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A Complexity Theory for VLSI
 TECHNICAL REPORT
, 1980
"... The established methodologies for studying computational complexity can be applied to the new problems posed by very largescale integrated (VLSI) circuits. This thesis develops a “VLSI model of computation” and derives upper and lower bounds on the silicon area and time required to solve the proble ..."
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Cited by 104 (1 self)
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The established methodologies for studying computational complexity can be applied to the new problems posed by very largescale integrated (VLSI) circuits. This thesis develops a “VLSI model of computation” and derives upper and lower bounds on the silicon area and time required to solve the problems of sorting and discrete Fourier transformation. In particular, the area A and time T taken by any VLSI chip using any algorithm to perform an $N$point Fourier transform must satisfy $AT^2 \geq c N^2 \log^2 N$, for some fixed $c > 0$. A more general result for both sorting and Fourier transformation is that $AT^{2x} = \Omega(N^{1+x} \log^{2x} N)$ for any $x$ in the range $0 < x < 1$. Also, the energy dissipated by a VLSI chip during the solution of either of these problems is at least $\Omega(N^{3/2} \log N)$. The tightness of these bounds is demonstrated by the existence of nearly optimal circuits for both sorting and Fourier transformation. The circuits based on the shuffleexchange interconnection pattern are fast but large: $T = O(\log^2 N)$ for Fourier transformation, $T = O(\log^3 N)$ for sorting; both have area $A$ of at most $O(N^2 / \log{1/2} N)$. The circuits based on the mesh interconnection pattern are slow but small: $T = O(N^{1/2} \log\log N)$, $A = O(N \log^2 N)$.
Simple PCPs with Polylog Rate and Query Complexity
, 2005
"... We give constructions of probabilistically checkable proofs (PCPs) of length n·poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters. Previous constr ..."
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Cited by 45 (15 self)
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We give constructions of probabilistically checkable proofs (PCPs) of length n·poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters. Previous constructions of short PCPs (from [5] to [9]) relied extensively on properties of low degree multivariate polynomials. In contrast, our constructions rely on new problems and techniques revolving around the properties of codes based on high degree polynomials in one variable (also known as ReedSolomon codes). We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a ReedSolomon codeword, i.e., a univariate polynomial of specified degree. This reduction is simpler than the corresponding steps in previous reductions, and gives a new alternative to using the popular “sumcheck protocol”. We then give a new PCP for the special task of proving that a function is close to being a ReedSolomon codeword. This step of the construction is by a selfcontained recursion, and the only ingredient needed in the analysis is the bivariate lowdegree test of Polischuk and Spielman [27]. Note that our constructions yield LTCs first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite (and less natural) direction of getting LTCs from PCPs.
SHORT PCPS WITH POLYLOG QUERY COMPLEXITY
 SIAM J. COMPUT. VOL. 38, NO. 2, PP. 551–607
, 2008
"... We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can be verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n ..."
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Cited by 19 (5 self)
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We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can be verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n · polylog n bit long codewords that are testable with polylog n queries. Our constructions rely on new techniques revolving around properties of codes based on relatively highdegree polynomials in one variable, i.e., Reed–Solomon codes. In contrast, previous constructions of
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 16 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 14 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Group automorphisms with few and with many periodic points
 Proc. Amer. Math. Soc
, 2005
"... Abstract. For any C ∈ [0, ∞] a compact group automorphism T: X → X is constructed with the property that 1 n log {x ∈ X  T n (x) = x}  − → C. This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological en ..."
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Cited by 6 (5 self)
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Abstract. For any C ∈ [0, ∞] a compact group automorphism T: X → X is constructed with the property that 1 n log {x ∈ X  T n (x) = x}  − → C. This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological entropy. 1.
Graphs of Prescribed Girth and BiDegree
"... We say that a bipartite graph Γ(V1 ∪ V2, E) has bidegree r, s if every vertex from V1 has degree r and every vertex from V2 has degree s. Γ is called an (r, s, t)–graph if, additionally, the girth of Γ is 2t. For t> 3, very few examples of (r, s, t)–graphs were previously known. In this paper we gi ..."
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Cited by 5 (2 self)
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We say that a bipartite graph Γ(V1 ∪ V2, E) has bidegree r, s if every vertex from V1 has degree r and every vertex from V2 has degree s. Γ is called an (r, s, t)–graph if, additionally, the girth of Γ is 2t. For t> 3, very few examples of (r, s, t)–graphs were previously known. In this paper we give a recursive construction of (r, s, t)–graphs for all r, s, t ≥ 2, as well as an algebraic construction of such graphs for all r, s ≥ t ≥ 3.
COUNTING CONGRUENCE SUBGROUPS
"... Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence ..."
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Cited by 4 (2 self)
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Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence
Fast Integer Multiplication Using Modular Arithmetic
 In Fortieth Annual ACM Symposium on Theory of Computing
, 2008
"... We give an O(N ·log N ·2 O(log ∗ N)) algorithm for multiplying two Nbit integers that improves the O(N · log N · log log N) algorithm by SchönhageStrassen [SS71]. Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 2 O(log ∗ N)) algorithm which however uses ..."
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Cited by 4 (0 self)
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We give an O(N ·log N ·2 O(log ∗ N)) algorithm for multiplying two Nbit integers that improves the O(N · log N · log log N) algorithm by SchönhageStrassen [SS71]. Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 2 O(log ∗ N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürer’s algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a padic version of Fürer’s algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar. 1