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Explicit Exclusive Set Systems with Applications to Broadcast Encryption ﺩ. ﻓﻴﺮﻭﺯ ﺗﺸﻴﺮ ﺍﻟﺒﻨﺪﺭﻱ ﺍﻟﺤﺮﺑﻲ Explicit Exclusive Set Systems with Applications to Broadcast Encryption ﺩ. ﻓﻴﺮﻭﺯ ﺗﺸﻴﺮ ﺍﻟﺒﻨﺪﺭﻱ ﺍﻟﺤﺮﺑﻲ ﺳﻤﻴﺔ ﺍﻟﻬﺰﺍﻉ ﻧﺠﻼﺀ ﺍﻟﺮﺷﻴﺪﻱ ﻫﺒﺔ ﺍﻟﻬﻠﻴﺲ ﻣﻨﺎﻝ ﺑﻦ ﻋﺎﻣﺮ

Broadcast Encryption Clients Server ØØ 1 server, n clients can understand broadcasts Only privileged Broadcast Encryption Clients Server ØØ 1 server, n clients can understand broadcasts Only privileged users ØØ Server broadcasts to all clients atbills E. g. , those who pay their monthly once Ø Need to encrypt broadcasts Ø E. g. , payperview TV, music, videos

Subset Cover Framework [NNL] Ø Offline stage: Ø For some S ½ [n], server Subset Cover Framework [NNL] Ø Offline stage: Ø For some S ½ [n], server creates a key K(S) and distributes it to all users in S Ø Let C be the collection of S Ø Server space complexity ~ |C| Ø ith user space complexity ~ # S containing i

Subset Cover Framework [NNL] Ø Online stage: Ø Given a set R ½ [n] Subset Cover Framework [NNL] Ø Online stage: Ø Given a set R ½ [n] of at most r revoked users Ø Server establishes a session key M that only users in the set [n] n R know Ø Finds S 1, …, St 2 C with [n] n R = S 1 [ … [ St Ø Ø Encrypt M under each of K(S 1), …, K(St) Content encrypted using session key M

Subset Cover Framework [NNL] Ø Communication complexity ~ t Ø Tolerate up to r Subset Cover Framework [NNL] Ø Communication complexity ~ t Ø Tolerate up to r revoked users Ø Tolerate any number of colluders Ø Information-theoretic security

The Combinatorics Problem Ø Find a family C of subsets of {1, …. , The Combinatorics Problem Ø Find a family C of subsets of {1, …. , n} such that any large set S µ {1, …, n} is the union of a small number of sets in C S = S 1 [ S 2 [ [ St Ø Parameters: Ø Universe is [n] = {1, …, n} Ø |S| >= n-r Ø Write S as a union of · t sets in C Ø Goal: Ø Minimize |C|

A Lower Bound Claim: Proof: 1. At least sets of size ¸ n-r 2. A Lower Bound Claim: Proof: 1. At least sets of size ¸ n-r 2. Only 3. Thus, 4. Solve for |C| different unions

Known Upper Bounds t |C| authors (r log n / log r)2 GSY r Known Upper Bounds t |C| authors (r log n / log r)2 GSY r log n/r 2 n LNN, ALO 2 r n log n LNN r 3 log n / log r r 3 log n /log r KRS Bad: once n and r are chosen, t and |C| are fixed

Known Upper Bounds Ø Only known general result: Ø Ø If r · t, Known Upper Bounds Ø Only known general result: Ø Ø If r · t, then |C| = O(t 3(nt)r/t log n) [KR] Drawbacks: Ø Ø Ø Probabilistic method To write S = S 1 [ S 2 [ … [ St , solve Set-Cover C has large description No way to verify C is correct Suboptimal size:

Our Results Ø Main result: tight upper bound |C| = poly(r, t) Ø Ø Our Results Ø Main result: tight upper bound |C| = poly(r, t) Ø Ø Match lower bound up to poly(r, t) Ø Ø Ø n, r, t all arbitrary In applications r, t << n When r, t << n, get |C| = O(rt ) Our construction is explicit Ø Find sets S = S 1 [ … [ St in poly(r, t, log n) time Ø Improved cryptographic applications

