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b19c43da8caf14349b719dfc5565b398ff717a6b | subsection | 11 | 73 | Prior on | Note that, by the independence of the components of the d-dimensional Brownian motion, the components \left\lbrace \left\lbrace {X}_{j}(t);\ 0 \le t \le T \right\rbrace ;\ j \in [d] \right\rbrace of \left\lbrace {X}(t);\ 0\le t\le T \right\rbrace are independent.More involved correlation models of \left\lbrace \left\lb... | {
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43b82245566c2f385aba7a117cdad09a58239292 | subsection | 12 | 73 | Prior on | In this case, the matrices A and Q from driftmatrix are given by{\normalcolor A(h)_{ij}}
&=
{\left\lbrace \begin{array}{ll}
\mathbb {I}_{i\le j} \frac{h^{j-i}}{(j-i)!}, & \mbox{if }j\ne q, \\ {\normalcolor \frac{h^{q-i}}{(q-i)!} - \theta \sum _{k=q+1-i}^{\infty } \frac{(-\theta )^{k+i-q-1} h^k}{ k! }} , & \mbox{if }j=q... | {
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75c1c0a76d5010ed3ecef4cd451e9773df379b98 | subsection | 13 | 73 | The algorithm | To avoid the introduction of additional indices, we will define the algorithm {\Psi } for d=1; for statements on the general case of d\in \mathbb {N} we will use the same symbols from eq:C-predict–eq:CupdateMM as vectors over the whole dimension—see e.g. bound:r for a statement about a general r \in \mathbb {R}^d.
By t... | {
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bd64fa0e3c20f17357fe1d87fd9444dd896dc03b | subsection | 14 | 73 | The algorithm | In the subsequent step, the following quantities are computed first\beta ^{(i)}(t+h)
&\frac{P^-(t+h)_{i1}}{(P^-(t+h))_{11} + R(t+h)} \in \mathbb {R}, \quad \text{(Kalman gain on $i$\textsuperscript {th} state)},
\\
y(t+h)
&f\left( m^{-,(0)}(t+h) \right) \in \mathbb {R}, \qquad \text{(measurement/data on $\dot{x}$)},
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8935572fcc954d8e16394bae7561944578acc3f3 | subsection | 15 | 73 | The algorithm | Concerning the data generation mechanism for y eq:defyMAP, we only consider the maximum-a-posteriori point estimate of \dot{x}(t) given \mathcal {N}(m^{-,(0)}(t),P_{00}^-(t)); a discussion of more inclined statistical models for y can be found in .
Next, for lack of such a discussion for R, we will examine different ch... | {
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f44297150a1ae880c6654270ac12961676537b88 | subsection | 16 | 73 | Measurement noise | Two sources of uncertainty add to R(t): noise from imprecise knowledge of x(t) and f.
Given f, previous integration steps of the filter (as well as an imprecise initial value) inject uncertainty about how close m^-(t) is to x(t) and how close y = f(m^-(t)) is to \dot{x}(t)) = f(x(t)).
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d11780c341f1b8971afa4227c911ecdc0ec46ef7 | subsection | 17 | 73 | Regularity of flow | Before we proceed to the analysis of {\Psi }, we provide all regularity results necessary for arbitrary q,d \in \mathbb {N} in this section.Assumption 1
The vector field f \in C^{q} ( \mathbb {R}^d; \mathbb {R}^d ) and all its derivatives of order up to q are uniformly bounded and globally Lipschitz, i.e. there exists... | {
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a72c8a0f1f0eb02be3406c65a7d9125ec3e9249a | subsection | 18 | 73 | Regularity of flow | Now, applying \Vert \Phi ^{(q+1)} \Vert \le L to the term \Phi _{\tau }^{(q+1)}(a) (for some \tau \in (0,h)) in the Lagrange remainder of the (q-i)th-order Taylor expansion of \Phi _h^{(i)}(a) yields eq:Taylorexpansion.
