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Reductions

Reductions. Christina Brzuska Tel-Aviv University. Limitations of Impossibility Results. Impagliazzo-Rudich: Standard techniques ? Certain ``types“ of reductions Goal of this talk: Define types of reductions. ?. References.

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Reductions

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  1. Reductions Christina Brzuska Tel-Aviv University

  2. Limitations of Impossibility Results • Impagliazzo-Rudich: • Standard techniques ? • Certain ``types“ of reductions • Goal of this talk: Define types of reductions ?

  3. References • Notions of Reducibility between Cryptographic Primitives Omer Reingold, Luca Trevisan, Salil Vadhan • Notions of Black-Box Reductions, Revisited Paul Baecher, CB, Marc Fischlin

  4. Reductions in Cryptography Goal: signature scheme from some assumption public key signature requests Adversary A scheme S Game C forgery Reduction R Reduction: if A breaks scheme S then RA wins game C

  5. One-Time-Signatures from OWFs (Lamport) Construction based on f just one OWF Game public key signature request Adversary A scheme S f y=f(x) x* forgery Reduction R OWFs  One-Time Signatures: Construction + Reduction

  6. Construction KeyGenf, Signf, Verifyf Assume f isone-way. Provesecurityofthisscheme. • KeyGenf: a1,…,an b1,…,bn f(a1),…,f(an) f(b1),…,f(bn) • Signf(sk,m): m=m1,…,mn=0010…0 a1a2 a3a4 … an b1b2 b3b4 … bn m 0 0 1 0 … 0 • Verifyf(pk,m,¾): Check whetherpre-images matchpk sk pk ¾

  7. Security Reduction RA,f A adversaryagainstsignaturescheme f(a1) f(a2) f(a3) f(a4) … f(an) f(b1) f(b2) f(b3) f(b4) … f(bn) a1 a2 a3 a4 … an b1 b2 b3 b4 … bn m 0 0 1 0 … 0 RA,f gets y=f(x), tries to compute a pre-image of y f(a1) f(a2) f(a3) f(a4) … f(an) f(b1) f(b2) f(b3) f(b4) … f(bn) a1 a2 a3 a4 … an b1 b2 b3 b4 … bn pk sk y Hope for Forgery m* ¾ ¾ ??? Hope for query m

  8. Fully Black-Box Reductions 9 PPT Construction (KeyGen, Sign, Verify) 9 PPT Reduction R 8 Adversary A 8 Function f Afbreaks (KeyGenf, Signf, Verifyf)  RA,fbreaks f 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Primitive f AfbreaksGf  RA,fbreaks f

  9. (Im)Possibility Results One-Way Functions Pseudorandom Generators Minicrypt Pseudorandom Functions Pseudorandom Permutations Message Authentication Codes Cryptomania Symmetric Encryption Signature Schemes Key Agreement [IR89] Signature Schemes[NY90, R91]

  10. Impagliazzo Rudich • This afternoon • Oracle result • Relative to O: OWFs, but no key agreement O(.)

  11. Which techniques are ruled out? • Thereexists an oracle O: • One-wayfunctionsexist relative to O, • KA does not exist relative to O. • Foranyoracle O: • Ifone-wayfunctionsexist relative to O, • then KA exists relative to O. Oracle Separation rules out ? Relativizing Reduction ? 9 PPT Construction KA 9 PPT Reduction R 8 Adversary A 8 Function f Fully Black-Box Reduction AfbreaksKAf  RA,fbreaks f

  12. Fully Black-Box Reduction implies Relativizing Reduction

  13. Relativizing Reductions • Foranyoracle O: • Ifone-wayfunctionsexist relative to O, • thenone-time signaturesexistrelative to O. • P1 is efficient algorithm • f= P1O is one-way. • No PPT A can invert f. • A also gets access to O f O(.) O(.) A P1

  14. Relativizing Reductions • Foranyoracle O: • Ifone-wayfunctionsexist relative to O, • thenone-time signaturesexistrelative to O. f Sig O(.) O(.) O(.) A P1 P2

  15. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW Take an Oracle O. Wehavetoshowthat: • Ifone-wayfunctionsexist relative to O, • thenone-time signaturesexists relative to O. f Sig O(.) O(.) O(.) A P1 P2

  16. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • Assume, OWFs exist relative to O. • Weshowthatone-time signaturesexist relative to O. Sig f O(.) P1 f Sig O(.) O(.) O(.) A P1 P2 G

  17. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • Assume, OWFs exist relative to O. • Weshowthatone-time signaturesexist relative to O. Sig O(.) P2 P1 f Sig O(.) O(.) O(.) A P1 P2 G

