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Computing with Polynomials over Composites Parikshit Gopalan Algorithms, Combinatorics & Optimization. Georgia Tech.

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Primes and Composites The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic … - C.F. Gauss Disquisitiones Arithmeticae (1801) Primality testing. Factoring. Many other facets to the prime vs. composite problem in computer science …

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ComplexityCombinatorics DerandomizationCoding Theory Polynomials over Primes

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Polynomials over Composites ? ? ? ?

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ComplexityCombinatorics Algorithms Cryptography Polynomials over Composites

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A Problem from Circuit Complexity Problem: Find a function that cannot be computed by small circuits with AND, OR and Mod-m gates. For Mod-p gates (p is prime) [Razborov, Smolensky] With p = 2, the Mod-3 function is hard. For Mod-m gates (m is composite, say 6) With m = 6, is the Mod-5 function hard ? No lower bounds known for any function. Circuit complexity: Show lower bounds on size of circuits computing a function. Is X i = 0 mod m ? Poly. Size Const. depth

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A Problem about Polynomials Computing Boolean functions by polynomials: Def: P(X 1,, X n ) over Z m represents f: {0,1} n ! {0,1} if f(x) f(y) ) P(x) P(y) mod m. Problem: What is the degree of OR mod m ? For p prime: (n). For m composite (say 6): Conjecture: (n) [Barrington] O(n 1/2 ) upper bound. [Barrington-Beigel-Rudich] (log n) lower bound. [Barrington-Tardos]

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A Problem about Set Systems Problem: Let F be a family of subsets S i of [n] where |S i | = 0 mod m |S i Å S j | 0 mod m How large can F be? RCW Thm. : For p prime, |F| · O(n p-1 ). Conjecture : For any m, |F| is at most polynomial. [Frankl] Thm. : If m = 6, |F| can be superpolynomial. [Grolumsz] Extremal Set Theory: How large can a set system satisfying certain conditions be?

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Polynomials over Composites Complexity: Circuits. Boolean function Representations. Combinatorics: Set systems. Ramsey graphs. Algorithms: Root-finding. Interpolation. Cryptography: RSA. Rabin cryptosystem.

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Primes versus Composites: The Prime Case: – Low degree polynomials have few zeroes. –Finite Fields, linear algebra. The Composite Case: –Proof techniques fail. –Problems behave differently. –Polynomials have (unexpected) structure. Primality testing. [Agrawal-Biswas, AKS] Complexity. [Barrington et al., Bhatnagar-G.-Lipton, Hansen]. Combinatorics. [Grolmusz, G.06].

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In This Talk: Complexity : Computing Boolean functions by polynomials. [Bhatnagar-G.-Lipton] Combinatorics : Explicit Ramsey graphs. Algorithms : Interpolation over Z m. Polynomials over Composites.

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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Motivation Def: P(X 1,, X n ) over Z m represents f: {0,1} n ! {0,1} if f(x) f(y) ) P(x) P(y) mod m [Razborov, Smolensky] : Small circuits low-degree polynomials. Prove degree lower bounds. [Barrington-Beigel-Rudich] : Degree lower bounds over Z m. (Simpler problem?) Applications to Combinatorics, Computational Learning.

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State of the Art [Barrington-Beigel-Rudich, Grolmusz, Tsai, Barrington-Tardos, Green, Alon-Beigel, …] : O(n) upper bound for OR, AND. [Barrington-Beigel-Rudich] Best lower bound is (log n). [Grolmusz, Barrington-Tardos] [Bhatnagar-G.-Lipton] : Symmetric Polynomials. Symmetric Polynomials ´ Communication Protocols. Number theory, Communication complexity. Tight bounds for most functions. [Hansen]

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Symmetric Polynomials over Z p f : {0,1} n ! {0,1}, P : {0,1} n ! Z p (both symmetric). Weight w(x) = no. of 1s in x. Hence f : {0, …, n} ) {0,1} P : {0,…, n} ) Z p. Q: What can we compute with low degree polynomials? A: Write w in base p as w = w 0 + w 1 p + w 2 p 2 … + w p Thm. : Polynomials of degree p t -1 compute all functions P: {0, …, n} ) Z p that depend on w 0, …, w t-1 (on w mod p t ).

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0 1 0 0 0 1 1 Mod-k functions over Z 2 1 0 0 0 0 0 0 1 0 0 0 1 1 Mod-2 Mod-3 Mod-4 O(1) (n) What happens over Z 6 ?