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\f... | {
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b1d81ce1a7d536dcc8ee0417a62b84a35532cd93 | subsection | 19 | 73 | Regularity of flow | The norm \cdot
h is used to make rigorous the intuition that the estimates of the solution^{\prime }s time derivative are `one order of h worse per derivative^{\prime }.We bound the second summand of{\left|\hspace{-1.0625pt}\left|\hspace{-1.0625pt}\left|{\Phi }_h(a) - {\Phi }_h(b)
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a2fdbd8283c5fb2315511ca318844a55f2a68b68 | subsection | 20 | 73 | Auxiliary bounds on intermediate quantities | The ODE filter \Psi iteratively computes the filtering mean {m}(nh)=( m^{(0)}(nh),\dots , m^{(q)}(nh) )^{\intercal } \in \mathbb {R}^{(q+1)} as well as error covariance matrices P(nh) \in \mathbb {R} on the mesh \lbrace nh \rbrace _{n=0}^{T/h}.
Ideally, the truncation error over all derivatives{\normalcolor {\varepsilo... | {
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efc54671b106d8b75dc2870239813969d1a93c1e | subsection | 21 | 73 | Auxiliary bounds on intermediate quantities | To capture this difference, let us first, for all i \in [q+1], recursively define f^{(i)} \colon \mathbb {R}^d \rightarrow \mathbb {R}^d by f^{(0)}(a) = a, f^{(1)}(a) = f(a) and f^{(i)}(a) = \left( \nabla _x f^{(i-1)} \cdot f \right)(a).
This implies{\normalcolor \Phi _0^{(i)} (m^{(0)}(nh) ) = f^{(i-1)}(m^{(0)}(nh)), }... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
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"Philipp Hennig"
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0e9c86d4308c98fab7b120de910c8218c471aae6 | subsection | 22 | 73 | Auxiliary bounds on intermediate quantities | We apply the triangle inequality to the definition of \Delta ^{-(i)}((n+1)h), as defined in def:Delta-, which, by Aprediction, yields\Delta ^{-(i)}((n+1)h)
&\le \sum _{k=i}^q \frac{h^{k-i}}{(k-i)!} \delta ^{(k)}(nh) + {\normalcolor K \theta \left|m^{(q)}(nh) \right|h^{q+1-i}}
\\
&\quad +
\underbrace{\left|\sum _{l=i}^q... | {
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5491bd12dc7367c117e93200e2e0f02019d4cbde | subsection | 23 | 73 | Auxiliary bounds on intermediate quantities | Application of the orders of A and Q from def:AIOUP,eq:QIOUP, the triangle inequality and ass:assumption2 to the definition of P^- in eq:C-predict yields\left|P^-(h)_{k,l} \right|&\stackrel{{eq:C^-_predict}}{\le }
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&\... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
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3e01df8a95abebe1ff742108ae814cc40b684598 | subsection | 24 | 73 | Local convergence rates | With the above bounds on intermediate algorithmic quantities (involving state misalignments \delta ^{(i)}) in place, we only need an additional assumption to proceed—via a bound on \delta ^{(i)}(0)—to our first main result on local convergence orders of {\Psi }.Assumption 3
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6dd5cfed85c4af90ab0a10bf7d3d9bd2a0682a14 | subsection | 25 | 73 | Local convergence rates | Now, we can bound the local truncation error \varepsilon ^{(0)}(h) as defined in eq:defvarpepsilon.[Local Truncation Error]
Under ass:fGlobalLipschitz,ass:assumption2,ass:eps0bound and for all sufficiently small h>0,\left\Vert \varepsilon ^{(0)}(h) \right\Vert \le {\left|\hspace{-1.0625pt}\left|\hspace{-1.0625pt}\left... | {
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c54cd288fed600321dd844b4302845e9ee9cb88e | subsection | 26 | 73 | Local convergence rates | \end{align}
The remaining bound on (i)(h), for all i [q+1] and sufficiently small h>0, is obtained by insertion of the bounds from {lemma:Delta^-i,lemma:r,lemma:Kalman_gain_local_order} (in the case of n=0), into {def:Delta^-}:
\begin{align}
\Delta ^{(i)}(h)
&\le K \left[ 1 + {\normalcolor \theta \left\Vert m^{(q)}(0) ... | {
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988cccf7f198c378824e49cf74027fe64b4942e7 | subsection | 27 | 73 | Local convergence rates | A global analysis for IOUP is therefore more complicated than for IBM:
Recall from {A_prediction} that, for q=1, the mean prediction for x((n+1)h) is
\begin{align}
\begin{pmatrix}
m^{-,(0)}((n+1)h)
\\
m^{-,(1)}((n+1)h)
\end{pmatrix}
\stackrel{{A_prediction}}{=}
\begin{pmatrix}
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05a8f9924889adf2c9281cbed0ca5218e6b75032 | subsection | 28 | 73 | Global analysis | As explained above, we only consider the case of the IBM prior, i.e. \theta = 0, in this section.