  18. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • P2 isefficient. • WecanimplementGfeff. rel. to O. • IsSig=GfsecureOT-Sig-scheme? Sig O(.) P2 P1 f Sig O(.) O(.) O(.) A P1 P2 G

  19. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • P2 isefficient. • WecanimplementGfeff. rel. to O. • IsSig=GfsecureOT-Sig-scheme? Sig f f Sig O(.) O(.) O(.) A P1 P2 G

  20. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • Assumetow. contr., thereis PPT A such that AObreaksGf. • Then, RA,fbreaks f. • RA,feff. implementable rel. to O? Sig f O(.) A G

  21. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • RA,fefficientlyimplementable relative to O: f f O(.) O(.) O(.) O(.) efficient P1 A A P1 R R

  22. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • Fullyblack-box reductionimpliesrelativizingreduction (in general). • Oracle separation à la Impagliazzo-Rudichrules out relativizingreductionsandthus also fullyblack-box reductions. I want to try to build a key agreement scheme from a one-way function. What shall I do? How can I get around Impagliazzo-Rudich?

  23. Circumventing Impossibility Results • C: Construction may work for all f (black-box) or for all f, there is a construction (non-black-b) 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f AfbreaksGf  RA,fbreaks f

  24. Example: weak OWF  OWF • Weakly OWF: Inverting probability is smaller than 1-(1/poly). • For every weakly OWF f, there is some poly n: Gf: (x1,…, xn)  (f(x1),…,f(xn)) is one way. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f AfbreaksGf  RA,fbreaks f

  25. Circumventing Impossibility Results • A: The reduction R may work for all A (black-box) or for all A, there is an R (non-black-box) 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f AfbreaksGf  RA,fbreaks f

  26. Example: Goldreich-Levin • OWF f: (x,r)  f’(x),r • Then, h(x,r):=<x,r> is a hardcore bit for f: Given f(x,r), it is hard to predict h(x,r) • Reduction from predicting b=h(x,r) to inverting f. 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f AfbreaksGf  RA,fbreaks f

  27. Example: Goldreich-Levin • Predicting to inverting (decision to search) • Uses amplification techniques • The reduction R depends on the success probability of A 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f AfbreaksGf  RA,fbreaks f

  28. Circumventing Impossibility Results • P: The reduction R may work for all primitives f (black-box) or for all f, there is an R (non-b-b) 9 PPT Construction G 9 PPT Reduction R 8 Adversary A 8 Function f AfbreaksGf  RA,fbreaks f

  29. CAP Notation BBB (fullyblack-box) 9 PPT Construction G 9 PPT Reduction R 8Adversary A 8 Primitive f Construction {B,N} Adversary  {B,N} Primitive  {B,N} AfbreaksGf  RA,fbreaks f 9 PPT Construction G 8 Primitive f 8 Primitive f 9 PPT Construction G 9 PPT Reduction R 8Adversary A 8Adversary A 9 PPT Reduction R 9 PPT Reduction R 8Primitive f 8Primitive f 9 PPT Reduction R

  30. Three Questions • Is the construction black-box with respect to the primitive? • Is the reduction black-box with respect to the adversary? • Is the reduction black-box with respect to the primitive? Construction {B,N} G: f Adversary  {B,N} R: A Primitive  {B,N} R: f

  31. As a Picture x CircumventImpagliazzo-Rudichwith an NNN-reduction! BBB BNB BBN NBB NNB BNN NBN NNN Relativizing Reductions

  32. 8 function f 9 PPT Construction G 8 Adversary A 9 PPT Reduction R Assumption AfbreaksGf  RA,fbreaks f Take an Oracle O. Wehavetoshowthat: • Ifone-wayfunctionsexist relative to O, • thenkeyagreementexistsrelative to O. Analogous Proof f KA O(.) O(.) O(.) A P1 P2 What now?

