-
Notifications
You must be signed in to change notification settings - Fork 13
/
lec28-F24.tex
440 lines (358 loc) · 16.1 KB
/
lec28-F24.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
\newcommand{\cO}{\ensuremath{\mathcal{O}}}
%\newcommand{\NC}[1]{\ensuremath{\mathbf{NC}^{#1}}}
\newcommand{\Ck}{\mathcal{C}_{\kappa}}
\chapter{Obfuscation}
\section{$\iO$ for Polynomial-sized Circuits}
\begin{definition}[Indistinguishability Obfuscator for $NC^1$]
Let $\Ck$ be the collection of circuits of size $O(\kappa)$ and depth
$O(\log{\kappa})$ with respect to gates of bounded fan-in.
Then a uniform PPT machine $\iO_{\NC{1}}$ is an
\emph{indistinguishability obfuscator} for circuit class $\NC{1}$ if it
is an indistinguishability obfuscator for $\Ck$.
\end{definition}
Given an indistinguishability obfuscator $\iO_{\NC{1}}$ for circuit
class $\NC{1}$, we shall demonstrate how to achieve an
indistinguishability obfuscator $\iO$ for all polynomial-sized circuits.
The amplification relies on fully homomorphic encryption (FHE).
\newcommand{\GEN}{\ensuremath{\mathsf{Gen}}}
%\newcommand{\Enc}{\ensuremath{\mathsf{Enc}}}
%\newcommand{\Dec}{\ensuremath{\mathsf{Dec}}}
%\newcommand{\Eval}{\ensuremath{\mathsf{Eval}}}
%\newcommand{\pk}{\ensuremath{\mathsf{pk}}}
%\newcommand{\sk}{\ensuremath{\mathsf{sk}}}
\begin{definition}[Homomorphic Encryption]
A \emph{homomorphic encryption scheme} is a tuple of PPT algorithms
$(\GEN, \Enc, \Dec, \Eval)$ as follows:
\begin{itemize}
\item
$(\GEN, \Enc, \Dec)$ is a semantically-secure public-key
encryption scheme.
\item
$\Eval(\pk, C, e)$ takes public key $\pk$, an arithmetic circuit
$C$, and ciphertext $e = \Enc(\pk, x)$ of some circuit input
$x$, and outputs $\Enc(\pk, C(x))$.
\end{itemize}
\end{definition}
As an example, the ElGamal encryption scheme is homomorphic over the multiplication function.
Consider a cyclic group $G$ of order $q$ and generator $g$, and let
$\sk = a$ and $\pk = g^a$.
For ciphertexts $\Enc(\pk, m_1) = (g^{r_1}, g^{a r_1} \cdot m_1)$
and $\Enc(\pk, m_2) = (g^{r_2}, g^{a r_2} \cdot m_2)$, observe that
\begin{equation*}
\Enc(\pk, m_1) \cdot \Enc(\pk, m_2) = (g^{r_1 + r_2}, g^{a (r_1 + r_2)}
\cdot m_1 \cdot m_2) = \Enc(\pk, m_1 \cdot m_2)
\end{equation*}
Note that this scheme becomes additively homomorphic by encrypting $g^m$
instead of $m$.
\begin{definition}[Fully Homomorphic Encryption]
An encryption scheme is \emph{fully homomorphic} if it is both compact
and homomorphic for the class of all arithmetic circuits.
Compactness requires that the size of the output of $\Eval(\cdot, \cdot,
\cdot)$ is at most polynomial in the security parameter $\kappa$.
\end{definition}
\subsection{Construction}
%We first present a simpler construction under the virtual black box
%model, assuming the existence of a circuit obfuscator $\cO_{\NC{1}}$ for
%$\NC{1}$.
\newcommand{\prog}[1]{\ensuremath{P_{\pk_1,\pk_2,\sk_{#1},e_1,e_2}}}
Let $(\GEN, \Enc, \Dec, \Eval)$ be a fully homomorphic encryption
scheme.
We require that $\Dec$ be realizable by a circuit in $\NC{1}$.
The obfuscation procedure accepts a security parameter $\kappa$ and
a circuit $C$ whose size is at most polynomial in $\kappa$.
\begin{enumerate}
\item
Generate $(\pk_1, \sk_1) \gets \GEN(1^\kappa)$ and
$(\pk_2, \sk_2) \gets \GEN(1^\kappa)$.
