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555099a019796676ec63edb755ea31b9f3825e6f | subsection | 65 | 77 | Lower-bound on the marginals | Thus, 0 holds with _{rn} instead of _{xt}.Since moreover ^2 \ge ^1 :
\quad \mu A_{+ ^2-^1} \in _{rn}.Since \mu A_{^2} = \mu A_{+ ^2-^1} A_{^1 -}, we
finally deduce from :
\quad \mu A_{^2} \ge [^1 - ].{1} Proof of Lemma REF :j \le J() - 1 means notably - [j+1]\, \ge , thus :& _{\nu _j} - j\, < = _{\nu _j} _{X_{}} - [j... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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6f9e24bd29ccc90dcbd828bc4460c71d3a5bb48b | subsection | 66 | 77 | Lower-bound on the marginals | Then :r_{j+1} > 0 \quad \text{ and }
{0< \le }
\nu _{j+1}(dx)
= \nu _j A_{}(dx) - \ (dx) \, /\, (1- ).Thanks to Lemma REF together with DBnuj : \nu _j \ge 0
thus \nu _j \in .Moreover, for any measurable set :&\hspace{56.9055pt}
\nu _{j+1}()
\ge \nu _j A_{}() - \, / (1- )
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&\hspac... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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89277fc4ea5f50a436d2bb9063ebcb669ce3ffde | subsection | 67 | 77 | Lower-bound on the marginals | \\&\text{Thus }
\hspace{14.22636pt}
1 = {k \le j} \; (k, [j+1]\, ) + \ell _j
\quad \text{i.e.}\quad r_{j+1}
= \ell _j - (j+1, [j+1]\, )
{eqd}\\&\hspace{28.45274pt}
\text{by evaluating \, on } \text{ and by \, --the definition of }r_{j+1}.By , 0 and by Lemma REF ::=& (j+1, [j+1]\, ) / \ell _j
{eqt}{}
\\&= 1- ^{-j}
\ti... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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b46223e1bdc01ff8dd0cf250fe77f41035805bd6 | subsection | 68 | 77 | Body | {1} Lemma REF implies Proposition REF :We obtain by induction and the Markov property :
\quad { y > 0} \quad _{y} (k\, t_D < )
\le ^k.Thus, by choosing sufficiently small (for any given value of t_D), we ensure :&\hspace{28.45274pt}
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extinction | [
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f2e08a98b2db94c06742f055564d1dc965c79350 | subsection | 69 | 77 | Body | It is the solution of the ODE :&\dfrac{dV_t}{dt}
= - \dfrac{1}{2 (V_t + B_t) } + \dfrac{r^D\; (V_t + B_t)}{2}
- c_Y\; (V_t + B_t)^3,
{Vt}{V_t}
\\&\text{Let }
y_\infty ^{B} \ge y_\infty ^{V} := \sup y >0, \;
\left|- 1/ (2 y) + r^Dy / 2 \; \right|
\ge c_Yy^3 / 2,
{wyinf}{y_\infty ^{V}}
\\&
T_B := \inf t >0, B_t \notin [-... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
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54fd01ef858ba3d33c513fa5c6d17dc29670c5c9 | subsection | 70 | 77 | Body | In any case, \le t_D. Hence :
\quad {y > 0} \quad _{y} (t_D < )
\le .{1} Lemma REF implies Proposition REF :Let \rho ,\, ^X,\, y_\infty > 0 (c_Y>0 is the same as for the definition of Y).
For simplicity,
we choose t_D := \log (2)/\rho >0 (i.e. \exp \rho \, t_D = 2).Let n_c^X> 0 and {r_D} \in
chosen according to Lemma ... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
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77e9aa3ea4cbcb2fc842869044bd96a943ee79a8 | subsection | 71 | 77 | Body | Thus, by our definitions of t_D, n_c^X, {r_D} :&_{(x, y)}[\exp (\rho \, V_{_c} )]
\le 2 \, 1+ _0 + ^X_\infty / 2
+ ^X
\, ^Y_\infty / 2.Taking the supremum over (x, y)\in
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\quad ^X_\infty \le 4 \, 1+ _0 + ^X\, ^Y_\infty .{1} Proof of Lemma REF :{2} Step 1: \sup _{y>0} \, \psi _D(y, r_D) {... | {
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959f5e13efe5bc419fc9b363aea50e30b0a8aec3 | subsection | 72 | 77 | Body | We can choose \Delta y > 0 such that, with N\sim 0, 1 :{t\le t_D} B_t \ge \Delta y= 2\, N \ge \Delta y \, /\,\sqrt{t_D}\le .
{DeltaY}{\Delta y}Then, we can choose A> 0 (sufficiently big) such that :
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\quad {r_D \ge {r_D}}\... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
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71386ec017a5b5cca0950dcedfab7d4b248caef6 | subsection | 73 | 77 | Body | Thanks to Lemma REF (for r^D =0 since (t_D < ) is decreasing with r^D)
we choose >0 such that :
\quad := \inf t\ge 0, \; Y^D_t \le&{r_D \le 0}
{ y_{\infty } > 0} \quad _{y_{\infty }} (t_D < )
\le {Eyd}{}Like in the previous step, we choose A^{\prime }>0 such that :
B_{t_D} \ge A^{\prime }\, t_D - \le .Again, we choose ... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
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79fed7a6052a8d0e1867f9814c3af54226d0506b | subsection | 74 | 77 | Body | In order to get an estimate of mortality in 0, we will use some coupling with branching processes and
consider the process after a time t_m.In practice, we prove :&{\rho > 0}
{{^0} > 0}
{n_c^0 > 0}
{ C_0^{\prime } \ge 1}
\\&\hspace{14.22636pt}
{y_{\infty }>0}
{n_c> y_{\infty }\vee n_c^0}
\qquad _0 \le C_0^{\prime } + {... | {
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4979f35ce5338f334aad6776e00c92d6ecb5bd4b | subsection | 75 | 77 | Body | \quadSo we can impose that \rho _0 > \rho ,
and even that \exp (- (\rho _0-\rho )\; t_m) is sufficiently small
to make transitions from 0
to 0, or of little incidence.&_{(x, y)}[\exp (\rho V_{_c} )]
\le _{(x, y)}\Big [
\exp (\rho V_{_c} ) V_{_c} < t_m
\Big ]
+ _{(x, y)}\Big [
\exp (\rho V_{_c} ) (X, Y)_{t_m} \in 0\cup ... | {
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"Aurélien Velleret"
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cbe1445f67927045c166c635278c7b3507df5626 | subsection | 76 | 77 | Aknowledgment | I am very grateful to Etienne Pardoux, my PhD supervisor,
for his wonderful support all along the redaction of this article.
