Digital signature cannot provide ______ for the message.

• (a) Integrity
• (b) Confidentiality
• (c) Non-repudiation
• (d) Authentication

Digital signature uses ______ for generating a valid signature.

• (a) Private key ($K_{priv}$)
• (b) Public key ($K_{pub}$)
• (c) Secret key ($K_{secret}$)
• (d) None of the above

The verification algorithm uses ______ for validating a digital signature.

• (a) Private key ($K_{priv}$)
• (b) Public key ($K_{pub}$)
• (c) Secret key ($K_{secret}$)
• (d) None of the above

Is a digital signature scheme possible without public-key cryptography?

• (a) Yes
• (b) No
• (c) May exist
• (d) None of the above

Explain the importance of hashing (using this experiment) and explain why hashing is needed.

Consider the digital signature process: $$\text{Signature} = \text{Sign}(H(M), K_{priv})$$ $$\text{Verification} = \text{Verify}(S, H(M), K_{pub})$$

Where:

• $M$ = Original message
• $H(M)$ = Hash of the message
• $K_{priv}$ = Private key
• $K_{pub}$ = Public key
• $S$ = Digital signature

Suggest a scheme that does not use any hashing scheme.

Propose an alternative digital signature scheme and discuss:

• Security implications
• Performance considerations
• Practical limitations

Mathematical representation: $\text{Signature} = \text{Sign}(M, K_{priv}) = M^{d} \bmod n$

Hint: consider what happens if an attacker who has already obtained valid signatures $S_1$ and $S_2$ for two messages $M_1$ and $M_2$ computes $S_1 \cdot S_2 \bmod n$ — what message does this forge a signature for, and why?

Explain why digital signature schemes work.

Discuss the mathematical foundations including:

• Asymmetric Key Properties: How $K_{pub}$ and $K_{priv}$ are mathematically related
• One-way Functions: The computational difficulty of deriving $K_{priv}$ from $K_{pub}$
• Hash Function Properties:
  • Collision resistance: it is computationally infeasible to find two distinct inputs $x \neq y$ such that $H(x) = H(y)$
  • Preimage resistance: given $h$, finding $x$ such that $H(x) = h$ is computationally infeasible
• RSA Digital Signature Security Argument (outline): why forging a signature without knowledge of $K_{priv}$ is believed to be as hard as solving the RSA problem, which is closely tied to the difficulty of factoring $n$