Boolean Functions and their Cryptographic Criteria
varies.
Lemma 3.1. If 𝑢 ∈ 𝔽 𝑛2
0, 𝑢 ≠ 0 𝑛
∑ (−1)<𝑥,𝑢> = {
2𝑛 , 𝑢 = 0 𝑛
𝑥∈𝔽 𝑛2
Proof. Firstly, we can easily see that if 𝑢 = 0 𝑛 , then all summands are 1 and the sum is 2𝑛 . Now, if
𝑢 ≠ 0 𝑛 , we can look at two sets𝐴 = {𝑥 ∈ 𝔽 𝑛2 ∶ < 𝑥, 𝑢 > = 0 } and 𝐴′ = {𝑥 ∈ 𝔽 𝑛2 ∶ < 𝑥, 𝑢 > = 1 }.
Obviously, those sets belong to 𝔽𝑛2 and their cardinalities are the same 2𝑛−1 . Since for any 𝑥 ∈ 𝐴 , the
summand is 1 and for any 𝑥 ∈ 𝐴 , the summand is −1, we get the equation. ∎
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