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Razborov (1996) recently proved that polynomial calculus proofs of the pigeonhole principle P HP m n must have degree at least n/2 + 1 over any field. We present a simplified proof of the same result.Furthermore, we show a matching upper bound on polynomial calculus proofs of the pigeonhole principle for any field of sufficiently large characteristic, and an n/2 +1 lower bound for any subset sum problem over the field of reals.We show that these degree lower bounds also translate into lower bounds on the number of monomials in any polynomial calculus proof, and hence on the running time of most implementations of the Gröbner basis algorithm.

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Abstract. The notion of traitor tracing was introduced by Chor, Fiat, and Naor [Tracing Traitors, Lecture Notes in Comput. Sci. 839, 1994, pp. 257-270] in order to combat piracy scenarios. Recently, Fiat and Tassa [Tracing Traitors, Lecture Notes in Comput. Sci. 1666, 1999 proposed a dynamic traitor tracing scenario, in which the algorithm adapts dynamically according to the responses of the pirate. Let n be the number of users and p the number of traitors.Our main result is an algorithm which locates p traitors, even if p is unknown, using a watermarking alphabet of size p + 1 and an optimal number of Θ(p 2 + p log n) rounds. This improves the exponential number of rounds achieved by Fiat and Tassa in this case. We also present two algorithms that use a larger alphabet: for an alphabet of size p + c + 1, c ≥ 1, an algorithm that uses O(p 2 /c + p log n) rounds; for an alphabet of size pc + 1, an algorithm that uses O(p log c n) rounds.Our final result is a lower bound of Ω(p 2 /c + p log c+1 n) rounds for any algorithm that uses an alphabet of size p + c, assuming that p is not known in advance.