\item
Encrypt $C$, encoded in canonical form, as
$e_1 \gets \Enc(\pk_1, C)$ and $e_2 \gets \Enc(\pk_2, C)$.
\item
Output an obfuscation
$\sigma = (\iO_{\NC{1}}(P), \pk_1, \pk_2, e_1, e_2)$
of program $\prog{1}$ as described below.
\end{enumerate}
The evaluation procedure accepts the obfuscation $\sigma$ and program
input $x$.
\begin{enumerate}
\item
Let $U$ be a universal circuit that computes $C(x)$ given a
circuit description $C$ and input $x$, and denote by $U_x$ the
circuit $U(\cdot, x)$ where $x$ is hard-wired.
Let $R_1$ and $R_2$ be the circuits which compute
$f_1 \gets \Eval(U_x, e_1)$ and $f_2 \gets \Eval(U_x, e_2)$,
respectively.
\item
Denote by $\omega_1$ and $\omega_2$ the set of all wires in $R_1$
and $R_2$, respectively.
Compute $\pi_1 : \omega_1 \to \{ 0, 1 \}$ and
$\pi_2 : \omega_2 \to \{ 0, 1 \}$, which yield the value of internal
wire $w \in \omega_1, \omega_2$ when applying $x$ as the input
to $R_1$ and $R_2$.
\item
Output the result of running $\prog{1}(x, f_1, \pi_1, f_2, \pi_2)$.
\end{enumerate}
Program $\prog{1}$ has $\pk_1$, $\pk_2$, $\sk_1$, $e_1$, and $e_2$
embedded.
\begin{enumerate}
\item
Check whether $R_1(x) = f_1 \land R_2(x) = f_2$.
$\pi_1$ and $\pi_2$ enable this check in logarithmic depth.
\item
If the check succeeds, output $\Dec(\sk_1, f_1)$;
otherwise output $\bot$.
\end{enumerate}
The use of two key pairs and two encryptions of $C$, similar to
CCA1-secure schemes seen previously, eliminates the virtual black-box
requirement for concealing $\sk_1$ within $\iO_{\NC{1}}(\prog{1})$.
\subsection{Proof of Security}
We prove the indistinguishability property for this construction
through a hybrid argument.
\newcommand{\Hyb}[1]{\ensuremath{\mathsf{H_{#1}}}}
\begin{proof}
Through the sequence of hybrids, we gradually transform an obfuscation
of circuit $C_1$ into an obfuscation of circuit $C_2$, with each
successor being indistinguishable from its antecedent.
\begin{description}
\item[$\Hyb{0}$]:
This corresponds to an honest execution of $\iO(C_1)$.
Recall that $e_1 = \Enc(\pk_1, C_1)$, $e_2 = \Enc(\pk_2, C_1)$,
and $\sigma = (\iO_{\NC{1}}(\prog{1}), \ldots)$.
\item[$\Hyb{1}$]:
We instead generate $e_2 = \Enc(\pk_2, C_2)$, relying on the
semantic security of the underlying fully homomorphic encryption
scheme.
\item[$\Hyb{2}$]:
We alter program $\prog{2}$ such that it instead embeds $\sk_2$
and outputs $\Dec(\sk_2, f_2)$.
The output of the obfuscation procedure becomes
$\sigma = (\iO_{\NC{1}}(\prog{2}, \ldots)$;
we rely on the properties of functional equivalence and
indistinguishability of $\iO_{\NC{1}}$.
\item[$\Hyb{3}$]:
We generate $e_1 = \Enc(\pk_1, C_1)$ since $\sk_1$ is now
unused, relying again on the semantic security of the fully
homomorphic encryption scheme.
\item[$\Hyb{4}$]:
We revert to the original program $\prog{1}$ and arrive
at an honest execution of $\iO(C_1)$.
\end{description}
\end{proof}
\section{Identity-Based Encryption}
Another use of indistinguishability obfuscation is to realize
identity-based encryption (IBE).
\newcommand{\SETUP}{\ensuremath{\mathsf{Setup}}}
\newcommand{\KEYGEN}{\ensuremath{\mathsf{KeyGen}}}
\newcommand{\mpk}{\ensuremath{\mathsf{mpk}}}
\newcommand{\msk}{\ensuremath{\mathsf{msk}}}
\newcommand{\id}{\ensuremath{\mathsf{id}}}
\begin{definition}[Identity-Based Encryption]
An \emph{identity-based encryption scheme} is a tuple of PPT algorithms
$(\SETUP, \KEYGEN, \Enc, \Dec)$ as follows:
\begin{itemize}
\item
$\SETUP(1^\kappa)$ generates and outputs a master public/private
key pair $(\mpk, \msk)$.