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Thm. [Bhatnagar-G.-Lipton] : Symmetric Polynomials are equivalent to Simultaneous Protocols. Simultaneous Protocols

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0 0 1 0 2 2 k3k3 P 2 (w) Z 2 P 3 (w) Z 3 0 1 0 0 0 1 1 k2k2 f(w) f : w {0,…,n} {0,1} w = 35 Cost = max(2 k 2, 3 k 3 )

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Thm. [Bhatnagar-G.-Lipton] : Symmetric Polynomials are equivalent to Simultaneous Protocols. Simultaneous Protocols If m has 3 prime factors, protocols involve 3 players.

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P 2 (X)P 3 (X) CRT 0 1 0 0 0 1 1 k2k2 k3k3 Deg(P 3 ) dDeg(P 2 ) d Deg(P) d 3 k 3 d 2 k 2 d Representations ) Protocols P(X) 0 0 1 0 2 2 CRT

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A Protocol for OR 0 0 1 0 2 2 k3k3 P 2 (w) P 3 (w) 0 1 0 0 0 1 1 k2k2 f(w) OR: f(w) = 0 iff w = 0 w 2 k 2 > n 3 k 3 > n If see 0, say 0. If not, say 1. If see 0, say 0. If not, say 1. Output 0 only if both say 0. Cost of protocol = O(n)

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Tight Bounds for OR [BBR94] Proof: Assume referee says 0. Then w 0 mod 2 k 2, w 0 mod 3 k 3. By the CRT, w 0 mod 2 k 2 3 k 3. But 2 k 2 3 k 3 > n. Hence w = 0. Lower Bound: Above protocol is optimal. Similar bounds for AND.

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Bounds for Threshold functions. Def: Threshold-k functionT k (w) =1 if w ¸ k. What is the degree of T k ? Thm [Bhatnagar-G.-Lipton] : Bound of O((nk) 1/2 ) assuming abc-conjecture. Unconditional for k constant. Uses results on Diophantine equations. For 2 · k · n-1, Degree bounds by symmetric polynomials imply that some Diophantine equations have no solutions.

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Bounds for Threshold-2 Def: Threshold-2 functionT 2 (w) =1 if w ¸ 2. Candidate Protocol: Both players read all but 1 digit. Output 1 if input is at least 2. 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 What is a bad input? ??11 Bad inputs are solutions to |3 k 3 - 2 k 2 | = 1. [BenGerson ~1400] : (9,8) is the only solution. Protocol is correct for large n.

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Previously : Best lower bound: (n 1/2 ). Thm [Bhatnagar-G.-Lipton] : ((nk) 1/2 ) lower bound for Threshold-k. (n) lower bound for Mod-k if k prime, k > min(p,q). Mod-5 has degree (n) over Z 6. Does Mod-2 have degree (n) over Z 15 ? [Hansen] : Yes, but not over Z 21 ! Lower Bounds from Communication Complexity Can asymmetry help compute a symmetric function?

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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Explicit Ramsey Graph Constructions [Erdös] : There exists a graph G on 2 n vertices with (G), (G) · 2n. Proof via the Probabilistic Method. $100 for explicit construction. [Ramsey] : Every graph on 2 n vertices has either an independent set or a clique of size n/2. Easy to construct G on 2 n vertices with (G), (G) · 2 n/2.

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A Brief History of Explicit Constructions [Nagy] : (G), (G) · 2 n/3. [Frankl-Wilson] : Gives (G), (G) · 2 n. [Grolmusz] : Using set system mod 6. Better polynomials ) better graphs. [Alon] : Similar to Frankl-Wilson. [G.] : Unified view of Frankl-Wilson, Grolmusz, Alon. [Barak-Rao-Shaltiel-Wigderson] : (G), (G) · 2 n.

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A Unified View [G.] New view of an OR representation: Two polynomials. Union of their zero sets is {0,1} n \ {0}. Simple construction based on OR polynomials. Unifies Frankl-Wilson, Grolmusz, Alon. All based on O(n) symmetric OR polynomials. Extends to multi-color Ramsey graphs.

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P = 0Q = 0

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A Unified View [G.] New view of an OR representation: Two polynomials. Union of their zero sets is {0,1} n \ {0}. Degree of representation = max(deg(P), deg(Q)). Both polynomials mod p. P mod p, Q mod q. Both polynomials mod p a. (n) O(n) [BBR, Alon] O(n) [FW] All constructions use symmetric polynomials.