Cryptographic Implications Ø Our explicit exclusive set system yield almost optimal information-theoretic broadcast encryption Cryptographic Implications Ø Our explicit exclusive set system yield almost optimal information-theoretic broadcast encryption and multicertificate revocation schemes Ø General n, r, t Ø Contrasts with previous explicit systems Ø Poly(r, t, log n) time to find keys for broadcast Ø Contrasts with probabilistic constructions Ø Parameters Ø For poly(r, log n) server storage complexity, we can set t = r log (n/r), but previously t = (r 2 log n)

Techniques Ø Case analysis: Ø r, t << n: algebraic solution Ø general r, Techniques Ø Case analysis: Ø r, t << n: algebraic solution Ø general r, t: use divide-and-conquer approach to reduce to previous case

Case: r, t << n Ø Find a prime p = n 1/t + Case: r, t << n Ø Find a prime p = n 1/t + Ø Users [n] are points in (Fp)t Ø Consider the ring Fp[X 1, …, Xt] Ø Goal: find set of polynomials C such that for any R ½ [n] with |R| · r, there exist p 1, …, pt 2 C such that R = Variety(p 1, …, pt)

Case: r, t << n Ø First design a polynomial collection so that for Case: r, t << n Ø First design a polynomial collection so that for any R ½ [n] with |R| · r such that for every coordinate i, 1 · i · t, All |R| points differ on the ith coordinate (*) Ø Then perform a few permutations : [n] -> [n] and construct new polynomial collections on ([n]). Take the union of these collections. Ø Can find the deterministically using MDS codes

Example Collection: r = 2, t = 3 For r = 2, t = Example Collection: r = 2, t = 3 For r = 2, t = 3, our collection is: 1. (X 1 – a)(X 1 – b) for all distinct a, b 2. a. X 1 + b – X 2 for any a, b 2 Fp 3. a. X 2 + b – X 3 for any a, b 2 Fp Revoke u = (u 1, u 2, u 3) and v = (v 1, v 2, v 3) u 1 v 1, u 2 v 2, and u 3 v 3 Let p 1 = (X 1 – u 1)(X 1 -v 1). Find p 2 by interpolating from au 1 + b – u 2 = 0, av 1 + b – v 2 = 0 Find p 3 by interpolation. Variety(p 1, p 2, p 3) = u, v We broadcast with keys K(pi), distributed to users which don’t vanish on pi If u 1 v 1, u 2 = v 2, and u 3 v 3, then (u 1, u 2, v 3) also in variety…

Our General Collection and Intuition: First type of polynomials implement a “base case”. Second Our General Collection and Intuition: First type of polynomials implement a “base case”. Second type of polynomials implement “AND”s.

Wrapping up the r, t << n case. Ø Using many tricks – balancing Wrapping up the r, t << n case. Ø Using many tricks – balancing techniques, expanders, etc. , can show even without distinct coordinates, can achieve size O(rt ). Ø Almost matches the (t Ø Open question: resolve this gap. ) lower bound.

General n, r, t 1 x xxx i j x x n Ø Problem! General n, r, t 1 x xxx i j x x n Ø Problem! n 2 term ? !? Ø Let m be such that r/m, t/m << n Ø Fix: - hash [n] to [r 2] first Ø For every interval [i, j], form an exclusive set system enough = j-i+1, so = r/m, t’an injective - do with n’ hashes r’ there is = t/m hash for every R Ø Given a set R, find intervals which evenly partition R. - apply construction above on [r 2]

Summary and Open Questions Ø Main result: tight explicit upper bound |C| = poly(r, Summary and Open Questions Ø Main result: tight explicit upper bound |C| = poly(r, t) Ø n, r, t arbitrary Ø Cover sets in poly(r, t, log n) time Ø Optimal # of keys per user Ø Other result: Slightly improve [LS] lower bound on keys per user in any scheme using a relaxed sunflower lemma: from ( )/(rt) to ( )/r Ø Open question: improve poly(r, t) factors