I wish to thank also Nicolas Champagnat for his advices and support.I would like finally to thank the very inspiring meetings and discussions brought about by the Chair “Modélisation Mathématique... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
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4abf915fa2fd2758f06437725b55bca2a45e94a6 | abstract | 0 | 95 | Abstract | We consider the setting of a Master server, M, who possesses confidential
data (e.g., personal, genomic or medical data) and wants to run intensive
computations on it, as part of a machine learning algorithm for example. The
Master wants to distribute these computations to untrusted workers who have
volunteered or are ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
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3dc65b0e6723471f1dc6f1456c338c1dfdaf436d | subsection | 1 | 95 | Introduction | We consider the setting of distributed computing in which a server {\fontfamily {cmtt}\selectfont \text{M}}, referred to as Master, possesses confidential data and wants to perform intensive computations on it. {\fontfamily {cmtt}\selectfont \text{M}} wants to divide these computations into smaller computational tasks ... | {
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"... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
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03fa413361556db28d451ea3ae72b9566ad75c8e | subsection | 2 | 95 | Introduction | Suppose that {\fontfamily {cmtt}\selectfont \text{M}} gets the help of 3 workers out of which at most 1 may be a straggler.
{\fontfamily {cmtt}\selectfont \text{M}} generates a random matrix R of same dimensions as A with entries drawn over the same alphabet as the entries of A.
{\fontfamily {cmtt}\selectfont \text{M}}... | {
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83623b912423d44ecbe76abb8ef76d9fa1a950ad | subsection | 3 | 95 | Introduction | The operations shown are in GF(5).]The Staircase code requires {\fontfamily {cmtt}\selectfont \text{M}} to divide the matrices A and R into A=\begin{bmatrix}A_1& A_2\end{bmatrix}^T and R=\begin{bmatrix}R_1& R_2\end{bmatrix}^T. In this setting, {\fontfamily {cmtt}\selectfont \text{M}} sends two sub-shares to each worker... | {
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f09e615d311a931f15830c7e5dd29d7f937bf56d | subsection | 4 | 95 | Introduction | We refer to such secure distributed computing system by an (n,k,z) system.
We make the following contributions:
General bounds for systems with any number of stragglers: We derive an upper and a lower bound on the Master's mean waiting time when using Staircase codes (Theorem REF ). Moreover, we derive the exact distr... | {
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629181653d4b782c57c70eaee46e9856fcee1497 | subsection | 5 | 95 | Introduction | The work that is closest to ours is that studies the problem of distributively multiplying two private matrices under information theoretic privacy constraints using classical secret sharing codes.
Our work can also be related to the work on privacy-preserving algorithms, e.g., , , , .
However, the privacy constraint ... | {
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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c72b4ced44c00a5105c1f81eadb2a8cc8e31ca08 | subsection | 6 | 95 | Introduction | Any k or more shares can decode A, and any collection of z workers obtain zero information about A.
For any set \mathcal {B}\subseteq \lbrace 1,\dots ,n\rbrace , let S_\mathcal {B}=\lbrace S_i, i\in \mathcal {B}\rbrace denote the collection of shares given to worker {\fontfamily {cmtt}\selectfont \text{W}}_i for all i\... | {
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"doi": "10.1145/359168.359176",
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Staircase Codes | [
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6a0271ad1351c9c7b39f829edd1c14dcdfe2cac5 | subsection | 7 | 95 | Introduction | It follows that F_{T_i} is a scaled distribution of F_{T_A}.
That is,
F_{T_i}(t)\triangleq F_{T_A}((k-z)t)=1-e^{ -\lambda (k-z)(t-\tfrac{c}{k-z})}, \quad \text{for } t\ge c/(k-z).
For an (n,k,z) system using Staircase codes, we assume that T_i is evenly distributed among the sub-tasksTherefore, we make two assumption... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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16fe77078f73958d16c99bb5e08845c33c849111 | subsection | 8 | 95 | Introduction | For an (n,k,z) system using classical secret sharing codes,
the Master's waiting time T_{\text{SS}}(n,k,z) is equal to the time spent by the fastest k workers to finish their individual tasks. Hence, we can write
T_{\text{SS}}(n,k,z)=T_{(k)}.
We drop the (n,k,z) notation from T_{\text{SC}}(n,k,z) and T_{\text{SS}}(n,... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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f9c65bf1f7bf963553f3f5d41591d1d75df0e6ae | subsection | 9 | 95 | Introduction | The mean waiting time of the Master \mathbb {E}[T_{\text{SC}}] for an (n,k,z) Staircase coded system is upper bounded by
\mathbb {E}[T_{\text{SC}}] &\le \min _{d\in \lbrace k,\dots ,n\rbrace }\left(\frac{H_n - H_{n-d}}{\lambda (d-z)}+\dfrac{c}{d-z}\right),
and lower bounded by
\mathbb {E}[T_{\text{SC}}] &\ge \frac... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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a84c79791cb1a7af6e9429a3444f67e04be0d7bb | subsection | 10 | 95 | Introduction | We also establish the comparison for fixed rate regimes, in particular rate k/n=1/2. Since here n\ge k+2, we compare in Figure REF the bounds to numerical results obtained by simulation and observe the same behavior as before. We also plot in the same figure the mean waiting time for classical secret sharing codes obta... | {
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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0.0... |
4d850e4796868ed9ce05de79fe9c61cf6b859e90 | subsection | 11 | 95 | Introduction | The i^{\text{th}} codeword, i.e., i^{\text{th}} share of the code, is the i^{\text{th}} row of the matrix C=VM_\text{SS}.
Decoding: The decoding of threshold secret sharing consists of taking any k codewords and inverting the corresponding encoding sub-matrix to obtain the secret and all the random keys.
In the setting... | {
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{
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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0... |
03468cd6b236c38d983f07e606ab0ab5c5427942 | subsection | 12 | 95 | Introduction | To construct the matrix M defined in Table REF , an (n,k,z) Staircase code requires dividing the data matrix A into b(k-z) matrices A_1,\dots ,A_{(k-z)b} each of dimensionIf the number of rows in A is not divisible by b, one can use zero padding or the representation of A in a smaller field GF(q_1) such that q=q_1^b. m... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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36db8d3891d6d7fc58f21f574a59e142a9f18c3b | subsection | 13 | 95 | Introduction | \normalsize {1}
{0.93}{!}{
\begin{}[baseline=(current bounding box.center)]
{stealth} = [draw=none,text=black]
\node [stealth] (1) at (0,0){
M_1=\hspace{8.5359pt} \begin{bmatrix}{2}{*}{\mathcal {S}} \\
\\
\mathcal {R}_1\\
\end{bmatrix}\quad ,};
\end{}\normalsize {1}
}
stealth] (2) [right=0.3cm of 1] M_2=\hspace{8.5359p... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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f7628f1ef970f5c47b967885a7bd2479fdfd1ec5 | subsection | 14 | 95 | Introduction | The \mathbf {0}'s are the all zero matrices used to complete the M_{i}'s to nm(k-z)b rows.
The structure of the matrix M_\text{SC}, called Staircase structure, allows the Master to decode the secret and achieve optimal communication and read overheads \fontfamily {euf}\selectfont \text{CO} and \fontfamily {euf}\selectf... | {
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{
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"raw": "R. Bitar and S. El Rouayheb, “Staircase codes for secret sharing with optimal communication and read overheads,” IEEE Transactions on Informat... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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1c3f3868ac9c807c6066eecf0bd6466c08288f6d | subsection | 15 | 95 | Introduction | Bounds on the Master's mean waiting time for all (n,k,z) systems
We derive an upper and a lower bound on the Master's mean waiting time \mathbb {E}[T_\text{SC}(n,k,z)] for all (n,k,z) systems, i.e., we prove Theorem REF . We restate Theorem REF for the sake of presentation.
