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On the Complexity of Parallel Hardness Amplification for One-Way Functions Chi-Jen Lu Academia Sinica, Taiwan

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Outline Motivation Motivation Our Results Our Results Proof Ideas Proof Ideas

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Motivation

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Fundamental Primitives One-way function (OWF): One-way function (OWF): –easy to compute, hard to invert Pseudo-random generator (PRG): Pseudo-random generator (PRG): –stretch a random seed into a long random looking string

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Relationship weak OWF weak OWF strong OWF [Yao] strong OWF [Yao] PRG [HILL] PRG [HILL] –in polynomial time –in lower complexity classes?

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Hardness Amplification OWF f has hardness : poly-time M OWF f has hardness : poly-time M Pr x [M fails to invert f(x)] >. 1-n - (1) strong OWF n -O(1) weak OWF 2 -n worst-case OWF

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Question 1 Worst-case OWF Strong OWF? Worst-case OWF Strong OWF? ??? 1-n - (1) strong OWF n -O(1) weak OWF 2 -n worst-case OWF

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Weak OWF Strong OWF [Yao] f f [Yao] f f f (x 1,x 2,…,x k ) = (f(x 1 ),f(x 2 ),…,f(x k )) good: simple, parallel good: simple, parallel bad: not security-preserving (blow up input size) bad: not security-preserving (blow up input size)

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Weak OWP Strong OWP [GILVZ] f f [GILVZ] f f f (x, w 1,…,w k ) = f(w k (…(f(w 1 (f(x)))

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[GILVZ] f f [GILVZ] f f f (x, w 1,…,w k ) = f(w k (…(f(w 1 (f(x))) good: security-preserving good: security-preserving bad: complex, sequential bad: complex, sequential walk on expander Weak OWP Strong OWP

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Question 2 Weak OWF Strong OWF: Weak OWF Strong OWF: security preserving + parallel (low complexity)? Weak OWF AC 0 strong OWF AC 0 : security preserving ? Weak OWF AC 0 strong OWF AC 0 : security preserving ? constant-depth poly-size circuits

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Bigger Question Low-complexity Crypto? Low-complexity Crypto? Crypto. constructions / reductions in low complexity classes? Theory vs. practice Theory vs. practice

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Attempt on Question 2 Derandomize [Yao]? Derandomize [Yao]? f (x 1,x 2,…,x k ) = (f(x 1 ),f(x 2 ),…,f(x k )) Generate x 1,x 2,…,x k in some pseudo- random way from a short seed x? Generate x 1,x 2,…,x k in some pseudo- random way from a short seed x? f (x) = (f(x 1 ),f(x 2 ),…,f(x k )) –[IW] some success w.r.t. hardness of computing functions (BPP vs. P) k independent inputs

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No success for OWF … Impossible task? Impossible task? Aim: hardness amplification is a high complexity task Aim: hardness amplification is a high complexity task What if strong OWF f AC 0 ? What if strong OWF f AC 0 ? hard. amp.: ignore f, compute f directly …

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Black-Box Hardness Amplification

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(Strongly) Black Box Transformation: Transformation: hard f harder f = hard f harder f = A MP f uses f as a black box A MP uses f as a black box Hardness proof: Hardness proof: A breaks f D EC A breaks f D EC uses A as a black box could be unbounded

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Weakly Black Box Transformation: Transformation: hard f harder f = hard f harder f = A MP f uses f as a black box A MP uses f as a black box Hardness proof: Hardness proof: A breaks f D EC A breaks f D EC uses A as a black box

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Complexity Transformation: Transformation: hard f harder f = hard f harder f = A MP f uses f as a black box A MP uses f as a black box Hardness proof: Hardness proof: A breaks f D EC A breaks f D EC uses A as a black box hardness A MP high complexity

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Previous Work

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Lin-Trevisan-Wee B.B. hardness t B.B. hardness t with A MP making s queries t = O(s). t = O(s).

