GF(2n)-Linear Tests versus GF(2)-Linear Tests

A small-biased distribution of bit sequences is defined as one withstanding GF (2)-linear tests for randomness, which are linear combinations of the bits themselves. We consider linear combinations over larger fields, specifically, GF (2) for n that divides the length of the bit sequence. Indeed, this means that we partition the bits to blocks of length n and treat each block as the representation of a field element. Various properties of the resulting field element can then be tested. We show that the latter GF (2)-linear tests are at least as powerful as the GF (2)-linear tests. This holds even for a very limited final test of the resulting field element (e.g., checking only the first bit). This is shown constructively in the sense that we show for each linear combination over GF (2), an explicit linear combination over GF (2) whose first bit (for instance) has the same bias. One corollary of the above is that the generator producing a random geometric series over GF (2), namely (a, b) 7→ (a · b)`i=0, is ` 2n -biased. Given the technical nature of the current work, we start with the formal setting (Section 1), to be followed by a discussion (Section 2). The proof of the main result appears in section 3. 1 Formal Setting We start with the notion of ε-bias, introduced in [7], which refers to GF (2)-linear tests: Definition 1 (ε-bias). For ε > 0, k, ` ∈ N, a generator G : {0, 1} → {0, 1} is called ε-biased if for every nontrivial GF (2)-linear combination α ∈ {0, 1}, Pr s∈{0,1} [〈G(s), α〉 = 0] = 1 2 ± ε. This research was partially supported by the Israel Science Foundation (grant No. 1041/08). Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. Email: [email protected]. 1 Electronic Colloquium on Computational Complexity, Report No. 18 (2009)