Moreover, as for q \ge 2 the proof of an analogue of proposition:steadystatesglobal would be very technically involved, we restrict our analysis to q=1 in order to maintain readability of this first global analysis.
(See s... | {
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ea704bf89296fc91f2078be7ddcf9efb2af058fd | subsection | 29 | 73 | Outline of global convergence proof | The goal of the following sequence of proofs in subsection:Globaltruncationerror is theorem:GlobalTruncation.
It is proved by a special version of the discrete Grönwall inequality whose prerequisite is provided in lemma:truncationerrorglobalbound.
This lemma:truncationerrorglobalbound follows from lemma:flowmapregulari... | {
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0c17150105e1fdcced5e467e1f4281e9d9c551b7 | subsection | 30 | 73 | Global bounds on Kalman gains | Since we will analyse the sequence of covariance matrices and Kalman gains using contractions in proposition:steadystatesglobal, we first introduce the following generalization of Banach fixed-point theorem (BFT).Let (\mathcal {X},d) be a non-empty complete metric space,
T_n \colon \mathcal {X} \rightarrow \mathcal {X}... | {
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d38d6a007eee3d05fedf78c7f3562f14673aa253 | subsection | 31 | 73 | Global bounds on Kalman gains | Then, for all x_0 \in \mathcal {X}, the recursive sequence x_n T_n(x_{n-1}) converges to u^{\ast } as n \rightarrow \infty .See Appendix:Banach.For constant R \equiv K h^p with p \in [0,\infty ], the unique (attractive) steady states for the following quantities areP_{11}^{-,\infty }
&\lim _{n \rightarrow \infty } P_{1... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
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990885a3e00ba0583c573ff46a2678928c9b6ec8 | subsection | 32 | 73 | Global bounds on Kalman gains | The recursions for P(nh) and P^-(nh) given by eq:C-predict,eq:CupdateMM follow a discrete algebraic Riccati equation (DARE)—a topic studied in many related settings .
While the asymptotic behavior lemma:steadystatesvaluesi of the completely detectable state X^{(1)} can also be obtained using classical filtering theory ... | {
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ed29769ac60eb329580fc9bafa5497ade5140168 | subsection | 33 | 73 | Global bounds on state misalignments | For the following estimates, we restrict the choice of p to be larger than q=1.Assumption 4
The noise model is chosen to be R\equiv Kh^p, for p \in [q,\infty ] = [1,\infty ], where Kh^{\infty } : = 0.Before bounding the added deviation of {\Psi } from the flow {\Phi } per step, a global bound on the state misalignment... | {
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7d2574f1fc15e6d284c44756a1b98c453447826a | subsection | 34 | 73 | Prerequisite for discrete Grönwall inequality | Equipped with the above bounds, we can now prove a bound on the maximal increment of the numerical error stemming from local truncation errors which is needed to prove ineq:lemma:developmentepsilon, the prerequisite for the discrete Grönwall inequality.Under ass:fGlobalLipschitz,ass:assumption2,ass:eps0bound,ass:R and ... | {
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d9d08d7332801c1e1792b2b686a2338a5e1b5528 | subsection | 35 | 73 | Global convergence rates | With the above bounds in place, we can now prove global convergence rates.