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Our Results

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Result (I) B.B. hardness t, with B.B. hardness t, with A MP realized in AC 0 (s) t (n/n) log O(1) s t (n/n) log O(1) s t n O(1) when n n O(1) & s 2 n O(1). t n O(1) when n n O(1) & s 2 n O(1). n: new input length n: init. input length PH NP P constant-depth circuits of size s

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Result (I) B.B. hardness t, with B.B. hardness t, with A MP realized in AC 0 (s) t (n/n) log O(1) s t (n/n) log O(1) s t log O(1) n when n=O(n) & s n O(1). t log O(1) n when n=O(n) & s n O(1). security preserving AC 0 n: new input length n: init. input length

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Result (II) Weakly B.B. hardness t, Weakly B.B. hardness t, with A MP realized in AC 0 & t > (n/n) log O(1) n A MP must embed a OWF with hardness t A MP must embed a OWF with hardness t

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Parallel Query Model

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Model [Vio] on input z: [Vio] A MP f on input z: –generates circuit C AC 0 (s) and non-adaptive queries x 1, …,x k –calls the oracle: (y 1, …,y k )=(f(x 1 ), …,f(x k )) –outputs (z) = C(y 1, …,y k ) –outputs A MP f (z) = C(y 1, …,y k )

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Proof Ideas

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Weakness of AC 0 circuits W.h.p. after a random restriction, W.h.p. after a random restriction, C AC 0 100 1** * w.p. 1 w.p. (1- )/2 0 w.p. (1- )/2. each bit independentlyreceived {

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Weakness of AC 0 circuits W.h.p. after a random restriction, any C AC 0 becomes biased W.h.p. after a random restriction, any C AC 0 becomes biased C AC 0 0, 1 100 **1 C(Y ) is the same for most Y

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B.B. Hard. Amp. z, (z) = C(f(x 1 ), …,f(x k )) AC 0 z, A MP f (z) = C(f(x 1 ), …,f(x k )) AC 0 Hardness t Hardness t Show: large t contradiction Show: large t contradiction Strategy: (follow closely [Vio]) find Strategy: (follow closely [Vio]) find –f: with hardness –f: with hardness –: with hardness < t –A MP f : with hardness < t

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Hardness Hardness W.h.p. a random function f is hard, W.h.p. a random function f is hard, even after a random restriction, if rate of * is high [Vio]. *1*1*00 …… 100*01* *01*11* 10*0*01 f (0 n ). f (1 n ) against inverter with poly queries

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kills A MP f kills A MP f [Vio] z, w.h.p. after a random, [Vio] z, w.h.p. after a random, (z) = C(f (x 1 ), …,f (x k )) AC 0 A MP f (z) = C(f (x 1 ), …,f (x k )) AC 0 is same for most f, if rate of * is low. W.h.p. over W.h.p. over, M A MP f for most f A =M breaks A MP f for most f D EC A inverts f well for most f.

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New Random Restriction Rate of * is low, but for a significant # of x, f (x) has enough *. Rate of * is low, but for a significant # of x, f (x) has enough *. is a (weak) OWF f is a (weak) OWF *1*1*00 …… 1001010 *01*11* 1010101 f (0 n ). f (1 n )

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Proof of Result (I) a restriction s.t. for most f, a restriction s.t. for most f, is hard to invert f is hard to invert kills kills A MP f some A inverts A MP f well D EC A inverts f well t in AC(s): large t, small s t in AC 0 (s): large t, small s

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Proof of Result (II) Derandomize Proof of Result (I) Derandomize Proof of Result (I)

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Other Result: PRG from OWF

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Result (III) B.B. PRG from OWF B.B. PRG from OWF P RG f : {0,1} r {0,1} m AC 0 (s) m-r o (r) when s 2 m o(1). sublinear stretch improving [Vio]: s m O(1).

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Conclusion & Questions

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High-Complexity Tasks Hard OWF harder OWF Hard OWF harder OWF OWF PRG of long stretch OWF PRG of long stretch

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Relation among Primitives –lower complexity? TDP TDFPKE PIROT KAOWF BC PRG … ZK

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