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Invertible Zero-Error Dispersers and Defective Memory with Stuck-At Errors Ariel Gabizon Ronen Shaltiel

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Defective memory with stuck-at errors msg z 000111 Enc *0**1*010 Dec 010 Reading proc. Dec must retrieve z, doesn’t know which bits are stuck. n-bit memory with ≤ p∙n “stuck bits” (can’t write) Goal: store msg z. Give explicit schemes. p: store n-p∙n-log c n bits. Rate > 1-p-o(1). Storage proc. Enc knows which bits are stuck. Earlier work [KuzTsy] shows existence of non-explicit schemes. Error-correcting codes where encoder knows in advance which bits will be corrupted. In standard codes we expect* rate 1-h(2p)-o(1).

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We can also handle few additional errs 000111 Enc Dec 010001111 Noise Give explicit schemes. Handle o(n ½ ) unknown errs. Rate > 1-p-o(1). Storage proc. Enc knows which bits are stuck. Reading proc. Dec must retrieve z, doesn’t know which bits are stuck. n-bit memory with ≤ p∙n “stuck bits” (can’t write) Goal: store msg z. msg z *0**1*010

4 `Defect pattern’:Subset S µ [n] of size ≤p∙n, String a 2 {0,1} p∙n. -Encoding proc. Enc, on message z 2 {0,1} m and pattern S,a produces x 2 {0,1} n with x| S = a. -Decoding proc. Dec such that Dec(x) = z Dec doesn’t depend on S,a m/n is the rate of the scheme. Explicit: Enc expected poly-time *, Dec poly-time. *Simply because that’s what we get. Dfn of memory scheme (for parameter 0

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Write Once Memory (k-WOM) n-bit memory initialized to zeros. If we raise a bit, it is “stuck-at” 1. Goal: Reuse memory k times. (think k=2). Stuck-at scheme => 2-WOM, gives opt. rate ≈ h(p) + (1-p). Intuitively WOM is easier to achieve. – Defect pattern is not adversarial. – Can pass o(n) bits from enc to dec by reserving first o(n) bits. (errorless channel from enc to dec). Starting point for this work: – Shpilka: Linear seeded extractors => 2-WOM. – We generalize: seedless zero-error disp. for bit-fixing sources schemes for stuck-at errors. 000000 11

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Bit-fixing sources and dispersers: S µ [n] – subset of size p∙n. a – string in {0,1} p∙n X S,a, set of strings in {0,1} n, s.t. x| S =a. Call such a set a bit-fixing source. |X S,a |=2 k for k=n-p∙n (# unfixed bits). Dfn: D:{0,1} n {0,1} m is a zero-error bit-fixing disperser for threshold k if S,a as above D(X S,a )={0,1} m. (Explicit: poly-time). Goal: achieve large m. Obviously m ≤ k = n-p∙n. Efficiently invertible: given S,a and output z, can find x 2 X S,a with D(x) = z in expected* poly(n)-time. *Simply because that’s what we get. *0**1**** D

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Dispersers => memory schemes (and vice-versa) Bit-fixing disperser D:{0,1} n {0,1} m. Encoding: Given message z 2 {0,1} m and defect pattern S,a, Enc finds x 2 X S,a with D(x) = z. Decoding: Given x, Dec computes D(x)=z. Output len. m=(1-o(1))∙k => optimal rate. Explicitness of scheme follows if D is explicit and efficiently invertibile. This work: construct such dispersers (for k>log c n). Improve [GabSha08] where m= (k).

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Construction of zero-error disperser

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Seeded zero-error dispersers Dfn: E(x,y) is a seeded bit-fixing disperser for threshold k, if z, and bit-fixing source X S,a with k unfixed bits, there exist: -a seed y, and -x 2 X S,a such that E(x,y) = z. E is efficiently invertible if x,y can be found in expected poly(n)-time given z,S,a. Known [RazReiVad] : m=(1-o(1))∙k, |y|=log 3 n. Shpilka: 2-WOM, Enc and Dec can share o(n) bits. Given z,S,a, Enc finds y and shares it.