[Global truncation error]
Under ass:fGlobalLipschitz,ass:assumption2,ass:eps0bound,ass:R and for all sufficiently small h>0,\max _{n \in [T/h + 1]} \left\Vert \varepsilon ^{(0)}(nh) \right\Vert \le \max _{n\in [T/h + 1]} {\left|\hspace{-1.0625p... | {
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30d9f799d8a096943970f536469540b981bfab94 | subsection | 36 | 73 | Global convergence rates | The convergence rate of this theorem is sharp in the sense that it cannot be improved over all f satisfying ass:fGlobalLipschitz since it is one order worse than the local convergence rate implied by lemma:Psiwithdelta.Using \left\Vert \varepsilon ^{(0)}(nh) \right\Vert \le {\left|\hspace{-1.0625pt}\left|\hspace{-1.062... | {
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e9461145db123d7b07bdf375b17b822f07759b7a | subsection | 37 | 73 | Global convergence rates | Recalling that \left\Vert \varepsilon ^{(0)}(nh) \right\Vert \le {\left|\hspace{-1.0625pt}\left|\hspace{-1.0625pt}\left|{\varepsilon }(nh)
\right|\hspace{-1.0625pt}\right|\hspace{-1.0625pt}\right|}_h, by def:hnorm, concludes the proof. | {
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3f48d3768b609fb8f84d03cfe1d342cc25332467 | subsection | 38 | 73 | Calibration of credible intervals | In PN, one way to judge calibration of a Gaussian output \mathcal {N}(m,V) is to check whether the implied 0.95 credible interval [m-2\sqrt{V},m+2\sqrt{V}] contracts at the same rate as the convergence rate of the posterior mean to the true quantity of interest. For the filter, this would mean that the rate of contract... | {
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457660507f8ddd623e2df11a0a5ca53510d905b5 | subsection | 39 | 73 | Calibration of credible intervals | By P_{01}^-(nh) \le P_{01}(nh) and \vert \beta ^{(1)} \vert \le 1 (due to defbetaMM), application of the triangle inequality to eq:recursionP00- yieldsP_{00}^-\left((n+1)h \right)
\le &
P_{00}^- ( n h )
+
\left|\beta ^{(0)}(nh) \right|\left|P_{01} (nh) \right|+
2hR \left|\beta ^{(0)}(nh) \right|+
h^2 R
+
\frac{\sigma ^... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
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] | [
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4754bfa3c1ab57d4494af28ec0c82f71234b4bb9 | subsection | 40 | 73 | Calibration of credible intervals | Hence, the sole negative summand - \beta ^{\infty ,(0)} P_{01}^{\infty } of eq:recursionP00- is in \Theta (h^{5 \wedge \frac{3p + 7}{2} }) and thereby of higher order than the remaining sum of positive summands:\underbrace{2hR}_{\in \Theta (h^{p+1})}
\underbrace{\beta ^{\infty ,(0)}(nh)}_{\in \Theta (h), \text{ by {lem... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
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a68c52de49ed0568ebe26fe0162de2b0bf174680 | subsection | 41 | 73 | Numerical experiments | In this section, we empirically demonstrate thatthe worst-case convergence rates from theorem:GlobalTruncation hold not only for q=1 but also for q \in \lbrace 2,3,4\rbrace and are sometimes even outpaced in practice (see subsec:exporiginalWPD),
the convergence rates of the credible intervals from theorem:contractiono... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
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96ad647e030fd12694d79b485ae4760c4444c907 | subsection | 42 | 73 | Global Convergence Rates for | We consider two test IVPs with closed-form solutions which allow the computation of exact global errors: the logistic equation\dot{x}(t)
=
\lambda _0 x(t)\left( 1 - x(t)/ \lambda _1 \right)
,
\ \forall t \in [0,1.5],
\qquad \textrm { with }
(\lambda _0,\lambda _1) = (3,1)
\textrm { and }
x(0)
=
0.1,which has the logist... | {
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dbf468c5955632bb996901ab8686f332a57fe3f2 | subsection | 43 | 73 | Global Convergence Rates for | The (dash-)dotted gray lines visualize idealized convergence rates of orders one to five. The upper and lower rows employ the minimal R\equiv 0 and maximal measurement variance R \equiv K_R h^q (K_R = 1) which are permissible under ass:R.]In the case of R \equiv 0, even (q+1)th order convergence rates appear to hold tr... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
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cd77c3248dfbc6d57b1b860082d93904091a7d0f | subsection | 44 | 73 | Calibration of Credible Intervals | To demonstrate the convergence rates of the posterior credible intervals proved in theorem:contractionofcredibleintervals, we now restrict our attention to the case of q=1, that was considered therein.
As in subsec:exporiginalWPD, we numerically solve the IVPs eq:exODElogistic,eq:exODElinear with the Gaussian ODE filte... | {
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b33bd6a50852819b6afb2959d8f76bf52bef9a38 | subsection | 45 | 73 | Necessity of ass:R | Having explored the asymptotic properties under ass:R in subsec:exporiginalWPD,subsec:exptheeffectofR, we now turn our attention to the question of whether this assumption is necessary to guarantee the convergence rates from theorem:GlobalTruncation,theorem:contractionofcredibleintervals.
This question is of significan... | {
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bfd75bafad632cf3060148216209f868a9242411 | subsection | 46 | 73 | Discussion of experiments | Before proceeding to our overall conclusions, we close this section with a comprehensive discussion of the above experiments.