Theorem 1 (Bounds on the Master's mean waitin... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... |
8a98ba9844c1b2ad80a2994624fa226773ced7c3 | subsection | 16 | 95 | Introduction | Theorem (Renyi )
The d^{\text{th}} order statistic T^{\prime }_{(d)} of n iid exponential random variables T^{\prime }_i
is equal to the following random variable in the distribution
T^{\prime }_{(d)} &\triangleq \sum _{j=1}^{d}\frac{T^{\prime }_j}{n-j+1}.
Using Renyi's Theorem, the mean of the d^{\text{th}} order... | {
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{
"arxiv_id": "",
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"raw": "A. Rényi, “On the theory of order statistics,” Acta Mathematica Academiae Scientiarum Hungarica, vol. 4, no. 3-4, pp. 191–231, 1953.",
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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48d8960f82fbf90d25d8355505423055b7944558 | subsection | 17 | 95 | Introduction | Since the minimum of the sum is greater than the sum of the minimums,
we can lower bound the waiting time T_{\text{SC}} in terms of residual waiting time T^{\prime }_{\text{SC}} \triangleq \min \lbrace \alpha _dT^{\prime }_{(d)}: d \in \lbrace k,\dots ,n\rbrace \rbrace , as
T_{\text{SC}} =\min _{d\in \lbrace k,\dots ,... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0b98d998d508bcc3815cd4b86f453e8a2d4d76dc | subsection | 18 | 95 | Introduction | Next, we evaluate \Pr (\mathcal {C}_d(t)) explicitly.
To this end, we first observe that {\alpha _j}^{-1}-{\alpha _{j-1}}^{-1}={(k-z)}^{-1} identically for each j \in \lbrace 1, \dots , n\rbrace .
Further, we apply Renyi's Theorem and independence of residual times T^{\prime }_is
to write
\Pr \left(\mathcal {C}_d(t) ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... |
4d4749fc89be9438753fc79145e35e2389a72a59 | subsection | 19 | 95 | Introduction | Exploiting the exponential form of \bar{F}(t), aggregating results from (REF ), (REF ) and (REF ), we can re-write (REF ) as
\Pr \left(\mathcal {C}_d(t)\right) &=
\sum _{i=0}^{k-1}\binom{n}{i}\sum _{j=0}^{i}\binom{i}{j}(-1)^{j}\bar{F}\big ( t(n-i+j)(d-z) +t (n-d)(n-d+1)/2\big ).
The proof follows from the integral \... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.029... |
d084efb23a57cb41d8eb8b09097df211e4befdbd | subsection | 20 | 95 | Introduction | Theorem 4 (Integral expression leading to F_{T_{\text{SC}}}(t))
The distribution of the Master's waiting time T_{\text{SC}} of an (n,k,z) system using Staircase codes is given by
F_{T_{\text{SC}}}\left( t \right)=
1-n!\int _{(y_k, \dots , y_n)\in A(t)}\frac{F_{T^{\prime }}(y_k)^{k-1}}{(k-1)!}dF_{T^{\prime }}(y_k)\do... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... |
e7956c06370de54b2131f2590563acb84177661b | subsection | 21 | 95 | Introduction | Both distributions are defined for t>0, and F_{T^{\prime }}(t)\triangleq 1-\exp (-\lambda (k-z)t).
We omit the proof of Corollary REF since it follows from simply integrating (REF ) and defer the proof of Theorem REF to the Appendix.
[Proof of Theorem REF ] Let T^{\prime }_i denote the residual service time of worker ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... |
10017603eb268761ac045daec8bddfbdba72641a | subsection | 22 | 95 | Introduction | For each k \le j \le n, we define t_j \triangleq \max \left\lbrace \frac{t}{\alpha _j} - \frac{c}{j-z},0\right\rbrace , y_{n+1} \triangleq \infty , and \hat{A}(t) \triangleq \cap _{j=k}^{n+1}\lbrace t_j < y_j \le y_{j+1}\rbrace \cap _{j=1}^{k-1}\lbrace 0 \le y_j \le y_{j+1} \rbrace .
In terms of t_j, y_{n+1} and \hat{A... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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0.0... |
450f244e7e2d565e806c5ed1be56bcf25ef6c718 | subsection | 23 | 95 | Introduction | This can be shown by writing the integral I_{k+1} in (k+1) integration variables y_1, \dots , y_{k+1} in terms of the integral I_k,
and evaluating the integral by substituting the induction hypothesis for I_k as follows
I_{k+1}
&=\int _{0}^{y_{k+1}}I_kdF_{T^{\prime }}(y_k) =\int _{0}^{y_{k+1}} \frac{F_{T^{\prime }}(y_... | {
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"doi": "10.1109/tit.2017.2723019",
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"raw": "R. Bitar and S. El Rouayheb, “Staircase codes for secret sharing with optimal communication and read overheads,” IEEE Transactions on Informa... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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] | 2,018 | en | Computer Science | [
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257c49130ce66f1737a7d490eb7294a516204225 | subsection | 24 | 95 | Introduction | Recall that d : \mathbb {R}_+^n \rightarrow \lbrace k, \ldots , n\rbrace is a function of the compute times T_1, \ldots , T_n.
d(T_1, T_2, \dots , T_n)\triangleq \arg \min \left\lbrace \frac{k-z}{i-z}T_{(i)}: i\in \lbrace k,\dots ,n\rbrace \right\rbrace .
Claim 8 The number of workers d that minimize the waiting time... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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83f27317a1bd5d2933b10673bcfce78298c299fe | subsection | 25 | 95 | Introduction | Therefore, we can apply the McDiarmid's inequality to obtain the concentration bound on d.
Simulations
We use the normalized difference between the mean waiting time of Staircase codes and classical secret sharing codes as a performance metric for Staircase codes. We refer to this metric as the savings. Using the resu... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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ee1e91368c5f156e283f2451fc5694ea28e31feb | subsection | 26 | 95 | Introduction | Similarly to Figure , we consider systems with z=1, \lambda =1 and vary c.][Table: Comparison of the performance of Staircase codes on Amazon EC2 to the theoretical bound in () and the value obtained by simulations assuming the shifted exponential model in Section . The shift c^* and the rate \lambda ^* of the workers ... | {
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} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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... |
88babd4f6affee6e7d4197e0287ad36a25c846c1 | subsection | 27 | 95 | Introduction | These results are also summarized in Table REF .
Note that for this set of implementation, the Master's data A is a matrix of size 378000\times 250 with entries generated uniformly at random from \lbrace 1,\dots ,255\rbrace . We run 1000 multiplications of A by a randomly generated vector \mathbf {x}.
