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Notes
- 1.
More precisely, each input is connected to an output with probability d/n and so the expected degree is d.
- 2.
Sample a matrix from ℳ a,b,p conditioned on the event that none of the columns is an all-zero column (a task which can be achieved efficiently by rejecting and resampling zero columns), and fail it if the resulting matrix has a row of weight larger than logn. To see that rejection happens with negligible probability, observe that for a fixed row, each column contributes 1 with probability p/(1−p)a=Θ(1/n), independently of the other columns, and therefore, by a Chernoff bound, the weight exceeds logn with negligible probability.
- 3.
Another line of work studied the pseudorandomness of the collection. See Sect. 7.6.
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Applebaum, B. (2014). One-Way Functions with Optimal Output Locality. In: Cryptography in Constant Parallel Time. Information Security and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17367-7_6
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