  33. Circumventing Impagliazzo-Rudich Also Impossible! Exploit efficiency! Let‘s try to find a NNNa reduction! efficient A 8 Primitive f 9 PPT Construction 8 Adversary A 9 PPT Reduction R PPT AfbreaksGf  RA,fbreaks f

  34. Proof is not straightforward Not PPT f O(.) 8 Primitive f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R A R Not PPT, if f is Inefficient. Can weembed O into f? AfbreaksGf  RA,fbreaks f

  35. Impagliazzo Rudich Oracles PSPACE • Add PSPACE oracle • Add a random function f. • Prove, f is one-way. • Prove, KA is easy to break. NP • Relative to • oracle O=(PSPACE,f) : • OWFs exist. • KA does not exist. Minicrypt Key Agreement Easy/P/BPP Key Agreement Minicrypt Easy/P/BPP

  36. Embed PSPACE oracle into f • Add PSPACE oracle • Add a random function f. • Prove, f is one-way. • Prove, KA is easy to break. • Relative to • oracle O=(PSPACE,f) : • OWFs exist. • KA does not exist. Still a One-Way function, becausetheprobabilitythattest=0…0 for a random (x,x‘,test) istiny. f‘: (x,x‘,test)  0||f(x), if test is not 0….0 1||PSPACE(x‘), if test is 0…0

  37. Access to f‘ and (f,SPACE) is the same f‘: (x,x‘,test)  0||f(x), iftestis not 0….0 1||PSPACE(x‘), iftestis 0…0 Not PPT f O(.) 8 Primitive f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R A R Not PPT, if f is Inefficient. Can weembed O into f? AfbreaksGf  RA,fbreaks f

  38. 8 Function f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • Assume, OWFs exist relative to O. • If f is a OWF relative to O, then so is f‘ • Use f‘ in proof f‘: (x,x‘,test)  0||f(x), iftestis not 0….0 1||O(x‘), iftestis 0…0 f‘ Sig O(.) O(.) O(.) A P1 P2

  39. 8 Function f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • f‘ is an OWF relative to O. • Weshowthatone-time signaturesexist relative to O. Sig f‘ O(.) P1 f‘ Sig O(.) O(.) O(.) A P1 P2 G

  40. 8 Function f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • f‘ is an OWF relative to O. • Weshowthatone-time signaturesexist relative to O. Sig O(.) P2 P1 f‘ Sig O(.) O(.) O(.) A P1 P2 G

  41. 8 Function f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • P2 isefficient. • WecanimplementGf‘ eff. rel. to O. • IsSig=Gf‘ secureOT-Sig-scheme? Sig O(.) P2 P1 f‘ Sig O(.) O(.) O(.) A P1 P2 G

  42. 8 Function f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • P2 isefficient. • WecanimplementGf‘ eff. rel. to O. • IsSig=Gf‘ secureOT-Sig-scheme? Sig f‘ f‘ Sig O(.) O(.) O(.) A P1 P2 G

  43. 8 Function f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • Assumetow. contr., thereis PPT A such that AObreaksGf‘. • Then, thereis PPT A‘ such thatA‘f‘breaksGf‘andRA‘,f‘ breaks f‘. Sig f‘ O(.) f‘: (x,x‘,test)  0||f(x), iftestis not 0….0 1||O(x‘), iftestis 0…0 A G

  44. 8 Function f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R OT-Sig Assumption AfbreaksGf  RA,fbreaks f OW • RA‘,f‘ efficientlyimplementable relative to O: f‘ f‘ O(.) O(.) O(.) O(.) efficient P1 A A P1 R R

  45. Theorem If there is an NNNa-reduction from key agreement to one-way functions, then there is relativizing reduction from key agreement to one-way functions. Corollary There is no NNNa-reduction from key agreement to one-way functions.

  46. Circumventing Impagliazzo-Rudich Also impossible? Exploit efficiency (of A and f)! Let‘s try to find an NNNap reduction! efficientA,f PPT 8 Primitive f 9 PPT Construction G 8 PPT Adversary A 9 PPT Reduction R AfbreaksGf  RA,fbreaks f

  47. A Trivial Reduction Showingimpossibilityresult forNNNap-reduction showing impossibilityofkeyagreement altogether Assume, secure key agreement exists. 8 PPT Primitive f 9 PPT Construction G (ignores f) 8 PPT Adversary A 9 PPT Reduction R (ignores everything) neverhappens AfbreaksGf  RA,fbreaks f

  48. A Non-Trivial Reduction Assume, secure key agreement exists. 9 PPT Construction G (ignores f) 9 PPT Reduction R (ignores everything) 8 Adversary A 8 PPT Primitive f neverhappens AfbreaksGf  RA,fbreaks f Not PPT, if A is not PPT

  49. A Non-Trivial Reduction Assume, secure key agreement exists. 9 PPT Construction G (ignores f) 9 PPT Reduction R (ignores everything) 8 PPT Adversary A 8 Primitive f AfbreaksGf  RA,fbreaks f Not PPT, if f is not PPT

  50. To Circumvent Impagliazzo Rudich • Try NNNp or NNNap • Exploit the efficiency of the primitive f • Else, impossible… • …if you have an idea, first check whether it falls into the impossibility result.

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