First and foremost, the experiments in subsec:exporiginalWPD suggest that theorem:GlobalTruncation, the main result of this paper, may be generalizable to q \ge 2.
Moreover, we demonstrated the ... | {
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ba36ea7a7c430b023e898116dc0cf5deed7c2fe7 | subsection | 47 | 73 | Discussion of experiments | As the observed convergence rates in practice sometimes outperform the proved worst-case convergence rates, we believe that an average-case analysis of the filter in the spirit of may shed more light upon the expected practical performance.
Furthermore, it appears that the UQ becomes asymptotically accurate as well as ... | {
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68f743df521a361ecead6b6ee0f34d4345064d9b | subsection | 48 | 73 | Conclusions | We presented a worst-case convergence rate analysis of the Gaussian ODE filter, comprising both local and global convergence rates.
While local convergence rates of h^{q+1} were shown to hold for all q \in \mathbb {N}, IBM and IOUP prior as well as any noise model R \ge 0, our global convergence results is restricted t... | {
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35b5db70f83f1beb4a98800ec3d1903fa6e32fff | subsection | 49 | 73 | Conclusions | To comprehend this idea, recall that the posterior filtering mean is the Bayes estimator with minimum mean squared error in linear dynamical systems with Gauss–Markov prior (as defined by the SDE SDE), i.e. when the data is not evaluations of f but real i.i.d. measurements, as well as in the special case of \dot{x}(t) ... | {
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4edc22614b6f831802478eafee26886ae027f2eb | subsection | 50 | 73 | Derivation of | As derived in the solution of the SDE SDE, i.e.\mathrm {d}{\normalcolor {X}(t)}
=
\begin{pmatrix} \mathrm {d}{\normalcolor X^{(0)}(t)} \\ \vdots \\ \mathrm {d}{\normalcolor X^{(q-1)}(t)} \\ \mathrm {d}{\normalcolor X^{(q)}(t)} \end{pmatrix}
=
\underbrace{\begin{pmatrix} 0 & 1 & 0 \dots & 0 \\ \vdots & \ddots & \ddots &... | {
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59dfb3e5fc49fd721083bef8db53bf9246741b27 | subsection | 51 | 73 | Derivation of | By matrix:Q and{\normalcolor
\begin{pmatrix} (Fh)^k \end{pmatrix}_{i,j}
=
h^k
\left[
\mathbb {I}_{j-i=k} + (- \theta )^{k+i-q} \mathbb {I}_{\lbrace j=q,\ i+k\ge q\rbrace }
\right], }it follows thatA(h)_{ij}
&=
\begin{pmatrix} \sum _{k=0}^{\infty } \frac{(hF)^k}{k!} \end{pmatrix}_{i,j}
{\normalcolor
=
{\left\lbrace \b... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
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e47c50fb955dad91d3551450fad353cd3e67d532 | subsection | 52 | 73 | Derivation of | \end{array}\right.} }Analogously, it follows that{\normalcolor
\exp (F(t-\tau ))
=
{\left\lbrace \begin{array}{ll}
\mathbb {I}_{i\le j} \frac{(t-\tau )^{j-i}}{(j-i)!}, & \mbox{if }j\ne q, \\ \frac{(t-\tau )^{q-i}}{(q-i)!} - \theta \sum _{k=q+1-i}^{\infty } \frac{(-\theta )^{k+i-q-1} (t-\tau )^k}{ k! } , & \mbox{if }j=... | {
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cd57c601eb0405ac11400f65f7258a287b57be0e | subsection | 53 | 73 | Extension to | The algorithm in Gaussian ODE filtering employs a prior X with independent dimensions {X}_j = \left( X_j^{(0)}, \dots , X_j^{(q)} \right)^ \intercal , j \in [d], by SDE.
While this constitutes a loss of generality for our new theoretical results, which do not immediately carry over to the case of x with dependent dimen... | {
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8c4843c0a49831c0665a584ce31d6f47ea216e4a | subsection | 54 | 73 | Extension to | Even more flexible correlation models (over all modeled derivatives) can be employed by inserting arbitrary matrices (of the same dimensionality) for K_x \otimes F and {K_{\varepsilon }} \otimes L in eq:SDEwithdependentdimensions, but such models seem hard to interpret.
