upgrade structures and migrate to nextra v4

content/CSE442T/CSE442T_L3.md

@@ -0,0 +1,115 @@
+# Lecture 3
+
+All algorithms $C(x)\to y$, $x,y\in \{0,1\}^*$ 
+
+P.P.T= Probabilistic Polynomial-time Turing Machine.
+
+## Chapter 2: Computational Hardness
+
+### Turing Machine: Mathematical model for a computer program
+
+A machine that can:
+
+1. Read in put
+2. Read/Write working tape move left/right
+3. Can change state
+
+### Assumptions
+
+Anything can be accomplished by a real computer program can be accomplished by a "sufficiently complicated" Turing Machine (TM).
+
+### Polynomial time
+
+We say $C(x),|x|=n,n\to \infty$ runs in polynomial time if it uses at most $T(n)$ operations bounded by some polynomials. $\exist c>0$ such that $T(n)=O(n^c)$
+
+If we can argue that algorithm runs in polynomially-many constant-time operations, then this is true for the T.M.
+
+$p,q$ are polynomials in $n$,
+
+$p(n)+q(n),p(n)q(n),p(q(n))$ are polynomial of $n$.
+
+Polynomial-time $\approx$ "efficient" for this course.
+
+### Probabilistic
+
+Our algorithm's have access to random "coin-flips" we can produce poly(n) random bits.
+
+$P[C(x)\text{ takes at most }T(n)\text{ steps }]=1$
+
+Our adversary $a(x)$ will be a P.P.T which is non-uniform (n.u.) (programs description size can grow polynomially in n)
+
+### Efficient private key encryption scheme
+
+#### Definition 3.2 (Efficient private key encryption scheme)
+
+The triple $(Gen,Enc,Dec)$ is an efficient private key encryption scheme over the message space $M$  and key space $K$ if:
+
+1. $Gen(1^n)$ is a randomized p.p.t that outputs $k\in K$
+2. $Enc_k(m)$ is a potentially randomized p.p.t that outputs $c$ given $m\in M$
+3. $Dec_k(c')$ is a deterministic p.p.t that outputs $m$ or "null"
+4. $P_k[Dec_k(Enc_k(m))=m]=1,\forall m\in M$
+
+### Negligible function
+
+$\epsilon:\mathbb{N}\to \mathbb{R}$ is a negligible function if $\forall c>0$, $\exists N\in\mathbb{N}$ such that $\forall n\geq N, \epsilon(n)<\frac{1}{n^c}$ (looks like definition of limits huh) (Definition 27.2)
+
+Idea: for any polynomial, even $n^{100}$, in the long run $\epsilon(n)\leq \frac{1}{n^{100}}$
+
+Example: $\epsilon (n)=\frac{1}{2^n}$, $\epsilon (n)=\frac{1}{n^{\log (n)}}$
+
+Non-example: $\epsilon (n)=O(\frac{1}{n^c})\forall c$
+
+### One-way function
+
+Idea: We are always okay with our chance of failure being negligible.
+
+Foundational concept of cryptography
+
+Goal: making $Enc_k(m),Dec_k(c')$ easy and $Dec^{-1}(c')$ hard.
+
+#### Definition 27.3 (Strong one-way function)
+
+$$
+f:\{0,1\}^n\to \{0,1\}^*(n\to \infty)
+$$
+
+There is a negligible function $\epsilon (n)$ such that for any adversary $\mathcal{A}$ (n.u.p.p.t)
+
+$$
+P[x\gets\{0,1\}^n;y=f(x):f(\mathcal{A}(y))=y]\leq\epsilon(n)
+$$
+
+_Probability of guessing a message $x'$ with the same output as the correct message $x$ is negligible_
+
+and
+
+there is a p.p.t which computes $f(x)$ for any $x$.
+
+- Hard to go back from output
+- Easy to find output
+
+$a$ sees output y, they wan to find some $x'$ such that $f(x')=y$.
+
+Example: Suppose $f$ is one-to-one, then $a$ must find our $x$, $P[x'=x]=\frac{1}{2^n}$, which is negligible.
+
+Why do we allow $a$ to get a different $x'$?
+
+> Suppose the definition is $P[x\gets\{0,1\}^n;y=f(x):\mathcal{A}(y)=x]\neq\epsilon(n)$, then a trivial function $f(x)=x$ would also satisfy the definition.
+
+To be technically fair, $\mathcal{A}(y)=\mathcal{A}(y,1^n)$, size of input $\approx n$, let them use $poly(n)$ operations. (we also tells the input size is $n$ to $\mathcal{A}$)
+
+#### Do one-way function exists?
+
+Unknown, actually...
+
+But we think so!
+
+We will need to use various assumptions. one that we believe very strongly based on evidence/experience
+
+Example:
+
+$p,q$ are large random primes
+
+$N=p\cdot q$
+
+Factoring $N$ is hard. (without knowing $p,q$)