3.The_Knowledge_of_Coefficient_Test_and_Assumption.md

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The Knowledge of Coefficient Test and Assumption

<<part II

the fact that Alice is able to compute E(P(s)) does not guarantee she will indeed send E(P(s)) to Bob, rather than some completely unrelated value.

📖 事实是Alice可以计算出E(P(s))但不能保证Alice一定会发送E(P(s))给Bob, 而不是发送一下完全不相关的值.

For α∈F∗p [1], let us call a pair of elements (a,b) in G an α-pair if a,b≠0 and b=α⋅a.

The KC Test proceeds as follows.

1. Bob chooses random α∈F∗p and a∈G. He computes b=α⋅a.
2. He sends to Alice the "challenge" pair (a,b). Note that (a,b) is an α-pair.
3. Alice must now respond with a different pair (a′,b′) that is also an α-pair.
4. Bob accepts Alice’s response only if (a′,b′) is indeed an α-pair. (As he knows α he can check if b′=α⋅a′.)

📖 对于α∈F∗p, 我们称在G域中的元素(a,b)在满足a,b≠0 and b=α⋅a时为α-pair

系数测试知识流程如下所示:

1. Bob从域中选择随机数α∈F∗p以及a∈G. 他计算了b=α⋅a.
2. 他发送Alice挑战对(a,b). 注意到 (a,b)是一个α-pair.
3. Alice必须使用不同的α-pair (a′,b′)来进行回应.
4. Bob接受了Alice的回应如果(a′,b′)确实也是α-pair(我们知道α可以用于检查, 如果b′=α⋅a′)

Appendix

[1] F∗p denotes the non-zero elements of Fp. It is the same as Z∗p described in Part I.

📖 ​F∗p是Fp的无零域, 它与part I中提到的Z*p是一样的.