For future research, it would be interesting to e... | {
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c1b66a6cd2d9ec33f55c2ab83a90c93010e20cd8 | subsection | 55 | 73 | Illustrative Example | To illustrate the algorithm defined in Gaussian ODE filtering, we apply it to a special case of the Riccati equation \frac{\mathrm {d}x}{\mathrm {d}t} (t)
=
f(x(t))
=
- \frac{(x(t))^3}{2},
\quad x(0)
=
1,
\qquad \qquad \left(\textrm {solution: } x(t)= (t+1)^{-1/2} \right),with step size h=0.1, measurement noise R=0.0 (... | {
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7f212c44b11dbe6cca5682349014b4cd88bf99d9 | subsection | 56 | 73 | Illustrative Example | In the subsequent update step defbetaMM,eq:defyMAP,def:r,eq:defpredictivemean, a Bayesian conditioning of the predictive distribution concreteexampleprediction:mean,concreteexampleprediction:covariance on this data is executed:{\beta } (h)
&=
\left( \beta ^{(0)}(h), \beta ^{(1)}(h) \right)^{\intercal }
=
\left( \frac{P... | {
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9b8ed355dc7a9f950776d8b3b26f5ce4b5f9be9a | subsection | 57 | 73 | The role of the state misalignments | Recall that, as in concreteexampleprediction:mean, the next step h \rightarrow 2h starts with computing m^{-,(i)}(2h) by a (q-i)th-order Taylor expansion of the ith state m^{(i)}(h), for all i \in [q+1], and note that—unlike in the previous step—there is now a non-zero state misalignment (recall def:deltai):\delta ^{(1... | {
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786125a2af5dc7759087f7d189f6158cfb384c5c | subsection | 58 | 73 | The role of the state misalignments | Moreover, for the IOUP prior, this effect can (in the worst case) become more pronounced since its predictions additionally deviate from the Taylor expansion of the state by adding a drift on the qth derivative—which can be seen by inserting def:AIOUP with \theta > 0, instead of \theta = 0, into the mean prediction con... | {
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7ca07284efa9f42e68046d03a5ef564998e7ff4a | subsection | 59 | 73 | Proof of eq:Phii=fi-1 | We prove the stronger statement\Phi _t^{(i+1)}(a)
=
f^{(i)} \left( \Phi _t^{(0)}(a) \right),from which eq:Phii=fi-1 follows by inserting t=0 and \Phi _0^{(0)}(a) = a.
Hence, it remains to show eq:appendixmoregeneral.[Proof of eq:appendixmoregeneral]
By induction over i \in \lbrace 0,\dots ,q\rbrace .
The base case (i=0... | {
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} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
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9d94747bb1c0d366b5bf259c2cc1b69910e06876 | subsection | 60 | 73 | Proof of lemma:r | Again, w.l.o.g. d=1.
Recall that, by def:r, r is implied by the values of m^{-,(0)} and m^{-,(1)}. | {
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f752254af0dab1c24922ce62c7077a9748847aec | subsection | 61 | 73 | Proof of lemma:r | By insertion of m^{-,(i)}((n+1)h)= \sum _{k=i}^q \frac{h^{k-i}}{(k-i)!} m^{(k)}(nh) + K \theta \left|m^{(q)}(nh) \right|h^{q+1-i} (due to Aprediction,def:Psi) into the definition def:r of r((n+1)h), we obtain the following equality which we then bound by repeated application of the triangle inequality:\left|r((n+1)h) \... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.04575177654623985,
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0.018358701840043068,
-0.021273506805300713,
0.0... | |
068c51e552f7eb6e10ef2f7ce4fb806b67c3c8ce | subsection | 62 | 73 | Proof of lemma:sequenceofcontractions | Let \tilde{u}_0 = u^{\ast } and \tilde{u}_n = T_n(\tilde{u}_{n-1}), for n \in \mathbb {N}.
Then,d(u^{\ast },x_n)
\le \underbrace{d(u^{\ast },u_n)}_{\rightarrow 0}
+
\underbrace{d(u_n,\tilde{u}_n)}_{a_n}
+
\underbrace{d(\tilde{u}_n,x_n)}_{= d\left((T_n \circ \dots \circ T_1) (u^{\ast }), (T_n \circ \dots \circ T_1)(x_0)... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.024678653106093407,
0.001443212851881981,
0.0013... | |
840a2800db61b07024913d87055508a1ab97e239 | subsection | 63 | 73 | Proof of proposition:steadystatesglobal | Again, w.l.o.g. d=1.
