Subset Sum \cubes" and the Complexity of Prime Testing

Suppose a 1 < a 2 < : : : < a Z are distinct integers in f1; : : :; N g. If a 0 +" 1 a 1 +" 2 a 2 + + " Z a Z is prime for all choices of " 1 ; " 2 ; : : : ; " Z 2 f0; 1g, then Z (9=2 + o(1)) log N= log log N. The argument uses a modiied form of Gallagher's Larger Sieve, and shows that if Z > (9=2 + o(1)) log N= log log N , then for some prime p < Z 2 , the subset sums " 1 a 1 + " 2 a 2 + + " Z a Z represent every residue class modulo p. Consequently, the smallest 2 3 Boolean circuit which tests primal-ity for any number given by n binary digits has size 2 n?o(n). Let A be a set of distinct positive integers a 1 < a 2 < : : : < a Z. The subset sums of A are the integers " 1 a 1 + " 2 a 2 + + " Z a Z , where all " i 2 f0; 1g. The set of all such subset sums will be denoted by A + so A + = n " 1 a 1 + " 2 a 2 + + " Z a Z : " i 2 f0; 1g o : A cube a 0 + A + = n a 0 + " 1 a 1 + " 2 a 2 + + " Z a Z : " i 2 f0; 1g o is obtained by translating the set A + by a xed positive integer a 0. By using a modiied version of Gallager's Larger Sieve 4], an upper bound will be proved below on the dimension Z of any cube comprised exclusively of prime numbers.