update notes
This commit is contained in:
Zheyuan Wu
@@ -2,6 +2,18 @@
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## Chapter 5: Authentication
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### One-Time Secure Digital Signature
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#### Definition 136.2 (Security of Digital Signature)
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A digital signature scheme is $(Gen, Sign, Ver)$ is secure if for all n.u.p.p.t. $\mathcal{A}$, there exists a negligible function $\epsilon(n)$ such that $\forall n\in\mathbb{N}$,
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$$
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P[(pk,sk)\gets Gen(1^n); (m,\sigma)\gets\mathcal{A}^{Sign_{sk}(\cdot)}(1^n); \mathcal{A}\textup{ did not query }m\textup{ and } Ver_{pk}(m,\sigma)=\textup{``Accept''}]\leq \frac{1}{p(n)}+\epsilon(n)
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$$
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A digital signature scheme is one-time secure if it is secure and the adversary makes only one query to the signing oracle.
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### Lamport's One-Time Signature
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Given a one-way function $f$, we can create a signature scheme as follows:
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@@ -82,7 +94,7 @@ $B$ inverts $f$ with prob $\geq \frac{1}{p(n)}$
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We now have one-time secure signature scheme.
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We want one-time secure signature scheme that increase the size of messages relative tothe keys.
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We want one-time secure signature scheme that increase the size of messages relative to the keys.
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Let $H:\{h_i:D_i\to R_i\}_{i\in I}$ be a family of CRHF if
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@@ -41,212 +41,3 @@ $$
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$$
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If $G_0$ and $G_1$ are indistinguishable, $H_s=\Pi(G_{b'})$ same distribution as $H_p=\Pi(G_0)$. (random permutation of $G_1$ is a random permutation of $G_0$)
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## Review
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### Assumptions used in cryptography (this course)
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#### Diffie-Hellman assumption
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The Diffie-Hellman assumption is that the following problem is hard.
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$$
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\text{Given } g,g^a,g^b\text{, it is hard to compute } g^{ab}.
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$$
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More formally,
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If $p$ is a randomly sampled safe prime.
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Denote safe prime as $\tilde{\Pi}_n=\{p\in \Pi_n:q=\frac{p-1}{2}\in \Pi_{n-1}\}$
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Then
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$$
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P\left[p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1;x\gets \mathbb{Z}_q;y=g^x\mod p:\mathcal{A}(y)=x\right]\leq \varepsilon(n)
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$$
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$p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1$ is the function condition when we do the encryption on cyclic groups.
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#### Discrete logarithm assumption
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> If Diffie-Hellman assumption holds, then discrete logarithm assumption holds.
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This is a corollary of the Diffie-Hellman assumption, it states as follows.
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This is collection of one-way functions
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$$
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p\gets \tilde\Pi_n(\textup{ safe primes }), p=2q+1
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$$
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$$
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a\gets \mathbb{Z}*_{p};g=a^2(\textup{ make sure }g\neq 1)
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$$
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$$
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f_{g,p}(x)=g^x\mod p
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$$
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$$
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f:\mathbb{Z}_q\to \mathbb{Z}^*_p
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$$
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#### RSA assumption
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The RSA assumption is that it is hard to factorize a product of two large primes. (no polynomial time algorithm for factorization product of two large primes with $n$ bits)
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Let $e$ be the exponents
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$$
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P[p,q\gets \Pi_n;N\gets p\cdot q;e\gets \mathbb{Z}_{\phi(N)}^*;y\gets \mathbb{N}_n;x\gets \mathcal{A}(N,e,y);x^e=y\mod N]<\varepsilon(n)
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$$
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#### Factoring assumption
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> If RSA assumption holds, then factoring assumption holds.
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The only way to efficiently factorize the product of prime is to iterate all the primes.
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### Fancy product of these assumptions
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#### Trapdoor permutation
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> RSA assumption $\implies$ Trapdoor permutation exists.
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Idea: $f:D\to R$ is a one-way permutation.