We prove the claims in the following order: lemma:steadystatesvaluesi, ineq:steadystatelemmai, lemma:steadystatesvaluesii, ineq:steadystatelemmaii, lemma:steadystatesvaluesiii, lemma:steadystatesvaluesv, lemma:steadystatesvaluesvi, lemma:steadystatesvaluesiv, ineq:ineq:steadystatelemmavi, ineq:ineq... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.0003963599447160959,
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... | |
4da5c34a6ee88b84773569dec0ab4b57f6cf80f7 | subsection | 64 | 73 | Proof of proposition:steadystatesglobal | As a start, for lemma:steadystatesvaluesi, we show that P_{11}^{-,\infty } is indeed the unique fixed point of the recursion for \lbrace P_{11}^{-}(nh) \rbrace _n by checking that, if P_{11}^-(nh) = \frac{1}{2} \left(\sigma ^2 h + \sqrt{4\sigma ^2 R h + \sigma ^4 h^2} \right), then also P_{11}^-((n+1)h) = \frac{1}{2} \... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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-0.015306338667869568,
0.0... | |
21a5eab608d5e860227eec9a48dac3e0f02af684 | subsection | 65 | 73 | Proof of proposition:steadystatesglobal | Since, by eq:CupdateMM, def:AIOUP with \theta = 0 and ass:assumption2,P_{11}^-(h)
=
P_{11}(0) + \sigma ^2 h
\le Kh,we can, using the reverse triangle inequality and the (by BFT) strictly decreasing sequence \lbrace \vert P_{11}^{-}(nh) - P_{11}^{-,\infty } \vert \rbrace _n , derive ineq:steadystatelemmai:\left|P_{11}^-... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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-0.029265111312270164,
... | |
7c4ef3092c1242fc42385540721fbfdd6a927ff5 | subsection | 66 | 73 | Proof of proposition:steadystatesglobal | Since P_{11}(nh) monotonically increases in P_{11}^-(nh) (because the derivative of P_{11}(nh) with respect to P_{11}^-(nh) is non-negative for all P_{11}^-(nh) due to R\ge 0; see eq:P11expressedbyP11-), we obtain ineq:steadystatelemmaii:P_{11}(nh)
&\stackrel{{eq:P_11_expressed_by_P_11^-}}{\le }
\frac{\left( \max _n P_... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0... | |
8dd5ef95ecd0f0da3052d03ef9fb067020063aa5 | subsection | 67 | 73 | Proof of proposition:steadystatesglobal | Hence, insertion of the limits lemma:steadystatesvaluesi,lemma:steadystatesvaluesii yield\lim _{n \rightarrow \infty } \left( 1 - \alpha (nh) \right)^{-1}
&=
\frac{\sigma ^2 h + \sqrt{4\sigma ^2 R h + \sigma ^4 h^2} + 2R}{\sigma ^2 h + \sqrt{4\sigma ^2 R h + \sigma ^4 h^2}}, \qquad \text{and}
\\
\lim _{n \rightarrow \i... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0... | |
ea85e3abbc19e892fa075d492f31a6c0bea2ada4 | subsection | 68 | 73 | Proof of proposition:steadystatesglobal | Next, recall thatP_{01}(nh)
\stackrel{{eq:C_update_MM}}{=}
\left( 1-\frac{P_{11}^{-}(nh)}{P_{11}^{-}(nh) + R} \right) P_{01}^{-}(nh)
=
R \frac{P_{01}^{-}(nh)}{P_{11}^{-}(nh) + R}
\stackrel{{def_beta_MM}}{=}
R \beta ^{(0)}(nh),which, since P_{01}(nh) depends continuously on \beta ^{(0)}(nh), implies the unique (attracti... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0... | |
1e2515935812340a9f7ccc8041cb9402b28b80e8 | subsection | 69 | 73 | Proof of proposition:steadystatesglobal | The basis (n=1) follows from\left|\beta ^{(0)}(h) \right|=
\frac{\left|P_{01}^-(h) \right|}{P_{11}^-(h) + R}
\stackrel{{eq:C^-_predict}}{\le }
\frac{ \left|P_{01}(0) \right|+ h P_{11}(0) + \frac{\sigma ^2}{2} h^2 }{\sigma ^2 h}
\stackrel{\text{Ass. }\ref {ass:assumption2}}{\le }
\left( \frac{2K_0}{\sigma ^2} + \frac{1}... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.0116... | |
2c9d841eb0ad14847e4ae8acd79d4e4f526cff35 | subsection | 70 | 73 | Proof of proposition:steadystatesglobal | P_{01}^- \le \hat{\beta }( P^-_{11} + R )) yields, after some rearrangements, that\left|\beta ^{(0)}(nh) \right|&\le \frac{\hat{\beta }\left( P^-_{11} + R \right) h \alpha (nh) + hP^-_{11} \alpha (nh) + \frac{\sigma ^2}{2} h^2}{P^-_{11}\alpha (nh) + \sigma ^2 h + R}
\\
&=