[Figure: An (n,k,... | {
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{
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"doi": "10.1007/978-3-319-25958-1_8",
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"raw": "E. Learned-Miller, G. B. Huang, A. Roy Chowdhury, H. Li, and G. Hua, “Labeled faces in the wild: A survey,” in Advances in face detection ... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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9a57bfca5882bcd1fe4750f04828b18b66cf4a70 | subsection | 28 | 95 | Introduction | \mathbb {E}\left[T_{\text{SC}}(k+1,k,z)\right] &= \frac{c}{k-z+1}+\frac{1}{\lambda }\sum _{i=1}^{k+1}(-1)^{i}\binom{k+1}{i}\left[\frac{i\exp \left(\frac{-\lambda c}{k-z}\right)}{(k-z)i+1}-\dfrac{1}{(k-z+1) i} \right].\\
\mathbb {E}[T_{\text{SC}}(k+2,k,z)]&= \mathbb {E}[T_\text{SC}(k+2,k+1,z)]+\frac{1}{\lambda }\sum _{... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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f90f675b123d2298c8d4439a058d971d07bcbb11 | subsection | 29 | 95 | Introduction | Since F_{T^{\prime }}(0) = 0, we can compute the Master's mean waiting time \mathbb {E}\left[ T(k+1,k,z)\right] as
\mathbb {E}\left[ T_{\text{SC}}(k+1,k,z)\right]&=\int _{0}^{\infty }(1- (1-\bar{F}_{T^{\prime }}(t_{k+1}))^{k+1})dt - \int _{0}^{\infty }(1-\bar{F}_{T^{\prime }}(t_k))^k\bar{F}_{T^{\prime }}\left(t_{k+1}\... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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a515e5c1b60d1c6fca2afee4ceff3f1b6d8a8842 | subsection | 30 | 95 | Introduction | Since F_{T^{\prime }}(0) = 0, we can compute the Master's mean waiting time \mathbb {E}\left[T_{\text{SC}}(k+2,k,z)\right] as
\mathbb {E}\left[T_{\text{SC}}(k+2,k,z)\right]&= \int _{0}^{\infty }(1 - F_{T^{\prime }}(t_{k+2})^{k+2}) dt - \int _{0}^{\infty }(k+2)\bar{F}_{T^{\prime }}(t_{k+2})F_{T^{\prime }}(t_{k+1})^{k+1... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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f85f70d809f6ea2ab7cfd9858e12d4891ffae908 | subsection | 31 | 95 | Introduction | Using the binomial expansion and integrating the exponential function \bar{F}_{T^{\prime }}(t)=\exp (-\lambda (k-z)t), we get
\mathbb {E}\left[ T_{\text{SC}}(k+2,k,z)\right]&
=\frac{c}{k-z+2}+\sum _{i=1}^{k+2}\frac{(-1)^i\binom{k+2}{i}}{\lambda }\left[\dfrac{i\exp \left(-\frac{\lambda c}{k-z+1}\right)}{(k-z+1)i+ 1}-\d... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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356b542e58f6e1cc9db630cb2fd5b54b5296ccae | subsection | 32 | 95 | Introduction | The idea is to divide the workers into two disjoint groups and ask each of them to securely multiply A by a vector that is statistically independent of \mathbf {x}. Then, the Master decodes A\mathbf {x} from the results of both multiplications, as described next. {\fontfamily {cmtt}\selectfont \text{M}} divides the wor... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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ede8c1567a9d3219002dbc1990d263777008fed7 | subsection | 33 | 95 | System Model | We consider a Master server {\fontfamily {cmtt}\selectfont \text{M}} which wants to perform intensive computations on confidential data represented by an m \times \ell matrix A (typically m>>\ell ). In machine learning applications m denotes the number of data points (examples) possessed by {\fontfamily {cmtt}\selectfo... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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0.... |
451545b95f0e7e1b1362585e7d557a53fcb996c1 | subsection | 34 | 95 | Computations model | We focus on linear computations. The motivation is that a building block in several iterative machine learning algorithms, such as gradient descent, is the multiplication of A by a sequence of \ell \times 1 attribute vectors \mathbf {x}^1, \mathbf {x}^2, \dots .
In the sequel, we focus on the multiplication A\mathbf {x... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
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... |
6e8b3df715304b4349f72d422b117a93866520a3 | subsection | 35 | 95 | Workers model | The workers have the following properties:
[label=0)]The workers incur random delays while executing the task assigned to them by {\fontfamily {cmtt}\selectfont \text{M}} resulting in what is known as the straggler problem , , .
We model all the delays incurred by each worker by an independent and identical shifted exp... | {
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{
"arxiv_id": "",
"doi": "10.21276/ijre.2018.5.5.4",
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"raw": "J. Dean and S. Ghemawat, “Mapreduce: simplified data processing on large clusters,” Communications of the ACM, vol. 51, no. 1, pp. 107–113, 20... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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7ed8c20f21ecc096c2c526fd1d3280ad742a3252 | subsection | 36 | 95 | Workers model | Decoding A requires a fraction \alpha _d b sub-shares, \alpha _d \triangleq \frac{(k-z)}{(d-z)}, from any of the d shares, d\in \lbrace k,\dots ,n\rbrace .
We show that Staircase codes outperform classical codes in terms of incurred delays.
Delay model
Let T_A be the random variable representing the time spent to comp... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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... |
2f5d8016ca1f8b2af0c2b3d9ad9c71c193ed77bc | subsection | 37 | 95 | Workers model | From this interpretation, it is easy to verify that the d^{\text{th}} order statistic T_{(d)} of (T_1, T_2, \dots , T_n) can be expressed as
T_{(d)}=T^{\prime }_{(d)}+c/(k-z),
where T^{\prime }_{(d)} is the d^{\text{th}} order statistic of n iid exponential random variables with rate \lambda (k-z). Therefore, we can ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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c474f0f601d16f6a10ad91f4c659d7abe61f54ee | subsection | 38 | 95 | Workers model | Theorem 1 (Bounds on the Master's mean waiting time \mathbb {E}{[T_\text{SC}] })
Let H_n be the n^{\text{th}} harmonic sum defined as H_n \triangleq \sum _{i=1}^n \frac{1}{i},
with the notation H_0 \triangleq 0.
The mean waiting time of the Master \mathbb {E}[T_{\text{SC}}] for an (n,k,z) Staircase coded system is upp... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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41cbe7b6e4aa075ff960e596fff3c62264cf9912 | subsection | 39 | 95 | Workers model | \mathbb {E}\left[T_{\text{SC}}(k+1,k,z)\right] &= \frac{c}{k-z+1}+\frac{1}{\lambda }\sum _{i=1}^{k+1}(-1)^{i}\binom{k+1}{i}\left[\frac{i\exp \left(\frac{-\lambda c}{k-z}\right)}{(k-z)i+1}-\dfrac{1}{(k-z+1) i} \right].\\
\mathbb {E}[T_{\text{SC}}(k+2,k,z)]&= \mathbb {E}[T_\text{SC}(k+2,k+1,z)]+\frac{1}{\lambda }\sum _{... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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0.0510590523481369,
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0.01808277890086174,
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0.02... |
b08c72dc70a7175498c4c1d4add6c042d9e37cc9 | subsection | 40 | 95 | Workers model | Classical secret sharing
Let A be an m \times \ell matrix with elements drawn uniformly at random from a finite alphabet, e.g., a finite field. An (n,k,z) classical secret sharing (a.k.a. threshold secret sharing) code , allows the Master to encode the data A into n shares and distribute them to n workers, such that an... | {
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{
"arxiv_id": "",
"doi": "10.1145/359168.359176",
"end": 501,
"openalex_id": "https://openalex.org/W2141420453",
"raw": "A. Shamir, “How to share a secret,” Communications of the ACM, vol. 22, no. 11, pp. 612–613, 1979.",
"source_ref_id": "a4fec1fa897e9ccba1b7... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... |
8a174a394564ccbb5e75b6f7a1a9c2022efc31c2 | subsection | 41 | 95 | Workers model | The addition and multiplication are element wise, e.g., for share 2 each element of R is multiplied by 2 and added to the correspondent element in A.