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$y\gets R$.
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* Finding $x$ such that $f(x)=y$ is hard.
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* With some secret info about $f$, finding $x$ is easy.
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$\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$
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1. $\forall i,f_i$ is a permutation
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2. $(i,t)\gets Gen(1^n)$ efficient. ($i\in I$ paired with $t$), $t$ is the "trapdoor info"
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3. $\forall i,D_i$ can be sampled efficiently.
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4. $\forall i,\forall x,f_i(x)$ can be computed in polynomial time.
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5. $P[(i,t)\gets Gen(1^n);y\gets R_i:f_i(\mathcal{A}(1^n,i,y))=y]<\varepsilon(n)$ (note: $\mathcal{A}$ is not given $t$)
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6. (trapdoor) There is a p.p.t. $B$ such that given $i,y,t$, B always finds x such that $f_i(x)=y$. $t$ is the "trapdoor info"
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_There is one bit of trapdoor info that without it, finding $x$ is hard._
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#### Collision resistance hash function
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> If discrete logarithm assumption holds, then collision resistance hash function exists.
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Let $h: \{0, 1\}^{n+1} \to \{0, 1\}^n$ be a CRHF.
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Base on the discrete log assumption, we can construct a CRHF $H: \{0, 1\}^{n+1} \to \{0, 1\}^n$ as follows:
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$Gen(1^n):(g,p,y)$
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$p\in \tilde{\Pi}_n(p=2q+1)$
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$g$ generator for group of sequence $\mod p$ (G_q)
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$y$ is a random element in $G_q$
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$h_{g,p,y}(x,b)=y^bg^x\mod p$, $y^bg^x\mod p \in \{0,1\}^n$
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$g^x\mod p$ if $b=0$, $y\cdot g^x\mod p$ if $b=1$.
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Under the discrete log assumption, $H$ is a CRHF.
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- It is easy to sample $(g,p,y)$
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- It is easy to compute
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- Compressing by 1 bit
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#### One-way permutation
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> If trapdoor permutation exists, then one-way permutation exists.
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A one-way permutation is a function that is one-way and returns a permutation of the input.
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#### One-way function
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> If one-way permutation exists, then one-way function exists.
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One-way function is a class of functions that are easy to compute but hard to invert.
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##### Weak one-way function
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A weak one-way function is
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$$
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f:\{0,1\}^n\to \{0,1\}^*
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$$
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1. $\exists$ a P.P.T. that computes $f(x),\forall x\in\{0,1\}^n$
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2. $\forall a$ adversaries, $\exists \varepsilon(n),\forall n$.
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$$
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P[x\gets \{0,1\}^n;y=f(x):f(a(y,1^n))=y]<1-\frac{1}{p(n)}
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$$
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_The probability of success should not be too close to 1_
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##### Strong one-way function
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> If weak one-way function exists, then strong one-way function exists.
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A strong one-way function is
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$$
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f:\{0,1\}^n\to \{0,1\}^*(n\to \infty)
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$$
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There is a negligible function $\varepsilon (n)$ such that for any adversary $a$ (n.u.p.p.t)
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$$
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P[x\gets\{0,1\}^n;y=f(x):f(a(y))=y,a(y)=x']\leq\varepsilon(n)
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$$
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_Probability of guessing correct message is negligible_
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#### Hard-core bits
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> Strong one-way function $\iff$ hard-core bits exists.
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A hard-core bit is a bit that is hard to predict given the output of a one-way function.
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#### Pseudorandom generator
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> If one-way permutation exists, then pseudorandom generator exists.
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We can also use pseudorandom generator to construct one-way function.
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And hard-core bits can be used to construct pseudorandom generator.
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#### Pseudorandom function
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> If pseudorandom generator exists, then pseudorandom function exists.
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A pseudorandom function is a function that is indistinguishable from a true random function.
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### Multi-message secure private-key encryption
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> If pseudorandom function exists, then multi-message secure private-key encryption exists.