\item
$\KEYGEN(\msk, \id)$ derives and outputs a secret key
$\sk_{\id}$ for identity $\id$.
\item
$\Enc(\mpk, \id, m)$ encrypts message $m$ under identity $\id$
and outputs the ciphertext.
\item
$\Dec(\sk_{\id}, c)$ decrypts ciphertext $c$ and outputs the
corresponding message if $c$ is a valid encryption under
identity $\id$, or $\bot$ otherwise.
\end{itemize}
\end{definition}
\newcommand{\SIGN}{\ensuremath{\mathsf{Sign}}}
\newcommand{\VERIFY}{\ensuremath{\mathsf{Verify}}}
We combine an indistinguishability obfuscator $\iO$ with a digital
signature scheme $(\GEN, \SIGN, \VERIFY)$.
\begin{itemize}
\item
Let $\SETUP \equiv \GEN$ and $\KEYGEN \equiv \SIGN$.
\item
$\Enc$ outputs $\iO(P_m)$, where $P_m$ is a program that
outputs (embedded) message $m$ if input $\sk$ is a secret key for
the given $\id$, or $\bot$ otherwise.
\item
$\Dec$ outputs the result of $c(\sk_{\id})$.
\end{itemize}
However, this requires that we have encryption scheme where the
``signatures'' do not exist.
We therefore investigate an alternative scheme.
%
%\newcommand{\Com}{\ensuremath{\mathsf{Com}}}
%
Let $(K, P, V)$ be a non-interactive zero-knowledge (NIZK) proof system.
Denote by $\Com(\cdot ; r)$ the commitment algorithm of a non-interactive
commitment scheme with explicit random coin $r$.
\begin{itemize}
\item
Let $\sigma$ be a common random string.
$\SETUP(1^\kappa)$ outputs $(\mpk = (\sigma, c_1, c_2), \msk =
r_1)$, where $c_1 = \Com(0 ; r_1)$ and
$c_2 = \Com(0^{|\id|} ; r_2)$.
\item
$\KEYGEN(\msk, \id)$ produces a proof
$\pi = P(\sigma, x_{\id}, s)$ for the following language $L$:
$x \in L$ if there exists $s$ such that
\begin{equation*}
\underbrace{c_1 = \Com(0 ; s)}_{\text{Type I witness}} \lor
\underbrace{\left( c_2 = \Com(\id^* ; s) \land \id^* \ne \id
\right)}_{\text{Type II witness}}
\end{equation*}
\item
Let $P_{\id,m}$ be a program which outputs $m$ if
$V(\sigma, x_{\id}, \pi_{\id}) = 1$ or outputs $\bot$ otherwise.
$\Enc(\mpk, \id, m)$ outputs $\iO(P_{\id,m})$.
\end{itemize}
We briefly sketch the hybrid argument:
\begin{description}
\item[$\Hyb{0}$]:
This corresponds to an honest execution as described above.
\item[$\Hyb{1}$]:
We let $c_2 = \Com(\id^* ; r_2)$, relying on the hiding property
of the commitment scheme.
\item[$\Hyb{2}$]:
We switch to the Type II witness using
$\pi_{\id_i} \forall i \in [q]$, corresponding to the queries
issued by the adversary during the first phase of the
selective-identity security game.
\item[$\Hyb{3}$]:
We let $c_1 = \Com(1 ; r_1)$.