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Linear seeded extractors => seeded bit- fixing disperser Known: Explicit constructions of E(x,y) s.t. bit-fixing source * X S,a with k unfixed bits, for most y’s E(X S,a,y) is close to uniform. y: T(x)=E(x,y) is linear over F 2 n. * In fact, distribution with min-entropy ≥ k. X S,a is an affine subspace of F 2 n. E is a seeded bit-fixing source disperser that is efficiently invertible. [Tre99,ShaUma01]: seed O(log n), output √k [RazReiVad00]: seed O(log 3 n), output (1-o(1))k

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(Seedless) disperser construction: Step 1: (seedless) disperser with short output m=O(log n) that is efficiently invertible. Step 2: (Using approach of [GabSha08]) Combine with seeded disperser with large output length to get (seedless) and efficiently invertible disperser with large output length.

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Thm: Any explicit bit-fixing extractor* D:{0,1} n {0,1} m with m=O(log n) is efficiently invertible. *Say, Pr X X S,a [D(X)=z] ¸ 2 -2m = 1/poly(n) Known Explicit constructions: [KempZuck03,GabRazSha04,Rao09]. Proof: Sample random x 2 X S,a and check if D(x) =z. Will take expected poly(n)-time. Step 1: Disperser with m=O(log n)

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Step 2: Increasing output length with threshold k’ rather than k => F is efficiently invertible

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Step 2: Increasing output length with threshold k’ rather than k affine subspace

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Dispersers for affine sources Get: zero-error disperser for affine sources. Used in memory scheme w/ additional errs. Have explicit D only for k=n/(log log n) ½. Final disperser has k > n/(log log n) ½. Output length m = k - O(n/(log log n) ½ ). Suffices for achieving rate 1-p-o(1). In paper: Subsource-hitter for bit-fixing. => Final disperser for bit-fixing sources. Works for any k > log c n. Output length m = k - O(log c n). Gives schemes with m = n-p∙n - O(log c n).

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Linear seeded extractors Subsource hitter for bit-fixing sources. Subsource hitter (Inspired by [GabRazSha] ) *0**1* X S,a *** seed y *0**1* X S,a *** seed y seed y’ y’ - seed for sampler, selects subset of [n]. y’ that partitions unfixed bits: Most unfixed bits are selected. At least k’ bits are not selected. Apply E(∙,y) only on selected portion of x.

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Handling stuck-at memory with noise msg z 000111 Enc *0**1*010 Dec 010001111 Noise Idea: Force Enc to output codeword in code for noise. Need code: defect pattern many consistent codewords. Linear code w/ dual dist p∙n. Code ∩ X S,a is affine space. dim n-p∙n-o(n) (for n ½ errs) Use affine source disp. k=n-p∙n-o(n) => m=n-p∙n-o(n). => rate 1-p-o(1). Given corrupted x’ that is close to x: Decode x’ to find x. Compute D(x) to get msg z.

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Open questions

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That’s it…

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Standrad coding theory (unknown errors) msg z 010101 Enc 010 Dec 010000111 Channel If p∙n bits are corrupted, rate can be* ≈ 1-h(p) Achieved by random codes. Not known to be tight. Explicit constructions achieve poorer rate. C(z)

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Suppose Enc\Dec know where errors are: msg z 00101 Enc *0**1010 Dec 010 Can trivially get get rate 1-p, by using `unstuck’ places Channel replaces ≤p∙n bits by fixed bits.

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What if only Enc knows errors? msg z 000111 Enc *0**1*010 Dec 010 Dec must retrieve z without knowing the error pattern Channel replaces ≤p∙n bits by fixed bits. [Kuznetsov-Tsybakov:] possible to get any rate R < 1-p. We give first explicit schemes. rate R > 1-p-o(1).

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Dfn: A subsource of X S,a is a set X S’,a’ ½ X S,a for -S’ ¾ S with |S’| · 2p ¢ n. *0**1* X S,a : 00**1* X S’,a’ :

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Dfn:E(x,y) is a subsource hitter if given z, there exists - a seed y - a subsource X S’,a’ of X S,a such that: E(X S’,a’,y) ´ z E is eff. invertible if y, X S’,a’ can be found in poly(n)-time.

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Theorem of [GabSha08] -D is a b.f.d (for sets |S’| · 2p ¢ n). -E is a subsource hitter. F(x), E(x,D(x)) is a bit-fixing disperser. Proof: -Given z, 9 X S’,a’ with E(X S’,a’,y) ´ z. -Also, 9 x 2 X S’,a’ with D(x)=y. For this x: E(x,D(x))=E(x,y)=z.

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Efficient Invertibilty of F Proof: -Given z, 9 X S’,a’ with E(X S’,a’,y) ´ z. -Also, 9 x 2 X S’,a’ with D(x)=y. For this x: E(x,D(x))=E(x,y)=z. E e.i. can efficiently find X S’,a’,y. D e.i. can eff. find x 2 X S’,a’ with D(x)=y.

