dp泄露问题
快速幂
\[\\5 * e \equiv 1 \pmod 7
\\e=5^{7-1-1}快速幂
\\ a \equiv a^{p} \pmod {p} ,assert \ p \ is \ prime 费马小定理
\\ a *a^{p-2} \equiv 1 \pmod {p}
\]
dp泄露
\[
已知e,dp,c,n
\\e*d \equiv 1 \pmod {phi}
\\dp = d \mod {p-1}
\\e*d - 1 = k_1*(q-1)*(p-1)
\\d = k_2*(p-1) + dp
\\e*dp + e *k_2 *(p-1) = k_1*(q-1)*(p-1) + 1
\\e*dp \equiv 1 \mod {p-1}
\\e *dp -1 = k *(p-1)
\\a^{(e *dp -1)} = a^{k *(p-1)}
\\a^{(e *dp -1)} \equiv a^{k *(p-1)} \pmod {p}
\\a^{(e *dp -1)} \pmod {p}= {a^{p-1}}^k \pmod {p}=1^k =1
\\a^{(e *dp -1)} \pmod {p} = 1
\\a^{e *dp -1} = 1 + k*p
\\a^{e *dp} - a = a*k*p
\\ n = p * q
\\gcd(n,(a^{e *dp}-a\mod n))=p
\]
爆破
\[\\e *dp -1 = k *(p-1)
\\e *dp \approx k *(p-1)
\\大*小 =小* 大
\\k \subset (1,e)
\]