The Master can decode the secret by contacting any k=2 workers, downloading their shares and decoding A and R. Secrecy is ensured, because A is padded with R in each shar... | {
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{
"arxiv_id": "",
"doi": "10.1109/tit.2017.2723019",
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"raw": "R. Bitar and S. El Rouayheb, “Staircase codes for secret sharing with optimal communication and read overheads,” IEEE Transactions on Informat... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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fb003bacb9f37849691fcace3e3f1b9f99e5e401 | subsection | 42 | 95 | Workers model | The matrix M_\text{SC} is the concatenation of h matrices M_i, i=1,\dots ,h, shownIn (REF ) the dimensions of the rows are scaled by m/(k-z)b for clarity of presentation. in (REF ). Each matrix M_i consists of the b_i sub-tasks downloaded by the Master when decoding from d_i workers, i.e., when there are n-d_i straggle... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.007472969591617584,
0.030... |
9c5f9525055a8fde472b0c33a5294b7ee4a63d2c | subsection | 43 | 95 | Workers model | \normalsize {1}
{0.93}{!}{
\begin{}[baseline=(current bounding box.center)]
{stealth} = [draw=none,text=black]
\node [stealth] (1) at (0,0){
M_1=\hspace{8.5359pt} \begin{bmatrix}{2}{*}{\mathcal {S}} \\
\\
\mathcal {R}_1\\
\end{bmatrix}\quad ,};
\end{}\normalsize {1}
}
stealth] (2) [right=0.3cm of 1] M_2=\hspace{8.5359p... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0... |
00efe143076c20c0c0d3129f55fd9c7ce4bb5ead | subsection | 44 | 95 | Workers model | The \mathbf {0}'s are the all zero matrices used to complete the M_{i}'s to nm(k-z)b rows.
The structure of the matrix M_\text{SC}, called Staircase structure, allows the Master to decode the secret and achieve optimal communication and read overheads \fontfamily {euf}\selectfont \text{CO} and \fontfamily {euf}\selectf... | {
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{
"arxiv_id": "",
"doi": "10.1109/tit.2017.2723019",
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"raw": "R. Bitar and S. El Rouayheb, “Staircase codes for secret sharing with optimal communication and read overheads,” IEEE Transactions on Informat... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... |
2d4d8a3c746211a7c192c82001a89f3783a6ad8b | subsection | 45 | 95 | Workers model | Bounds on the Master's mean waiting time for all (n,k,z) systems
We derive an upper and a lower bound on the Master's mean waiting time \mathbb {E}[T_\text{SC}(n,k,z)] for all (n,k,z) systems, i.e., we prove Theorem REF . We restate Theorem REF for the sake of presentation.
Theorem 1 (Bounds on the Master's mean waitin... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... |
f48124258e29708eb9cc4b717a3259e9c90624dc | subsection | 46 | 95 | Workers model | Theorem (Renyi )
The d^{\text{th}} order statistic T^{\prime }_{(d)} of n iid exponential random variables T^{\prime }_i
is equal to the following random variable in the distribution
T^{\prime }_{(d)} &\triangleq \sum _{j=1}^{d}\frac{T^{\prime }_j}{n-j+1}.
Using Renyi's Theorem, the mean of the d^{\text{th}} order... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 259,
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"raw": "A. Rényi, “On the theory of order statistics,” Acta Mathematica Academiae Scientiarum Hungarica, vol. 4, no. 3-4, pp. 191–231, 1953.",
"source_ref_id": "d6059617eda807f43ad75c951f00a91f7777886... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.02... |
859e6f010651a2d7c5bf7c179d4fc1e3c090afac | subsection | 47 | 95 | Workers model | Since the minimum of the sum is greater than the sum of the minimums,
we can lower bound the waiting time T_{\text{SC}} in terms of residual waiting time T^{\prime }_{\text{SC}} \triangleq \min \lbrace \alpha _dT^{\prime }_{(d)}: d \in \lbrace k,\dots ,n\rbrace \rbrace , as
T_{\text{SC}} =\min _{d\in \lbrace k,\dots ,... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.04275... |
f03cfc2c9e530075fbf9596a9bb49133857b5365 | subsection | 48 | 95 | Workers model | Next, we evaluate \Pr (\mathcal {C}_d(t)) explicitly.
To this end, we first observe that {\alpha _j}^{-1}-{\alpha _{j-1}}^{-1}={(k-z)}^{-1} identically for each j \in \lbrace 1, \dots , n\rbrace .
Further, we apply Renyi's Theorem and independence of residual times T^{\prime }_is
to write
\Pr \left(\mathcal {C}_d(t) ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... |
4a9888b559240b76d15a82a13186de5180dfc9b9 | subsection | 49 | 95 | Workers model | Exploiting the exponential form of \bar{F}(t), aggregating results from (REF ), (REF ) and (REF ), we can re-write (REF ) as
\Pr \left(\mathcal {C}_d(t)\right) &=
\sum _{i=0}^{k-1}\binom{n}{i}\sum _{j=0}^{i}\binom{i}{j}(-1)^{j}\bar{F}\big ( t(n-i+j)(d-z) +t (n-d)(n-d+1)/2\big ).
The proof follows from the integral \... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.029... |
ed1c83621eadd0141ba97747f22885149ee1d4d2 | subsection | 50 | 95 | Workers model | Theorem 4 (Integral expression leading to F_{T_{\text{SC}}}(t))
The distribution of the Master's waiting time T_{\text{SC}} of an (n,k,z) system using Staircase codes is given by
F_{T_{\text{SC}}}\left( t \right)=
1-n!\int _{(y_k, \dots , y_n)\in A(t)}\frac{F_{T^{\prime }}(y_k)^{k-1}}{(k-1)!}dF_{T^{\prime }}(y_k)\do... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... |
ea8727b485fab3e0dc73f58adbdd428236f1542b | subsection | 51 | 95 | Workers model | Both distributions are defined for t>0, and F_{T^{\prime }}(t)\triangleq 1-\exp (-\lambda (k-z)t).
We omit the proof of Corollary REF since it follows from simply integrating (REF ) and defer the proof of Theorem REF to the Appendix.
[Proof of Theorem REF ] Let T^{\prime }_i denote the residual service time of worker ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... |
4b903ad707165498ec51b5e1ac0903d4fea2647f | subsection | 52 | 95 | Workers model | For each k \le j \le n, we define t_j \triangleq \max \left\lbrace \frac{t}{\alpha _j} - \frac{c}{j-z},0\right\rbrace , y_{n+1} \triangleq \infty , and \hat{A}(t) \triangleq \cap _{j=k}^{n+1}\lbrace t_j < y_j \le y_{j+1}\rbrace \cap _{j=1}^{k-1}\lbrace 0 \le y_j \le y_{j+1} \rbrace .
