NoteNextra-origin/content/CSE442T/CSE442T_L4.md

Lecture 4

Recap

Negligible function \epsilon(n) if \forall c>0,\exist N such that n>N, \epsilon (n)<\frac{1}{n^c}

Example:

\epsilon(n)=2^{-n},\epsilon(n)=\frac{1}{n^{\log (\log n)}}

Chapter 2: Computational Hardness

One-way function

Strong One-Way Function

1. \exists a P.P.T. that computes f(x),\forall x\in\{0,1\}^n
2. \forall \mathcal{A} adversaries, \exists \epsilon(n),\forall n.

P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]<\epsilon(n)

That is, the probability of success guessing should decreasing (exponentially) as encrypted message increase (linearly)...

To negate statement 2:

P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]=\mu(n)

is a negligible function.

Negation:

\exists \mathcal{A}, P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]=\mu(n) is not a negligible function.

That is, \exists c>0,\forall N \exists n>N \epsilon(n)>\frac{1}{n^c}

\mu(n)>\frac{1}{n^c} for infinitely many n. or infinitely often.

Keep in mind: P[success]=\frac{1}{n^c}, it can try O(n^c) times and have a good chance of succeeding at least once.

Definition 28.4 (Weak one-way function)

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 \mathcal{A} adversaries, \exists \epsilon(n),\forall n.

P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]<1-\frac{1}{p(n)}

The probability of success should not be too close to 1

Probability

Useful bound 0<p<1

1-p<e^{-p}

(most useful when p is small)

For an experiment has probability p of failure and 1-p of success.

We run experiment n times independently.

P[\text{success all n times}]=(1-p)^n<(e^{-p})^n=e^{-np}

Theorem 35.1 (Strong one-way function from weak one-way function)

If there exists a weak one-way function, there there exists a strong one-way function

In particular, if f:\{0,1\}^n\to \{0,1\}^* is weak one-way function.

\exists polynomial q(n) such that

g(x):\{0,1\}^{nq(n)}\to \{0,1\}^*

and for every n bits x_i

g(x_1,x_2,..,x_{q(n)})=(f(x_1),f(x_2),...,f(x_{q(n)}))

is a strong one-way function.

Proof:

1. Since \exist P.P.T. that computes f(x),\forall x we use this q(n) polynomial times to compute g.

2. (Idea) a has to succeed in inverting f all q(n) times. Since x is a weak one-way, \exists polynomial p(n). \forall q, P[q inverts f]<1-\frac{1}{p(n)} (Here we use < since we can always find a polynomial that works)

   Let q(n)=np(n).

   Then P[a inverting g]\sim P[a inverts f all q(n)] times. <(1-\frac{1}{p(n)})^{q(n)}=(1-\frac{1}{p(n)})^{np(n)}<(e^{-\frac{1}{p(n)}})^{np(n)}=e^{-n} which is negligible function.

QED

we can always force the adversary to invert the weak one-way function for polynomial time to reach the property of strong one-way function

Example: (1-\frac{1}{n^2})^{n^3}<e^{-n}

Some candidates of one-way function

Multiplication

Mult(m_1,m_2)=\begin{cases}
    1,m_1=1 | m_2=1\\
    m_1\cdot m_2
\end{cases}

But we don't want trivial answers like (1,1000000007)

Idea: Our "secret" is 373 and 481, Eve can see the product 179413.

Not strong one-way for all integer inputs because there are trivial answer for \frac{3}{4} of all outputs. Mult(2,y/2)

Factoring Assumption:

The only way to efficiently factorizing the product of prime is to iterate all the primes.

In other words:

\forall a\exists \epsilon(n) such that \forall n. P[p_1\gets \prod n_j]

We'll show this is a weak one-way function under the Factoring Assumption.

\forall a,\exists \epsilon(n) such that \forall n,

P[p_1\gets \Pi_n;p_2\gets \Pi_n;N=p_1\cdot p_2:a(n)=\{p_1,p_2\}]<\epsilon(n)

where \Pi_n=\{p\text{ all primes }p<2^n\}