这是一份Sydney悉尼大学MATH1004/MATH1904的成功案例
A party $B$ sends a message to a party $A$ in the McEliece encryption scheme as follows:$B$ does the following to encrypt the message:
look up $A$ ‘s public key $(\widehat{G}, t)$;
$\diamond$ represent the message $m$ as a binary string of length $k$; if the message is too big, break it into blocks;
$\diamond$ choose a random error vector $z$ of length $n$ and Hamming weight $\leq t$;
compute $c=m \widehat{G}+z$;
send $c$ to $A$.
A does the following to decrypt the received message:
$\diamond$ compute $\widehat{c}=c P^{-1}$;
$\diamond$ use the decoding algorithm for the code generated by $G$ to decode $\widehat{c}$ to $\widehat{m}$;
$\diamond$ compute $m=\widehat{m} S^{-1}$.
MATH1004/MATH1904 COURSE NOTES ：
Circuit axioms: A subset $\mathcal{C}$ of $2^{E}$ is the set of circuits of a matroid on $E$ if and only if $\mathcal{C}$ satisfies:
- $\emptyset / \in \mathcal{C}$
- no member of $\mathcal{C}$ is a proper subset of another;
- circuit elimination: if $C_{1}, C_{2}$ are distinct members of $\mathcal{C}$ and $e \in C_{1} \cap C_{2}$, then $\mathcal{C}$ has a member $C_{3}$ such that $C_{3} \subseteq\left(C_{1} \cup C_{2}\right)-{e}$.
Note: The circuit elimination axiom can be strengthened to the following: - strong circuit elimination: if $C_{1}, C_{2} \in \mathcal{C}, f \in C_{1}-C_{2}$, and $e \in C_{1} \cap C_{2}$, then $\mathcal{C}$ has a member $C_{3}$ such that $f \in C_{3} \subseteq\left(C_{1} \cup C_{2}\right)-{e}$.