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A multi-message secure private-key encryption is a function that is secure against an adversary who can see multiple messages.
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#### Single message secure private-key encryption
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> If multi-message secure private-key encryption exists, then single message secure private-key encryption exists.
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#### Message-authentication code
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> If pseudorandom function exists, then message-authentication code exists.
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### Public-key encryption
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> If Diffie-Hellman assumption holds, and Trapdoor permutation exists, then public-key encryption exists.
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### Digital signature
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#### One-time secure digital signature
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#### Fixed-length one-time secure digital signature
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> If one-way function exists, then fixed-length one-time secure digital signature exists.
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222
pages/CSE442T/Exam_reviews/CSE441T_E2.md
Normal file
222
pages/CSE442T/Exam_reviews/CSE441T_E2.md
Normal file
@@ -0,0 +1,222 @@
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# CSE442T Exam 2 Review
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## Review
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||||
|
||||
### Assumptions used in cryptography (this course)
|
||||
|
||||
#### Diffie-Hellman assumption
|
||||
|
||||
The Diffie-Hellman assumption is that the following problem is hard.
|
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|
||||
$$
|
||||
\text{Given } g,g^a,g^b\text{, it is hard to compute } g^{ab}.
|
||||
$$
|
||||
|
||||
More formally,
|
||||
|
||||
If $p$ is a randomly sampled safe prime.
|
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|
||||
Denote safe prime as $\tilde{\Pi}_n=\{p\in \Pi_n:q=\frac{p-1}{2}\in \Pi_{n-1}\}$
|
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|
||||
Then
|
||||
|
||||
$$
|
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P\left[p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1;x\gets \mathbb{Z}_q;y=g^x\mod p:\mathcal{A}(y)=x\right]\leq \varepsilon(n)
|
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$$
|
||||
|
||||
$p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1$ is the function condition when we do the encryption on cyclic groups.
|
||||
|
||||
#### Discrete logarithm assumption
|
||||
|
||||
> If Diffie-Hellman assumption holds, then discrete logarithm assumption holds.
|
||||
|
||||
This is a corollary of the Diffie-Hellman assumption, it states as follows.
|
||||
|
||||
This is collection of one-way functions
|
||||
|
||||
$$
|
||||
p\gets \tilde\Pi_n(\textup{ safe primes }), p=2q+1
|
||||
$$
|
||||
$$
|
||||
a\gets \mathbb{Z}*_{p};g=a^2(\textup{ make sure }g\neq 1)
|
||||
$$
|
||||
$$
|
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f_{g,p}(x)=g^x\mod p
|
||||
$$
|
||||
$$
|
||||
f:\mathbb{Z}_q\to \mathbb{Z}^*_p
|
||||
$$
|
||||
|
||||
#### RSA assumption
|
||||
|
||||
The RSA assumption is that it is hard to factorize a product of two large primes. (no polynomial time algorithm for factorization product of two large primes with $n$ bits)
|
||||
|
||||
Let $e$ be the exponents
|
||||
|
||||
$$
|
||||
P[p,q\gets \Pi_n;N\gets p\cdot q;e\gets \mathbb{Z}_{\phi(N)}^*;y\gets \mathbb{N}_n;x\gets \mathcal{A}(N,e,y);x^e=y\mod N]<\varepsilon(n)
|
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$$
|
||||
|
||||
#### Factoring assumption
|
||||
|
||||
> If RSA assumption holds, then factoring assumption holds.
|
||||
|
||||
The only way to efficiently factorize the product of prime is to iterate all the primes.
|
||||
|
||||
### Fancy product of these assumptions
|
||||
|
||||
#### Trapdoor permutation
|
||||
|
||||
> RSA assumption $\implies$ Trapdoor permutation exists.
|
||||
|
||||
Idea: $f:D\to R$ is a one-way permutation.
|
||||
|
||||
$y\gets R$.
|
||||
|
||||
* Finding $x$ such that $f(x)=y$ is hard.
