ) , for h ← H and z ← { 0 , 1 } ℓ . Algorithm 13 ( D ) Input: y ∈ { 0 , 1 } n , h ∈ H and z 1 ∈ { 0 , 1 } ℓ . For all z 2 ∈ { 0 , 1 } k + 2 − ℓ : 1. Let ( x , h ) = A ( y , h , z 1 ◦ z 2 ) . 2. If f ( x ) = y , return x . ◮ B’s size is (( s ′ + O ( n )) · 2 2 log ε ′ + 2 = Θ( s ′ /ε 2 ) 1 Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 19 / 23

  80. g is one-way Let A be an s ′ -size algorithm that inverts g w.p ε ′ and let ℓ = k − 2 log 1 � � . ε ′ Consider the following inverter for f Algorithm 12 ( B ) Input: y ∈ { 0 , 1 } n . Return D ( y , h , z ) , for h ← H and z ← { 0 , 1 } ℓ . Algorithm 13 ( D ) Input: y ∈ { 0 , 1 } n , h ∈ H and z 1 ∈ { 0 , 1 } ℓ . For all z 2 ∈ { 0 , 1 } k + 2 − ℓ : 1. Let ( x , h ) = A ( y , h , z 1 ◦ z 2 ) . 2. If f ( x ) = y , return x . ◮ B’s size is (( s ′ + O ( n )) · 2 2 log ε ′ + 2 = Θ( s ′ /ε 2 ) 1 D ( f ( x ) , h , h ( x ) 1 ,...,ℓ ) ∈ f − 1 ( f ( x )) = ε ′ ◮ Pr x ←{ 0 , 1 } n ; h ←H � � Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 19 / 23

  81. g is one-way, cont. We saw that D ( f ( x ) , h , h ( x ) 1 ,...,ℓ ) ∈ f − 1 ( f ( x )) = ε ′ � � Pr (1) x ←{ 0 , 1 } n ; h ←H Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 20 / 23

  82. g is one-way, cont. We saw that D ( f ( x ) , h , h ( x ) 1 ,...,ℓ ) ∈ f − 1 ( f ( x )) = ε ′ � � Pr (1) x ←{ 0 , 1 } n ; h ←H By the leftover hash lemma SD (( f ( x ) , h , h ( x ) 1 ,...,ℓ ) x ←{ 0 , 1 } , h ←H , ( f ( x ) , h , U ℓ ) x ←{ 0 , 1 } , h ←H ) ≤ ε ′ / 2 (2) Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 20 / 23

  83. g is one-way, cont. We saw that D ( f ( x ) , h , h ( x ) 1 ,...,ℓ ) ∈ f − 1 ( f ( x )) = ε ′ � � Pr (1) x ←{ 0 , 1 } n ; h ←H By the leftover hash lemma SD (( f ( x ) , h , h ( x ) 1 ,...,ℓ ) x ←{ 0 , 1 } , h ←H , ( f ( x ) , h , U ℓ ) x ←{ 0 , 1 } , h ←H ) ≤ ε ′ / 2 (2) Hence, ≥ ε ′ − ε ′ / 2 = ε ′ / 2 . B ( f ( x )) ∈ f − 1 ( f ( x )) � � Pr x ←{ 0 , 1 } n Iftach Haitner (TAU) Application of Information Theory, Lecture 11 January 6, 2015 20 / 23

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