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Seeded dispersers subsource hitters: First, sample a random subset of the input bits. Apply seeded disperser only on this subset. *0**1* Preimage set now includes all settings of non-fixed bits outside of the subset. apply E here

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Similar results for `mixed’ scenario ‘stuck errors’ known to encoder+ noise added after encoding uses affine dispersers and error correcting codes rather than bit-fixing dispersers. We get similar improvements for output length of affine dispersers.

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Open questions -Invertible dispersers for `Hamming sources’: Find mapping D:{0,1} n {0,1} m such that for all large enough hamming balls B D(B) = {0,1} m, and there is efficient way of finding preimage of z in B. Thanks

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Dispersers imply memory schemes Bit-fixing disperser D:{0,1} n {0,1} m. - Encoding: Given message z 2 {0,1} m and defect pattern S,a, Enc finds x 2 X S,a with D(x) = z. - Decoding: Given x, Dec computes D(x)=z. Output length m = (1-o(1))∙k => optimal rate. Explicitness of scheme follows if D is explicit and efficiently invertibile.

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m. -Message z 2 {0,1} m -Codeword C(z) 2 {0,1} n Decoder must retrieve z from C(z) after p∙n bits of C(z) have been corrupted. (will look at different dfns next) m/n, the rate of C. Coding Theory

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What if only Enc knows errors? msg z 000111 Enc *0**1*010 Dec 010 [Kuznetsov-Tsybakov:] possible to get any rate R < 1-p. We give first explicit schemes. `defect pattern’- p∙n `stuck’ bits

33 Memory with n cells. 0

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`Defect pattern’:Subset S µ [n] of size p∙n, String a 2 {0,1} p∙n. Message z 2 {0,1} m. -Encoding function E, given S,a,z, produces x 2 {0,1} n with x| S = a. -Decoding function D such that D(x) = z D doesn’t depend on S,a m/n is the rate of the scheme Formal dfn of memory scheme:

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[Kuznetsov-Tsybakov] : Introduced model. Non- explicit schemes with any rate R<1-p [Tsybakov] : Explicit schemes using linear codes with rate <<1-p Our result: Explicit* schemes with rate 1-p- o(1). *-Our encoding is runs in expected polynomial time. Previous results and Ours

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Bit-fixing dispersers: *0**1**** Given all strings with a certain `defect pattern’ a bit-fixing disperser produces all possible outputs D

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Bit-fixing dispersers: *0**1**** Given all strings with a certain `defect pattern’ a bit-fixing disperser produces all possible outputs D Example: D(x 1,…,x n ), i x i (mod 2)

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Bit-fixing dispersers: S µ [n] – subset of size p∙n. a – string in {0,1} p∙n X S,a, set of strings in {0,1} n, with x| S =a. Call such a set a bit-fixing source. Dfn: D:{0,1} n {0,1} m is a bit-fixing disperser if for every S,a as above D(X S,a )={0,1} m (this is actually a zero-error disperser)

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Bit-fixing dispersers: *0**1**** Another example: Suppose k=n-p∙n. Take the sum of input bits (mod k).. gives us logk output bits. D

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Dispersers to memory schemes: Bit-fixing disperser D(x). Message z -Given defect pattern S,a, Enc finds x 2 X S,a with D(x) = z. -Dec simply computes D. Efficiency of scheme requires `efficient invertibility’ of D.

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Dfn: We say D is efficiently invertible if given S,a and output z, we can find in expected* poly(n)-time x 2 X S,a with D(x) = z. *Simply because that’s what we get

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Our results: Bit-fixing dispersers with output length m=k-o(k), where k = n-p ¢ n. Previous result m= (k). [GabSha08] Also, our dispersers are efficiently invertible – (previous constructions seem not to be)

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Disperser Construction: Step 1: Disperser with short output. Step 2: Seeded disperser with large output length. Step 3: Combining the above to get (seedless) disperser with large output length. (Follows [GabSha08])

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Thm: Any explicit bit-fixing extractor* D:{0,1} n {0,1} m with m=O(log n) is efficiently invertible. *Say, Pr x X S,a [D(x)=z] ¸ 2 -2m =1/poly(n) Known Explicit constructions: [KempZuck03,GabRazSha04,Rao09]. Proof: Sample random x 2 X S,a and check if D(x) =z. Will take expected poly(n)-time. Step 1: Disperser with m=O(log n)

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2 nd step: we construct efficiently invertible seeded bit-fixing dispersers with large output length.