|
||||
* With some secret info about $f$, finding $x$ is easy.
|
||||
|
||||
$\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$
|
||||
|
||||
1. $\forall i,f_i$ is a permutation
|
||||
2. $(i,t)\gets Gen(1^n)$ efficient. ($i\in I$ paired with $t$), $t$ is the "trapdoor info"
|
||||
3. $\forall i,D_i$ can be sampled efficiently.
|
||||
4. $\forall i,\forall x,f_i(x)$ can be computed in polynomial time.
|
||||
5. $P[(i,t)\gets Gen(1^n);y\gets R_i:f_i(\mathcal{A}(1^n,i,y))=y]<\varepsilon(n)$ (note: $\mathcal{A}$ is not given $t$)
|
||||
6. (trapdoor) There is a p.p.t. $B$ such that given $i,y,t$, B always finds x such that $f_i(x)=y$. $t$ is the "trapdoor info"
|
||||
|
||||
_There is one bit of trapdoor info that without it, finding $x$ is hard._
|
||||
|
||||
#### Collision resistance hash function
|
||||
|
||||
> If discrete logarithm assumption holds, then collision resistance hash function exists.
|
||||
|
||||
Let $h: \{0, 1\}^{n+1} \to \{0, 1\}^n$ be a CRHF.
|
||||
|
||||
Base on the discrete log assumption, we can construct a CRHF $H: \{0, 1\}^{n+1} \to \{0, 1\}^n$ as follows:
|
||||
|
||||
$Gen(1^n):(g,p,y)$
|
||||
|
||||
$p\in \tilde{\Pi}_n(p=2q+1)$
|
||||
|
||||
$g$ generator for group of sequence $\mod p$ (G_q)
|
||||
|
||||
$y$ is a random element in $G_q$
|
||||
|
||||
$h_{g,p,y}(x,b)=y^bg^x\mod p$, $y^bg^x\mod p \in \{0,1\}^n$
|
||||
|
||||
$g^x\mod p$ if $b=0$, $y\cdot g^x\mod p$ if $b=1$.
|
||||
|
||||
Under the discrete log assumption, $H$ is a CRHF.
|
||||
|
||||
- It is easy to sample $(g,p,y)$
|
||||
- It is easy to compute
|
||||
- Compressing by 1 bit
|
||||
|
||||
#### One-way permutation
|
||||
|
||||
> If trapdoor permutation exists, then one-way permutation exists.
|
||||
|
||||
A one-way permutation is a function that is one-way and returns a permutation of the input.
|
||||
|
||||
#### One-way function
|
||||
|
||||
> If one-way permutation exists, then one-way function exists.
|
||||
|
||||
One-way function is a class of functions that are easy to compute but hard to invert.
|
||||
|
||||
##### Weak one-way function
|
||||
|
||||
A weak one-way function is
|
||||
|
||||
$$
|
||||
f:\{0,1\}^n\to \{0,1\}^*
|
||||
$$
|
||||
|
||||
1. $\exists$ a P.P.T. that computes $f(x),\forall x\in\{0,1\}^n$
|
||||
2. $\forall a$ adversaries, $\exists \varepsilon(n),\forall n$.
|
||||
|
||||
$$
|
||||
P[x\gets \{0,1\}^n;y=f(x):f(a(y,1^n))=y]<1-\frac{1}{p(n)}
|
||||
$$
|
||||
_The probability of success should not be too close to 1_
|
||||
|
||||
|
||||
##### Strong one-way function
|
||||
|
||||
> If weak one-way function exists, then strong one-way function exists.
|
||||
|
||||
A strong one-way function is
|
||||
|
||||
$$
|
||||
f:\{0,1\}^n\to \{0,1\}^*(n\to \infty)
|
||||
$$
|
||||
|
||||
There is a negligible function $\varepsilon (n)$ such that for any adversary $a$ (n.u.p.p.t)
|
||||
|
||||
$$
|
||||
P[x\gets\{0,1\}^n;y=f(x):f(a(y))=y,a(y)=x']\leq\varepsilon(n)
|
||||
$$
|
||||
|
||||
_Probability of guessing correct message is negligible_
|
||||
|
||||
#### Hard-core bits
|
||||
|
||||
> Strong one-way function $\iff$ hard-core bits exists.
