updates

content/CSE442T/CSE442T_L20.md

@@ -26,7 +26,8 @@ Under the discrete log assumption, $H$ is a CRHF.
 - It is easy to compute
 - Compressing by 1 bit
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 The hash function $h$ is a CRHF
 
@@ -72,7 +73,7 @@ So $\mathcal{B}$ can break the discrete log assumption with non-negligible proba
 
 So $h$ is a CRHF.
 
-QED
+</details>
 
 To compress by more, say $h_k:{0,1}^n\to \{0,1\}^{n-k},k\geq 1$, then we can use $h: \{0,1\}^{n+1}\to \{0,1\}^n$ multiple times.
 
@@ -106,7 +107,8 @@ One-time secure:
 
 Then ($Gen',Sign',Ver'$) is one-time secure.
 
-Ideas of Proof:
+<details>
+<summary>Ideas of Proof</summary>
 
 If the digital signature scheme ($Gen',Sign',Ver'$) is not one-time secure, then there exists an adversary $\mathcal{A}$ which can ask oracle for one signature on $m_1$ and receive $\sigma_1=Sign'_{sk'}(m_1)=Sign_{sk}(h_i(m_1))$.
 
@@ -119,7 +121,7 @@ Case 1: $h_i(m_1)=h_i(m_2)$, Then $\mathcal{A}$ finds a collision of $h$.
 
 Case 2: $h_i(m_1)\neq h_i(m_2)$, Then $\mathcal{A}$ produced valid signature on $h_i(m_2)$ after only seeing $Sign'_{sk'}(m_1)\neq Sign'_{sk'}(m_2)$. This contradicts the one-time secure of ($Gen,Sign,Ver$).
 
-QED
+</details>
 
 ### Many-time Secure Digital Signature