116 lines
3.6 KiB
Markdown
116 lines
3.6 KiB
Markdown
# CSE442T Introduction to Cryptography (Lecture 3)
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All algorithms $C(x)\to y$, $x,y\in \{0,1\}^*$
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P.P.T= Probabilistic Polynomial-time Turing Machine.
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## Chapter 2: Computational Hardness
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### Turing Machine: Mathematical model for a computer program
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A machine that can:
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1. Read in put
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2. Read/Write working tape move left/right
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3. Can change state
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### Assumptions
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Anything can be accomplished by a real computer program can be accomplished by a "sufficiently complicated" Turing Machine (TM).
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### Polynomial time
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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 $T(n)=O(n^c)$
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If we can argue that algorithm runs in polynomially-many constant-time operations, then this is true for the T.M.
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$p,q$ are polynomials in $n$,
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$p(n)+q(n),p(n)q(n),p(q(n))$ are polynomial of $n$.
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Polynomial-time $\approx$ "efficient" for this course.
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### Probabilistic
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Our algorithm's have access to random "coin-flips" we can produce poly(n) random bits.
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$P[C(x)\text{ takes at most }T(n)\text{ steps }]=1$
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Our adversary $a(x)$ will be a P.P.T which is non-uniform (n.u.) (programs description size can grow polynomially in n)
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### Efficient private key encryption scheme
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#### Definition 3.2 (Efficient private key encryption scheme)
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The triple $(Gen,Enc,Dec)$ is an efficient private key encryption scheme over the message space $M$ and key space $K$ if:
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1. $Gen(1^n)$ is a randomized p.p.t that outputs $k\in K$
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2. $Enc_k(m)$ is a potentially randomized p.p.t that outputs $c$ given $m\in M$
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3. $Dec_k(c')$ is a deterministic p.p.t that outputs $m$ or "null"
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4. $P_k[Dec_k(Enc_k(m))=m]=1,\forall m\in M$
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### Negligible function
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$\epsilon:\mathbb{N}\to \mathbb{R}$ 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)
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Idea: for any polynomial, even $n^{100}$, in the long run $\epsilon(n)\leq \frac{1}{n^{100}}$
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Example: $\epsilon (n)=\frac{1}{2^n}$, $\epsilon (n)=\frac{1}{n^{\log (n)}}$
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Non-example: $\epsilon (n)=O(\frac{1}{n^c})\forall c$
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### One-way function
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Idea: We are always okay with our chance of failure being negligible.
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Foundational concept of cryptography
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Goal: making $Enc_k(m),Dec_k(c')$ easy and $Dec^{-1}(c')$ hard.
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#### Definition 27.3 (Strong one-way function)
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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 $\epsilon (n)$ such that for any adversary $\mathcal{A}$ (n.u.p.p.t)
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$$
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P[x\gets\{0,1\}^n;y=f(x):f(\mathcal{A}(y))=y]\leq\epsilon(n)
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$$
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_Probability of guessing a message $x'$ with the same output as the correct message $x$ is negligible_
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and
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there is a p.p.t which computes $f(x)$ for any $x$.
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- Hard to go back from output
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- Easy to find output
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$a$ sees output y, they wan to find some $x'$ such that $f(x')=y$.
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Example: Suppose $f$ is one-to-one, then $a$ must find our $x$, $P[x'=x]=\frac{1}{2^n}$, which is negligible.
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Why do we allow $a$ to get a different $x'$?
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> 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.
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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 $\mathcal{A}$)
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#### Do one-way function exists?
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Unknown, actually...
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But we think so!
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We will need to use various assumptions. one that we believe very strongly based on evidence/experience
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Example:
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$p,q$ are large random primes
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$N=p\cdot q$
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Factoring $N$ is hard. (without knowing $p,q$)
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