In terms of t_j, y_{n+1} and \hat{A... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... |
d3c614aeb9e871554670b4034d9d9b3b7fc9919b | subsection | 53 | 95 | Workers model | This can be shown by writing the integral I_{k+1} in (k+1) integration variables y_1, \dots , y_{k+1} in terms of the integral I_k,
and evaluating the integral by substituting the induction hypothesis for I_k as follows
I_{k+1}
&=\int _{0}^{y_{k+1}}I_kdF_{T^{\prime }}(y_k) =\int _{0}^{y_{k+1}} \frac{F_{T^{\prime }}(y_... | {
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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0... |
ce31443b02fbaa886f99a9701bee0730136365d7 | subsection | 54 | 95 | Workers model | Recall that d : \mathbb {R}_+^n \rightarrow \lbrace k, \ldots , n\rbrace is a function of the compute times T_1, \ldots , T_n.
d(T_1, T_2, \dots , T_n)\triangleq \arg \min \left\lbrace \frac{k-z}{i-z}T_{(i)}: i\in \lbrace k,\dots ,n\rbrace \right\rbrace .
Claim 8 The number of workers d that minimize the waiting time... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.04106731340289116,
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0.... |
453a0ee626177bfbc30aa187524a2dfd144f6159 | subsection | 55 | 95 | Workers model | Therefore, we can apply the McDiarmid's inequality to obtain the concentration bound on d.
Simulations
We use the normalized difference between the mean waiting time of Staircase codes and classical secret sharing codes as a performance metric for Staircase codes. We refer to this metric as the savings. Using the resu... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... |
c52692eb71409770aa82b9fb1ad0dc4bb4688664 | subsection | 56 | 95 | Workers model | Similarly to Figure , we consider systems with z=1, \lambda =1 and vary c.][Table: Comparison of the performance of Staircase codes on Amazon EC2 to the theoretical bound in () and the value obtained by simulations assuming the shifted exponential model in Section . The shift c^* and the rate \lambda ^* of the workers ... | {
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} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... |
260cfa092c150285af96ab0e51ad21812bebccc5 | subsection | 57 | 95 | Workers model | These results are also summarized in Table REF .
Note that for this set of implementation, the Master's data A is a matrix of size 378000\times 250 with entries generated uniformly at random from \lbrace 1,\dots ,255\rbrace . We run 1000 multiplications of A by a randomly generated vector \mathbf {x}.
[Figure: An (n,k,... | {
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"doi": "10.1007/978-3-319-25958-1_8",
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"raw": "E. Learned-Miller, G. B. Huang, A. Roy Chowdhury, H. Li, and G. Hua, “Labeled faces in the wild: A survey,” in Advances in face detection ... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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bff5811a7ddf2cdec842d5ea7b1889c29ec53cc1 | subsection | 58 | 95 | Workers model | \mathbb {E}\left[T_{\text{SC}}(k+1,k,z)\right] &= \frac{c}{k-z+1}+\frac{1}{\lambda }\sum _{i=1}^{k+1}(-1)^{i}\binom{k+1}{i}\left[\frac{i\exp \left(\frac{-\lambda c}{k-z}\right)}{(k-z)i+1}-\dfrac{1}{(k-z+1) i} \right].\\
\mathbb {E}[T_{\text{SC}}(k+2,k,z)]&= \mathbb {E}[T_\text{SC}(k+2,k+1,z)]+\frac{1}{\lambda }\sum _{... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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129d084fffedc30c71e8e98c506ed15ebb938983 | subsection | 59 | 95 | Workers model | Since F_{T^{\prime }}(0) = 0, we can compute the Master's mean waiting time \mathbb {E}\left[ T(k+1,k,z)\right] as
\mathbb {E}\left[ T_{\text{SC}}(k+1,k,z)\right]&=\int _{0}^{\infty }(1- (1-\bar{F}_{T^{\prime }}(t_{k+1}))^{k+1})dt - \int _{0}^{\infty }(1-\bar{F}_{T^{\prime }}(t_k))^k\bar{F}_{T^{\prime }}\left(t_{k+1}\... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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f6e276c250aa7710e61033231caf61cbbe5838ce | subsection | 60 | 95 | Workers model | Since F_{T^{\prime }}(0) = 0, we can compute the Master's mean waiting time \mathbb {E}\left[T_{\text{SC}}(k+2,k,z)\right] as
\mathbb {E}\left[T_{\text{SC}}(k+2,k,z)\right]&= \int _{0}^{\infty }(1 - F_{T^{\prime }}(t_{k+2})^{k+2}) dt - \int _{0}^{\infty }(k+2)\bar{F}_{T^{\prime }}(t_{k+2})F_{T^{\prime }}(t_{k+1})^{k+1... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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82d6e20a6fb89e5b5f9b8a266ca6774fedb39ab4 | subsection | 61 | 95 | Workers model | Using the binomial expansion and integrating the exponential function \bar{F}_{T^{\prime }}(t)=\exp (-\lambda (k-z)t), we get
\mathbb {E}\left[ T_{\text{SC}}(k+2,k,z)\right]&
=\frac{c}{k-z+2}+\sum _{i=1}^{k+2}\frac{(-1)^i\binom{k+2}{i}}{\lambda }\left[\dfrac{i\exp \left(-\frac{\lambda c}{k-z+1}\right)}{(k-z+1)i+ 1}-\d... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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c3d87130442c4d09fdfed9ff42d60d43fc5555e8 | subsection | 62 | 95 | Workers model | The idea is to divide the workers into two disjoint groups and ask each of them to securely multiply A by a vector that is statistically independent of \mathbf {x}. Then, the Master decodes A\mathbf {x} from the results of both multiplications, as described next. {\fontfamily {cmtt}\selectfont \text{M}} divides the wor... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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5cba4da9af8201efb4254d62050edc4983505567 | subsection | 63 | 95 | Our Results | Our results characterize the delay performance of secure coded computing when using Staircase codes and compare it to classical secret sharing codes. The performance of Staircase codes is reflected in the Master's waiting time T_{\text{SC}}.