\end{description}
%\nocite{*}
%\printbibliography
% !TEX root = collection.tex
\newcommand{\extline}{$\scriptsize$-$\normalsize$\!}
\newcommand{\lextlineend}{$\scriptsize$\lhd\!$\normalsize$}
\newcommand{\rextlineend}{$\scriptsize\rule{.1ex}{0ex}$\rhd$\normalsize$}
\newcounter{index}
\newcommand\extlines[1]{%
\setcounter{index}{0}%
\whiledo {\value{index}< #1}
{\addtocounter{index}{1}\extline}
}
\newcommand\rextlinearrow[2]{$
\setbox0\hbox{$\extlines{#2}\rextlineend$}%
\tiny$%
\!\!\!\!\begin{array}{c}%
\mathrm{#1}\\%
\usebox0%
\end{array}%
$\normalsize$\!\!%
}
\newcommand\lextlinearrow[2]{$
\setbox0\hbox{$\lextlineend\extlines{#2}$}%
\tiny%
$%
\!\!\!\!\begin{array}{c}%
\mathrm{#1}\\%
\usebox0%
\end{array}%
$\normalsize$\!\!%
}
\renewcommand\lextlinearrow[2]{%
}
\renewcommand\rextlinearrow[2]{%
}
\renewcommand\lextlinearrow[2]{%
% \setbox0\hbox{$\lextlineend\extlines{#2}$}%
$\stackrel{\mathrm{#1}}{\leftarrow}$%
}
\renewcommand\rextlinearrow[2]{%
%\setbox0\hbox{$\extlines{#2}\rextlineend$}%
$\stackrel{\mathrm{#1}}{\rightarrow}$%
}
%\section{Indistinguishable Obfuscation Constructions using Puncturing}
\section{Digital Signature Scheme via Indistinguishable Obfuscation}
A digital signature scheme can be constructed via indistinguishable obfuscation (iO). A digital signature scheme is made up of $(\SETUP, \SIGN, \VERIFY)$.\\
%\newcommand{\vk}{\mathsf{vk}}
\noindent $(\vk, \sk) \leftarrow \SETUP(1^k)$:\\
\indent $\sk$ = key of puncturable function and the seed of the PRF $F_k$\\
\indent $\vk = iO(P_k)$ where $P_k$ is the program:\\
\indent \indent $P_k(m, \sigma)$:\\
\indent \indent \indent for some OWF function $f$\\
\indent \indent \indent \indent return 1 if $f(\sigma) = f(F_k(m))$\\
\indent \indent \indent \indent return 0 otherwise\\
\noindent $\sigma \leftarrow \SIGN(\sk, m)$: Output $F_k(m)$.\\
\noindent $\VERIFY(\vk, m, \sigma)$: Output $P_k(m, \sigma)$.\\
\noindent Our security requirements will be that the adversary does wins the following game negligibly:\\
\begin{tabular}{llc}
{\large Challenger} & & {\large Adversary}\\
$(\vk, \sk) = \SETUP(1^k)$ and&&\\
picks random $m$&&\\
& \rextlinearrow{P_{k},m}{46} &\\
& \lextlinearrow{\sigma, m^*}{46} &\\
& Adversary wins game if $\VERIFY(\vk, m^*, \sigma) = 1$&
\end{tabular}\\
\noindent To prove the security of this system, we use a hybrid argument. $H_0$ is as above.
\noindent $H_1$: Adjust $\vk$ so that $\vk = iO(P_{k, m, \alpha})$ where $\alpha = F_k(m)$ and $P_{k, m, \alpha}$ is the program such that:\\
\indent $P_{k,m, \alpha}(m^*, \sigma)$:\\
\indent \indent for some OWF $f$\\
\indent \indent \indent if $m = m^*$:\\
\indent \indent \indent \indent if $f(\sigma) = f(\alpha)$ return 1\\
\indent \indent \indent \indent otherwise return 0\\
\indent \indent \indent else proceed as $P_{k}$ from before\\
\indent \indent \indent \indent if $f(\sigma) = f(F_k(m^*))$ return 1\\
\indent \indent \indent \indent otherwise return 0\\
\noindent Note that this program does not change its output for any value. This is indistinguishable from $H_0$ by indistinguishability obfuscation.\\
\noindent $H_2$: Adjust $\alpha$ so that it is a randomly sampled value. The indistinguishability of $H_2$ and $H_1$ follows from the security of PRG. \\
\noindent $H_3$: Adjust the program such that instead of $\alpha$ it relies on some $\beta$ that is compared instead $f(\alpha)$ in the third line.\\
Any adversary that can break $H_3$ non-negligibly can break the OWF $f$ with at the value $\beta$.
