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Short seed extractors against quantum storage Amnon Ta-Shma Tel-Aviv University 1

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Privacy amplification [BB] Alice and Bob share information that is partially secret towards an eavesdropper Eve. Their goal is to extract a shorter string that is completely secret. They may use a short, public random string.

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More formally: Alice and Bob share x {0,1} n. x has a-priori distribution X that has a lot of entropy. H (X) k a Pr[X=a] 2 -k Eve holds a random variable W on {0,1} b that holds partial information about x. 3

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A (k,b, ) extractor - classical case E:{0,1} n {0,1} t {0,1} m is a (k,b, ) extractor, if: For every X with H (X) k, and, For every W=W(X) distributed on {0,1} b |U t E(X, U t ) W(X) – U t U m W(X) | Sample: x X, y {0,1} t Output: y,E(x,y),W(x) Sample: x X, y {0,1} t,u {0,1} m Output: y,u,W(x) 4

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In the classical world The problem can be solved almost optimally using extractors. Solutions give: t=O(log(n/ )) m= (k-b) 5

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A (k,b, ) extractor - quantum case E:{0,1} n {0,1} t {0,1} m is a (k,b, ) extractor against quantum storage, if: For every X with H (X) k, and, For every = (X) on b qubits |U t E(X, U t ) (X) – U t U m (X) | tr Sample: x X, y {0,1} t Output: y,E(x,y), (x) Sample: x X, y {0,1} t,u {0,1} m Output: y,u, (x) 6

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In the quantum world Some extractors fail. [GKKRWJ] show an extractor against b bits that fails against polylog(b) qubits. Some extractors work. Konig, Maurer,Renner 04 Fehr, Schaffner 08 Konig Terhal 08 7

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Previous extractors - quantum case TechniqueSeed lengthAuthor Pair-wise independence, Collisionst= (n)Konig, Maurer, Renner Almost pair-wise independencet= (m)Variation on KMR Z 2 n Fourier transformt= (b)Fehr, Schaffner Any one-output extractor is goodt= (m)Konig Terhal Any extractor is good with error 2 b t= (b)Konig Terhal Several methodst=O(log(n))Classical E : {0,1} n {0,1} t {0,1} m 8

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Our result A (k,b, ) extractor E:{0,1} n {0,1} t {0,1} m against quantum storage, with: Optimal t=O(log n) when m=n (1) Trevisan: m=(k-b) (1) Optimal: (k-b) 9

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The basic paradigm Reconstruction algorithms Reconstruction Extraction in the classical world [Trevisan] Reconstruction with few queries Extraction against quantum storage. 10

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Distinguisher A test is a function T : {0,1} m {0,1} A test T -distinguishes D 1 from D 2 if | Pr x D1 [T(x)=1] – Pr x D2 [T(x)=1] | 11

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Reconstruction algorithms A function E:{0,1} n {0,1} t {0,1} m has a reconstruction algorithm R if For every x {0,1} n, and every T that distinguishes U t E(x,U t ) from U t+m There exists a string adv=adv(x) of a bits, s.t. R T (adv(x))=x 12

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Reconstruction Extraction [Tre] Suppose E has reconstruction with a advice bits, Suppose E is not a (k,b, ) extractor. Then, there exist: X with H (X) k, Eve storing b bits of information, -distinguishing E from uniform. B={x| Eve -dist W(x) U t E(x, U t ) from W(x) U t+m } |B| ε|X| 13

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For every x B The test T: Gets advice W(x). Applies Eve( W(x), y, w). -distinguishes U t E(x, U t ) from U t+m. The reconstruction algorithm: Makes oracle calls to T. Gets additional a bits of advice adv(x). Reconstructs x. Thus x B can be reconstructed using a+b bits. 14

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Reconstruction Extraction [Tre] |B| 2 a+b and 2 k |X| |B|/. Thus, ka+b+log(1/ ). 15