Towards our goal, we establish in Theorem REF general bounds on the Master's m... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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d1b750275b73940a0bf58fa06581b45664982a35 | subsection | 64 | 95 | Our Results | Using the general integral expression, we derive the exact expression of the CDF F_{T_\text{SC}}(t) for systems with n=k+1 and n=k+2 as stated in the next Theorem.Theorem 2 (Exact expression of \mathbb {E}{[T_\text{SC}]} for systems with up to 2 stragglers)
The mean waiting time of the Master for (k+1,k,z) and (k+2,k... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... |
9ea217976338dcdcdfd2198a923d53ad20951fff | subsection | 65 | 95 | Staircase codes | Staircase codes are the main ingredient of our scheme. The goal of this section is to explain the encoding and decoding of Staircase codes that are necessary for our delay analysis. Before we explain Staircase codes, we start by briefly explaining the encoding and decoding of classical secret sharing codes, which can b... | {
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{
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
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22218f6b5a7d22e6d47a90583b5be6539e265285 | subsection | 66 | 95 | Staircase codes | The random matrices R_1,\dots ,R_{zb} are partitioned into h matrices \mathcal {R}_i, i=1,\dots ,h, each of dimension zm/(k-z)b \times \ell (k-z)b/b_ib_{i-1} with b_0=1.The matrix M_\text{SC} is the concatenation of h matrices M_i, i=1,\dots ,h, shownIn (REF ) the dimensions of the rows are scaled by m/(k-z)b for clari... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.026... |
071eb87c9c7866167146ca4bd54179d5139b29f0 | subsection | 67 | 95 | Staircase codes | Each matrix M_i consists of the b_i sub-tasks downloaded by the Master when decoding from d_i workers, i.e., when there are n-d_i stragglers.\normalsize {1}
{0.93}{!}{
\begin{}[baseline=(current bounding box.center)]
{stealth} = [draw=none,text=black]
\node [stealth] (1) at (0,0){
M_1=\hspace{8.5359pt} \begin{bmatrix}{... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.027815675362944603,
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b3c60028202ed4a1a545ae044ce19a1dad430b74 | subsection | 68 | 95 | Staircase codes | The \mathbf {0}'s are the all zero matrices used to complete the M_{i}'s to nm(k-z)b rows.The structure of the matrix M_\text{SC}, called Staircase structure, allows the Master to decode the secret and achieve optimal communication and read overheads \fontfamily {euf}\selectfont \text{CO} and \fontfamily {euf}\selectfo... | {
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{
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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f397673dafbbde12d22a621d5f0b0ffb8d3a4032 | subsection | 69 | 95 | Classical secret sharing | Let A be an m \times \ell matrix with elements drawn uniformly at random from a finite alphabet, e.g., a finite field. An (n,k,z) classical secret sharing (a.k.a. threshold secret sharing) code , allows the Master to encode the data A into n shares and distribute them to n workers, such that any set of z, z<k<n, worker... | {
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{
"arxiv_id": "",
"doi": "10.1145/359168.359176",
"end": 476,
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"raw": "A. Shamir, “How to share a secret,” Communications of the ACM, vol. 22, no. 11, pp. 612–613, 1979.",
"source_ref_id": "a4fec1fa897e9ccba1b7... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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83d511081e63e2a16600fa00cd88d1c466b05edf | subsection | 70 | 95 | Classical secret sharing | The addition and multiplication are element wise, e.g., for share 2 each element of R is multiplied by 2 and added to the correspondent element in A.
The Master can decode the secret by contacting any k=2 workers, downloading their shares and decoding A and R. Secrecy is ensured, because A is padded with R in each shar... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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4c2084bb1271cecdaef21968642c867e80b1ab92 | subsection | 71 | 95 | Bounds on the Master's mean waiting time for all | We derive an upper and a lower bound on the Master's mean waiting time \mathbb {E}[T_\text{SC}(n,k,z)] for all (n,k,z) systems, i.e., we prove Theorem REF . We restate Theorem REF for the sake of presentation.Theorem 1 (Bounds on the Master's mean waiting time \mathbb {E}{[T_\text{SC}] }) Let H_n be the n^{\text{th}} h... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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714b69e4f911fa37975e8b59d3f61554874454f0 | subsection | 72 | 95 | Proof of the upper bound on the mean waiting time | We use Jensen's inequality to upper bound the mean waiting time \mathbb {E}[T_{\text{SC}}]. Since \min is a convex function, we can use Jensen's inequality to upper bound the mean waiting time,\mathbb {E}[T_{\text{SC}}] &= \mathbb {E}\left[\min _{d\in \lbrace k,\dots ,n\rbrace } \left\lbrace \alpha _dT^{\prime }_{(d)}+... | {
"cite_spans": [
{
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"raw": "A. Rényi, “On the theory of order statistics,” Acta Mathematica Academiae Scientiarum Hungarica, vol. 4, no. 3-4, pp. 191–231, 1953.",
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Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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6f28d1dec953ee2ee1bcfd468af8850db6b9ff64 | subsection | 73 | 95 | Proof of the upper bound on the mean waiting time | Alternatively, we can use the upper and lower bounds \log (n)< H_n < \log (n+1) on the Harmonic number H_n,
to upper bound the mean waiting time\mathbb {E}[T_{\text{SC}}]<\min &\left\lbrace \min _{d\in \lbrace k,\dots ,n-1\rbrace }\left\lbrace \dfrac{1}{\lambda (d-z)}\log \left(\dfrac{n+1}{n-d}\right)+\frac{c}{d-z}\rig... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
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ef55dd8ba32892ea1e71a4de370a5d6c58329f55 | subsection | 74 | 95 | Proof of the lower bound on the mean waiting time | Recall that T_{SC}= \min \lbrace \alpha _d T_{(d)}: d \in \lbrace k,\dots ,n\rbrace \rbrace =\min \lbrace \alpha _d T^{\prime }_{(d)}+\dfrac{c}{d-z}: d \in \lbrace k,\dots ,n\rbrace \rbrace . Since the minimum of the sum is greater than the sum of the minimums,
we can lower bound the waiting time T_{\text{SC}} in terms... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05297177657485008,
-0.003339342772960663,
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0.03771492838859558,
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0.020047498866915703,
-0.015928149223327637,
0.0... |
b121e820daac48e9d431afe19452cd961a1502d0 | subsection | 75 | 95 | Proof of the lower bound on the mean waiting time | For the residual service times T^{\prime }_1,\dots , T^{\prime }_n, we consider the following set\mathcal {C}_d(t) &\triangleq \left\lbrace T^{\prime }_{(k)} > \frac{t}{\alpha _d}\right\rbrace \bigcap _{i=d+1}^n\left\lbrace T^{\prime }_{(i)}-T^{\prime }_{(i-1)} > \frac{t}{\alpha _i} - \frac{t}{\alpha _{i-1}}\right\rbra... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.062014639377593994,
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0.03198392689228058,
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0.012108419090509415,
-0.04260454699397087,
0.06... |
9daf908114badbdb0b3f2fd00bbff0aa0b6e1f3a | subsection | 76 | 95 | Proof of the lower bound on the mean waiting time | Utilizing the exponential form, we can write\prod _{j=d+1}^{n}\Pr \left\lbrace \frac{T^{\prime }_j}{n-j+1} > \frac{t}{(k-z)}\right\rbrace &=\bar{F}\left(\sum _{j=d+1}^{n}{(n-j+1)}t\right)=\bar{F}\left(\dfrac{(n-d)(n-d+1)t}{2}\right).From definition, it follows that \alpha _k = 1.