\section{Public Key Encryption via Indistinguishable Obfuscation}
A public key encryption scheme can be constructed via indistinguishable obfuscation. A public key encryption scheme is made up of $(Gen, Enc, Dec)$. The PRG used below is a length doubling PRG.\\
\noindent $(\pk, \sk) \leftarrow Gen(1^k)$\\
\indent $\sk$ = key of puncturable function and the seed of the PRF $F_k$\\
\indent $\pk = iO(P_k)$ where $P_k$ is the program:\\
\indent \indent $P_k(m, r)$:\\
\indent \indent \indent $t = PRG(r)$\\
\indent \indent \indent Output $c = (t, F_k(t) \oplus m)$\\
\noindent $Enc(\pk, m)$: Sample $r$ and output $(\pk(m,r))$.\\
\noindent $Dec(\sk = k, c = (c_1, c_2))$: Output $F_k(\sk,c_1) \oplus c_2$.\\
\noindent Our security requirements will be that the adversary does wins the following game negligibly:\\
\begin{tabular}{llc}
{\large Challenger} & & {\large Adversary}\\
$(\pk, \sk) = Gen(1^k)$ and&&\\
Randomly sample $b$ from $\{0,1\}$ and&&\\
$c^* = Enc(\pk, b)$ and&&\\
& \rextlinearrow{P_{k},c^*}{26} &\\
& \lextlinearrow{b^*}{26} &\\
& Adversary wins game if $b=b^*$&
\end{tabular}\\
\noindent To prove the security of this system, we use a hybrid argument. $H_0$ is as above.
\noindent $H_1$: Adjust $\pk$ so that $\pk = iO(P_{k, \alpha, t})$ where $\alpha = F_k(t)$ and $P_{k, \alpha, t}$ is the program such that:\\
\indent $P_{k, \alpha, t}(m, r)$:\\
\indent \indent $t^* = PRG(r)$\\
\indent \indent if $t^* = t$, output $(t^*, \alpha \oplus m)$\\
\indent \indent else output $(t^*, F_k(t^*) \oplus m)$\\
\noindent Note that this program does not change its output for any value. This is indistinguishable from $H_0$ by indistinguishability obfuscation.\\
\noindent $H_2$: Adjust $\alpha$ so that it is a randomly sampled value.\\
\noindent $H_3$: Adjust the program such that $t^*$ is randomly sampled and is not in the range of the PRG.\\
Any adversary that can win $H_3$ can guess a random value non-negligibly.\\
\section{Indistinguishable Obfuscation Construction from $NC^1$ $iO$}
A construction of indistinguishable obfuscation from $iO$ for circuits in $NC^1$ is as follows:\\
Let $P_{k,C}(x)$ be the circuit that outputs the garbled circuit $\widetilde{UC(C,x)}$ with randomness $F_k(x)$ which is a punctured (at $k$) PRF in $NC^1$\\
\indent Note that $UC(C,x)$ outputs $C(x)$ ($UC$ is the ``universal'' circuit)\\
$iO(C) \rightarrow $ sample $k$ randomly from $\{0,1\}^{|x|}$ and output $iO_{NC^1}(P_{k,C})$ padded to a length $l$\\
As before, we use a hybrid argument to show the security for $iO$.\\
\noindent $H_0$: $iO(C) = iO_{NC^1}(P_{k,C})$ as above.\\
\noindent $H_{final} = H_{2^n}$: $iO(\pk, c_2)$\\
\noindent $H_1 \cdots H_i$: Create a program $Q_{k, c_1, c_2, i}(x)$ and obfuscate it.\\
$Q_{k,c_1,c_2,i}(x)$:\\
\indent Sample $k$ randomly\\
\indent if $x \ge i$, return $P_{k,c_1}(x)$\\
\indent else , return $P_{k,c_2}(x)$\\
\noindent Note that $H_i$ and $H_{i+1}$ are indistinguishable for any value other than $x=i$.\\
\noindent $H_{i,1}$ (between $H_i$ and $H_{i+1}$): Create a program $Q_{k, c_1, c_2, i, \alpha}(x)$, where $\alpha = Q_{k, c_1, c_2, i}(x)$ and obfuscate it.\\
$Q_{k, c_1, c_2, i, \alpha}(x)$:\\
\indent Sample $k$ randomly\\
\indent if $x = i$, return $\alpha$\\
\indent else , return $Q_{k,c_1,c_2, i}(x)$\\
\noindent $H_{i,2}$: Replace $\alpha$ with a random $\beta$ using fresh coins\\
\noindent $H_{i,3}$: Create the $c_2(x)$ value using fresh coins\\
\noindent $H_{i,4}$: Create the $c_2(x)$ value using $F_k(x)$\\
\noindent $H_{i,5}$: Finish the migration to $Q_{k,c_1,c_2,i+1}$\\
Note that at $H_{final}$, the circuit being obfuscated is completely changed from $c_1$ to $c_2$.