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Extractor against quantum storage Suppose E has reconstruction with q queries. Suppose E is not a (k,b, ) extractor. Then, there exist: X with H (X) k, Eve storing b qubits of information, B={x| Eve -dist (x) U t E(x, U t ) from (x) U t+m } |B| ε|X| 16

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For every x B The test T: Gets advice (x). Applies Eve( (x), y, w). -distinguishes U t E(x, U t ) from U t+m. The reconstruction algorithm: Makes oracle calls to T. Gets additional a bits of advice adv(x). Reconstructs x. Thus x B can be reconstructed using a+qb bits For the classical advice adv(x) For q queries to Eve 17

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Extractor against quantum storage |B| 2 a+qb. Thus, 2 k |X| 2 a+qb /. ka+qb+log(1/ ). 18

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Conclusions so far A function E:{0,1} n {0,1} t {0,1} m that has a reconstruction algorithm with A short classical advice adv(x), and, A few queries to the distinguisher Yields a good extractor against quantum storage. 19

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An extractor with reconstruction The NW generator List decoding Trevisans extractor The quantum case Trevisans work 20

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The NW Generator NW:{0,1} n {0,1} t {0,1} m has reconstruction that is correct on average. Given a distinguisher T, and The right advice adv(x) R T (adv(x),i) = x i For most i [n] 21 The NW generator uses a single query

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List decoding 22

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Trevisans extractor Uses: NW and its reconstruction algorithm, A code C : {0,1} n {0,1} N that is (L=poly(n),p=1/2- ) list-decodable. T(x,y)= NW( C(x), y) 23

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Reconstruction for Trevisans ext. T(x,y)= NW( C(x), y) Find a word w {0,1} N that is 1/2+ close to C(x) using the NW reconstruction algorithm. Apply list decoding. Get a List L of all code words close to w, x L. The advice tells us which is x. Works well, but requires N queries. 24

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The way around NW generator – learns a single bit of C(x), with one query, on average over i [N] 25 Learn the whole of x, with poly(n) queries. Trevisan: List decoding Learn a single bit of x, with polylog(n) queries, for any i [n] of our choice. Us: Local list decoding

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Two questions 1.How do we achieve that? Answer: using local list decoding. 2. Does this suffice for the analysis? Answer: Yes, using lower bounds on random access codes. 26

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The new extractor Uses: NW generator and its reconstruction algorithm, A code C : {0,1} n {0,1} N that is (L=poly(n),p=1/2+ ) locally list-decodable with q=polylog(n) queries. E(x,y)= NW( C(x), y) 27

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The Analysis Suppose E(x,y)= NW( C(x), y) is not a (k,b, ) ext, violated with X and = (X). For any x B Advice: a+qb qubits We can learn any bit of x, with succ. prob. 2/3. |B| 2 (a+qb) log n. 2 k |X| 2 (a+qb) log n /. k(a+qb) log n+log(1/ ). 28 a RAC for B using a+qb qubits

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Random access code for X RAC : X density matrix over m qubits such that for every x X: For all i [n], one can recover x i from RAC(x) with success probability at least 2/3. For most i [n], one can recover x i from RAC(x). Average-case RAC Worst-case RAC 29

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RAC for X Arbitrary XX={0,1} n (n) Worst case RAC 0 (n) Average case RAC 30

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Summary For the construction, we use: Trevisan extractor, with Local, list-decodable error correcting codes For the analysis, we use: Reconstruction algorithms together with Random access codes 31

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Local decoding A code C:{0,1} n {0,1} N has (q,, ) a local Decoding algorithm D, if For every x {0,1} n, y {0,1} N, d(y,C(x)) N For every i [n] Pr [ D y (i)=x i ] 1- and D makes at most q queries to y. 32

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Challenge 1.Find an extractor that Works against quantum storage With optimal parameters. 2. Generalize the construction to Eve that holds more qubits but has few information about X.

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List decoding A code C:{0,1} n {0,1} N is (L,p) list-decodable, if for every w {0,1} N there are at most L codewords that are p-close to w. |{i | y i =w i }| pN 34

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Unique decoding 35

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