Further, the k^{\text{th}} order statis... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... |
40b4813d30d7e056350ae1e6f3e918cb28946ed7 | subsection | 77 | 95 | Proof of the lower bound on the mean waiting time | That is,\Pr \left\lbrace T^{\prime }_{(k)}> t\right\rbrace &=\sum _{i=0}^{k-1}\binom{n}{i}F\left({(k-z)t} \right)^i \bar{F}\left({(k-z)t}\right)^{n-i}.Since F(t) = 1 - \bar{F}(t), using the binomial expansion, we haveF\left({(k-z)t}\right)^i=
\sum _{j=0}^{i}\binom{i}{j}(-1)^{j}\bar{F}\left({(k-z)t}\right)^{j}.Exploitin... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.054086800664663315,
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0.02496783435344696,
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0.026845000684261322,
-0.056772828102111816,
0.... |
d822be474ce2c0b71c464b0d291a1c3f97189c0b | subsection | 78 | 95 | Proof of the lower bound on the mean waiting time | The d^\text{th} order statistic is greater than t, if and only if at most d-1 out of n iid random variables (T_1,\dots , T_n) can be less than t, and the rest are greater than t. | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05450298637151718,
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0.020858224481344223,
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-0.028273804113268852,
... |
2de6aa0e7106fa5d1629d9c7b11f12496e1f81c4 | subsection | 79 | 95 | Distribution of the Master's waiting time for all | Now we are ready to derive an integral expression for the probability distribution of T_{\text{SC}}, the Master's waiting time when using Staircase codes.Theorem 4 (Integral expression leading to F_{T_{\text{SC}}}(t))
The distribution of the Master's waiting time T_{\text{SC}} of an (n,k,z) system using Staircase cod... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04393567144870758,
-0.009626183658838272,
-0.05266178399324417,
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0.023081481456756592,
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0.02382899820804596,
-0.00019212320330552757,
0.035148534923791885,
0.0025857973378151655,
0.024896880611777306,
-0.02559863030910492,... |
b251d949f60890e18b3e4d2246c8738ece98ae74 | subsection | 80 | 95 | Distribution of the Master's waiting time for all | \\
F_{T_{\text{SC}}(k+2,k,z)}(t) &= F_{T^{\prime }}(t_{k+2})^{k+2} + (k+2)\bar{F}_{T^{\prime }}(t_{k+2})\Big [F_{T^{\prime }}(t_{k+1})^{k+1} + (k+1)F_{T^{\prime }}(t_k)^k(\bar{F}_{T^{\prime }}(t_{k+1}) -\frac{1}{2}\bar{F}_{T^{\prime }}(t_{k+2}))\Big ].Both distributions are defined for t>0, and F_{T^{\prime }}(t)\trian... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.06871503591537476,
0.007307837717235088,
-0.040154967457056046,
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0.048942681401968,
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0.003590986132621765,
-0.024669673293828964,
0.... |
56d683070e0e791c74dde75e6bf11e08dac2ba85 | subsection | 81 | 95 | Distribution of the Master's waiting time for all | The product form of joint density follows from the independence of the residual service times.In terms of \alpha _j = \frac{k-z}{j-z}, the order statistics of residual times T^{\prime }_{(j)}, and the offset \frac{c}{k-z}, we can write\left\lbrace T_{\text{SC}} > t \right\rbrace &= \bigcap _{j = k}^n\left\lbrace T^{\pr... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02182072401046753,
0.009773547761142254,
-0.03897211700677872,
-0.0051118480041623116,
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0.01040680706501007,
0.031083086505532265,
0.015213469974696636,
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0.035798195749521255,
-0.029465606436133385,
0.034974195063114166,
-0.03390604630112648,
0.... |
b21dbada46702e13829c2c094fe202aa35859776 | subsection | 82 | 95 | Distribution of the Master's waiting time for all | In terms of t_j, y_{n+1} and \hat{A}(t),
we can write the tail distribution\Pr \lbrace T_{\text{SC}} > t\rbrace = \int _{y \in \hat{A}(t)}dF_{T^{\prime }_{(1)},\dots ,T^{\prime }_{(n)}}(y) = n!\int _{t_n}^{\infty }\cdots \int _{t_k}^{y_{k+1}}\prod _{i=k}^ndF_{T^{\prime }}(y_i)\left(\int _{0}^{y_k}\cdots \int _{0}^{y_{2... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.07129694521427155,
0.014863704331219196,
0.006871029268950224,
-0.0208000298589468,
0.020037004724144936,
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0.001789290807209909,
0.005894359201192856,
0.05023748800158501,
-0.033145755529403687,
0.03390877693891525,
-0.018221009522676468,
0.03... |
b0b95824941a54413d3730e63a0090435b9acf2e | subsection | 83 | 95 | Interplay between code design and latency | Universal Staircase codes allows the master to decode A\mathbf {x} from any random number d of workers, k\le d \le n. The downside is that the universal construction requires a large number of sub-tasks b=\text{LCM}\lbrace k-z+1,\dots ,n-z\rbrace . In many applications, there may be an overhead associated with excessiv... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tit.2017.2723019",
"end": 706,
"openalex_id": "https://openalex.org/W2731854684",
"raw": "R. Bitar and S. El Rouayheb, “Staircase codes for secret sharing with optimal communication and read overheads,” IEEE Transactions on Informat... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.011159174144268036,
0.004729110281914473,
-0.0792049691081047,
-0.008085252717137337,
-0.0001381310139549896,
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0.04899968206882477,
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0.024087950587272644,
0.020426703616976738,
... |
9a89037e285cfb62a6d24f5add0c18398ff9b113 | subsection | 84 | 95 | Interplay between code design and latency | That is, for each i \in [n] taking t,t^{i} \in \mathbb {R}_+^n such that t_j = t^i_j for each j \in [n] \setminus \lbrace i\rbrace and t_i \ne t^i_i,\sup \lbrace |g(t) - g(t^i)|: t, t^i \in \mathbb {R}_+^n\rbrace \le {n-k}.The claim follows from the fact that d\in \lbrace k,\dots ,n\rbrace .
We prove the tightness of (... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.026892177760601044,
0.012583768926560879,
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0.007608258165419102,
0.021428273990750313,
-0.007509053219109774,
-0.008279799483716488,
-0.05073189735412598... |
f4289d667a677aa371cc50fbe94ee5ccdd567723 | subsection | 85 | 95 | Simulations | We use the normalized difference between the mean waiting time of Staircase codes and classical secret sharing codes as a performance metric for Staircase codes. We refer to this metric as the savings. Using the result of Theorem REF , we can get a lower and an upper bound on the savings brought by Staircase codes. The... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.029349887743592262,
0.022729381918907166,
-0.0726119875907898,
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0.0020650941878557205,
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0.020685262978076935,
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0.036733124405145645,
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0.02196665108203888,
-0.02082255482673645,
0... |
9b6f72fd03191e423a1a9d6872e48fbc215f63a8 | subsection | 86 | 95 | Simulations | Similarly to Figure , we consider systems with z=1, \lambda =1 and vary c.][Table: Comparison of the performance of Staircase codes on Amazon EC2 to the theoretical bound in () and the value obtained by simulations assuming the shifted exponential model in Section . The shift c^* and the rate \lambda ^* of the workers ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0578073114156723,
0.019869355484843254,
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0.015390357002615929,
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0.029422517865896225,
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0.02475276030600071,
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0.007069492246955633,
-0.02511901594698429,
... |
f1fb294c18f2563f6b6fd7815e17ca35e8a2c5e6 | subsection | 87 | 95 | Implementation and Validation of the Theoretical Model | We describe a representative sample of our implementation on Amazon EC2 clusters and discuss our observations. In Section REF , we present traces for systems with fixed rate k/n=1/2 (Figure REF ). We noticed that the straggler behavior, and therefore the savings, can depend on the date and time of the implementation. T... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.040168046951293945,
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0.023548364639282227,
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0.013613183051347733,
-0.0017064170679077506,
... |
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