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The Ramsey Graph Construction Ramsey Construction: Vertices: {0,1} n. Edges: Add edge (x,y) if P(x © y) = 0. Thm: Degree d OR representation gives (G), (G) · n d. Consider a clique: x 1, …, x k We have: P(x i © x j ) = 0. Claim : Polynomials P(X © x 1 ), …, P(X © x k ) are LI. Dimension of vector space O(n d ). Hence k · n d.Plug in X = x 1 : P(0,…,0) 0, P(x 1 © x 2 ) = 0, …, P(x 1 © x k ) = 0.

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Symmetry is the Barrier P mod p, Q mod q. [BBR, Alon] Gives a representation of OR over Z pq. Known lower bound: (n/pq). When n < pq [Alon] … X i represents OR mod pq. Both polynomials mod p a.[FW] Based on interpolation algorithm mod p a. Theorem [G.] : (n) lower bound for symmetric polynomials.

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High-Level Idea 1.Algebraic Step: Characterize zero-sets of low- degree polynomials. 2.Combinatorial Step: Show that there is no good partition of the hypercube. Symmetry: Multivariate polynomials ! Univariate polynomials {0,1} n ! {0, …, n}.

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Partition Lemma Partition Problem: Adversary gets number n. Picks 1. Primes p and q where p ¢ q > n. 2. A µ {1,…, p-1} and B µ {1, …, q-1} Every x 2 {1, …, n} is covered by A or B. Minimize |A| ¢ |B|. x mod p lies in A Trivial Solutions : A = {1,…, p-1} and B = {p, 2p, …, } A = {q, 2q, …} and B = {1, …, q-1} Gives |A| ¢ |B| = n. Partition Lemma: Any solution gives |A| ¢ |B| ¸ n/4.

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Symmetry is the Barrier Symmetry versus asymmetry question also applies to Ramsey graph constructions. Symmetric polynomials give graphs on {0,1} n based on distances. Q : Are graphs on {0,1} n based on distances not good Ramsey graphs?

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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Polynomial Interpolation mod m Low degree polynomials mod m have many roots. Eg: X 6 = 0 mod 64 (0,2,4 …, 62) How many evaluations are needed to interpolate P(X) over Z m ? Values at various points are dependent over Z m. Eg:Let x, y Z 64 If x y mod 2 then P(x) P(y) mod 2 What is the min. degree of a polynomial which vanishes on Z m ?

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Polynomial Interpolation mod m Problem: Given a set I µ Z m, compute P(X) from its evaluations at points in I. Minimize degree, query complexity. Previous Work: [Bshouty-Tamon-Wilson, Karpinski-van der Poorten-Shparlinski, …] Restrictions on m, degree, coefficients...

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Polynomial Interpolation mod m Problem: Given a set I µ Z m, compute P(X) from its evaluations at points in I. Thm. [G.05] : Interpolation algorithm over Z m : Minimizes degree. Minimizing queries: NP-complete. Algorithm within factor log m of optimal. Algorithm gives m = h 1 h 2 … h t (h i, h j ) = 1 Approximation factor bounded by t. Cor. : PAC-learning, Uniform learning, Zero-testing.

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New Structural Result What is the min. degree of a monic polynomial which vanishes on I µ Z m ? How many queries are needed to interpolate P(X) over I µ Z m ? Thm. [G05]: Algorithm to compute d(I). Computing q(I) is NP-complete. d(I) q(I) d(I)·(#factors of m) d(I) q(I)

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Interpolation over Prime Powers 603 mod 9 01 2 mod 3 09121114 |P(x) – P(y)| |x – y| Let m = 27. I

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Ultrametric: d(x,z) max(d(x,y), d(y,z)) Prime Powers and Ultrametric Spaces

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Ultrametric: d(x,z) max(d(x,y), d(y,z)) Prime Powers and Ultrametric Spaces Algebraic properties of polynomials ) Combinatorial properties of Ultrametric spaces. Find k points that are farthest apart. Greedy algorithm works for ultrametrics. Ultrametrics form a Greedoid.

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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In This Talk: Computing Boolean functions by polynomials. Explicit Ramsey graphs. Interpolation over Z m. Conclusions. Polynomials over Composites.

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Lower bounds for Circuits with Mod gates. Better (simpler?) explicit Ramsey graphs. Future Directions Polynomial representations over Z m. Set systems with restricted intersections mod m. Tractable Open Problems. Main Open Problems.

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Future Directions Do low degree OR polynomials exist? Symmetric polynomials for Symmetric functions. CRT. Hard explicit construction problem ? Algebraic step: Characterize zero-sets of low-degree multivariate polynomials over Z p. Symmetry versus Asymmetry. Better Lower Bounds.

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