|
||||
|
||||
A hard-core bit is a bit that is hard to predict given the output of a one-way function.
|
||||
|
||||
#### Pseudorandom generator
|
||||
|
||||
> If one-way permutation exists, then pseudorandom generator exists.
|
||||
|
||||
We can also use pseudorandom generator to construct one-way function.
|
||||
|
||||
And hard-core bits can be used to construct pseudorandom generator.
|
||||
|
||||
#### Pseudorandom function
|
||||
|
||||
> If pseudorandom generator exists, then pseudorandom function exists.
|
||||
|
||||
A pseudorandom function is a function that is indistinguishable from a true random function.
|
||||
|
||||
### Multi-message secure private-key encryption
|
||||
|
||||
> If pseudorandom function exists, then multi-message secure private-key encryption exists.
|
||||
|
||||
A multi-message secure private-key encryption is a function that is secure against an adversary who can see multiple messages.
|
||||
|
||||
#### Single message secure private-key encryption
|
||||
|
||||
> If multi-message secure private-key encryption exists, then single message secure private-key encryption exists.
|
||||
|
||||
#### Message-authentication code
|
||||
|
||||
> If pseudorandom function exists, then message-authentication code exists.
|
||||
|
||||
|
||||
### Public-key encryption
|
||||
|
||||
> If Diffie-Hellman assumption holds, and Trapdoor permutation exists, then public-key encryption exists.
|
||||
|
||||
### Digital signature
|
||||
|
||||
A digital signature scheme is a triple $(Gen, Sign, Ver)$ where
|
||||
|
||||
- $(pk,sk)\gets Gen(1^k)$ is a p.p.t. algorithm that takes as input a security parameter $k$ and outputs a public key $pk$ and a secret key $sk$.
|
||||
- $\sigma\gets Sign_{sk}(m)$ is a p.p.t. algorithm that takes as input a secret key $sk$ and a message $m$ and outputs a signature $\sigma$.
|
||||
- $Ver_{pk}(m, \sigma)$ is a deterministic algorithm that takes as input a public key $pk$, a message $m$, and a signature $\sigma$ and outputs "Accept" if $\sigma$ is a valid signature for $m$ under $pk$ and "Reject" otherwise.
|
||||
|
||||
For all $n\in\mathbb{N}$, all $m\in\mathcal{M}_n$.
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||||
|
||||
$$
|
||||
P[(pk,sk)\gets Gen(1^k); \sigma\gets Sign_{sk}(m); Ver_{pk}(m, \sigma)=\textup{``Accept''}]=1
|
||||
$$
|
||||
|
||||
#### One-time secure digital signature
|
||||
|
||||
#### Fixed-length one-time secure digital signature
|
||||
|
||||
> If one-way function exists, then fixed-length one-time secure digital signature exists.
|
||||
|
||||
|
||||
4
pages/CSE442T/Exam_reviews/_meta.js
Normal file
4
pages/CSE442T/Exam_reviews/_meta.js
Normal file
@@ -0,0 +1,4 @@
|
||||
export default {
|
||||
CSE442T_E1: "CSE442T Exam 1 Review",
|
||||
CSE442T_E2: "CSE442T Exam 2 Review"
|
||||
}
|
||||
@@ -27,7 +27,5 @@ export default {
|
||||
CSE442T_L21: "Lecture 21",
|
||||
CSE442T_L22: "Lecture 22",
|
||||
CSE442T_L23: "Lecture 23",
|
||||
CSE442T_L24: {
|
||||
display: 'hidden'
|
||||
}
|
||||
CSE442T_L24: "Lecture 24"
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user