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Seeded Bit-fixing dispersers: *0**1**** We `invest’ a short seed to obtain a longer output. E * auxiliary short random input a.k.a `seed’

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Dfn:E(x,y) is a seeded bit-fixing disperser if given z, there exist: -a seed y, and -x 2 X S,a such that: E(x,y) = z. E is efficiently invertible if x,y can be found in poly(n)-time...idea is that |y|<<|z|.

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Observation: If T(x)=E(x,y) is linear function for every fixed seed y, then E is efficiently invertible: For every y: (assume |y|=O(log n) ) -Check if exists x with -x 2 X S,a -E(x,y) =z (This is a set of affine equations and can be check efficiently)

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`Linear seeded extractors’ exactly give us such an E with large output length Such constructions by [Trevisan\RazReingoldVadhan\ShaltielUmans,..]

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Step 3: Get rid of seed: -D(x) is a b.f.d. with output length d -E(x,y) is a seeded b.f.d. with seed length d [GabSha08] : F(x) = E(x,D(x)) is a bit-fixing disperser assuming E is also a `subsource hitter’. Also: If D and E are efficiently invertible so is F.

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Subsource hitters [GabSha08] `a seeded bit-fixing disperser where the set of preimages of z contains a bit-fixing source’

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Dfn: A subsource of X S,a is a set X S’,a’ ½ X S,a for -S’ ¾ S with |S’| · 2p ¢ n. *0**1* X S,a : 00**1* X S’,a’ :

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Dfn:E(x,y) is a subsource hitter if given z, there exists - a seed y - a subsource X S’,a’ of X S,a such that: E(X S’,a’,y) ´ z E is eff. invertible if y, X S’,a’ can be found in poly(n)-time.

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Theorem of [GabSha08] -D is a b.f.d (for sets |S’| · 2p ¢ n). -E is a subsource hitter. F(x), E(x,D(x)) is a bit-fixing disperser. Proof: -Given z, 9 X S’,a’ with E(X S’,a’,y) ´ z. -Also, 9 x 2 X S’,a’ with D(x)=y. For this x: E(x,D(x))=E(x,y)=z.

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Efficient Invertibilty of F Proof: -Given z, 9 X S’,a’ with E(X S’,a’,y) ´ z. -Also, 9 x 2 X S’,a’ with D(x)=y. For this x: E(x,D(x))=E(x,y)=z. E e.i. can efficiently find X S’,a’,y. D e.i. can eff. find x 2 X S’,a’ with D(x)=y.

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Seeded dispersers subsource hitters: First, sample a random subset of the input bits. Apply seeded disperser only on this subset. *0**1* Preimage set now includes all settings of non-fixed bits outside of the subset. apply E here

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Similar results for `mixed’ scenario ‘stuck errors’ known to encoder+ noise added after encoding uses affine dispersers and error correcting codes rather than bit-fixing dispersers. We get similar improvements for output length of affine dispersers.

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Open questions -Invertible dispersers for `Hamming sources’: Find mapping D:{0,1} n {0,1} m such that for all large enough hamming balls B D(B) = {0,1} m, and there is efficient way of finding preimage of z in B. Thanks

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Pseudorandomness Course Dispersers are related to Extractors that have applications in TCS, e.g.: -Running randomized algorithms with weak randomness -Generating pseudorandom bits for small space algorithms.

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Increasing the output length of dispersers [GabSha08] D – a bit-fixing disperser with output length d E – a subsource hitter with seed length d, output length m F(x) = E(x,D(x)) is a bit-fixing disperser with output length m.

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Efficient invertibility F(x) = E(x,D(x)) D has output length O(log n) Finding preimage of z: -choose random y – find subspace X’ of X S,a with E(X’,y) ´ z (system of affine equations) Find x 2 X’ with D(x) = y. We get F(x) = E(x,D(x)) = E(x,y) = z. (Last equality because x 2 X’)

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Efficient invertibility When D has output length O(logn) random sampling will succeed w.h.p in poly(n)-time.

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Dispersers to memory schemes: Message z 000111 Encoder finds x 2 X S,a with D(x) = z *0**1*010 Decoder computes D(x) to retrieve z 010 must always decode correctly without knowing the defect pattern `defect pattern’- p∙n `stuck’ bits

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Standard coding: Message z 010101 Encoder 010 Decoder 010000111 Noise Codeword C(z)

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