\frac{2 \hat{\beta }P^-_{11} R + \sigma ^2 h \l... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.02971583977341652,
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0.... | |
90e04280db14c0c9bbfea965daf1d51105fc3419 | subsection | 71 | 73 | Proof of lemma:deltaglobalbound | For all n \in [T/h + 1], we can estimate\delta ^{(1)}(nh)
&=
\left\Vert m^{(1)}\left(nh\right) - f\left( m^{(0)}\left(nh\right) \right) \right\Vert =
\left\Vert \Psi _{\normalcolor h}^{(1)}({m}((n-1)h) - f\left( m^{(0)}\left(nh\right) \right) \right\Vert \\
&\le \underbrace{\left\Vert \Psi _{\normalcolor h}^{(1)}({m}((... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.024077123031020164,
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0.008201174437999725,
0.00915479939430952,
... | |
64c7d2ea57c69d9a2d68ef6cd5c1ecdd8acbecf7 | subsection | 72 | 73 | Proof of lemma:deltaglobalbound | For the inductive step, we distinguish two cases.
If \delta ^{(1)}((n-1)h) \le \delta ^{\infty }, then \bar{T}(\delta ^{(1)}((n-1)h)) < 2 \delta ^{\infty }, since\bar{T}(\delta ^{(1)}((n-1)h)) - \delta ^{\infty }
\le \left|\delta ^{\infty } - \bar{T}(\delta ^{(1)}((n-1)h)) \right|<
\delta ^{\infty } - \underbrace{\delt... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.042904943227767944,
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-0.... | |
f38402d8c2a180faa068b555b8d3c63d7b9e5ad7 | abstract | 0 | 60 | Abstract | Raziel combines secure multi-party computation and proof-carrying code to
provide privacy, correctness and verifiability guarantees for smart contracts
on blockchains. Effectively solving DAO and Gyges attacks, this paper describes
an implementation and presents examples to demonstrate its practical viability
(e.g., pr... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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-... | |
e31ad91b4c0e8ee0fbd410a0bdfc9af72fc55e4d | subsection | 1 | 60 | Introduction | The growing demand for blockchain and smart contract, , , , ,
technologies sets the challenge of protecting them from intellectual
property theft and other attacks: security,
confidentiality and privacy are the key issues holding back their
adoption. As shown in this paper, the
solution must be inter-disciplinary (i.e... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 251,
"openalex_id": "",
"raw": "Nick Szabo. The Idea of Smart Contracts. Nick Szabo's Papers and Concise Tutorials, 1997. https://web.archive.org/web/20150812055200/http://szabo.best.vwh.net/idea.html.",
"source_ref_id": "99... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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0.006753527093678713... | |
c2685398b38e0e214e58f6b7ff4d4f4c49ab61d3 | subsection | 2 | 60 | Contributions | We propose a system for securely computing smart contracts guaranteeing
their privacy, correctness and verifiability. Our main and novel contributions
are:Practical formal verification of smart contracts: the proofs accompanying
the smart contracts can be used to prove functional correctness of
a computation as well as... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 620,
"openalex_id": "",
"raw": "Phil Daian. Analysis of the DAO exploit. Hacking Distributed, 2016. http://hackingdistributed.com/2016/06/18/analysis-of-the-dao-exploit.",
"source_ref_id": "0930a560fee456dc4fc2f98e02a8360d01... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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7077cd128cf266a3027f707038b9956e1d58f0cb | subsection | 3 | 60 | Minimal set of functionalities | We argue that the combined features here described are indeed the
minimal set of functionalities that must be offered by a secure solution
to protect computations on blockchains:A pervasive goal of blockchain technologies is the removal of trusted
third parties: towards this end, the preferred solution in cryptography
... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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d1e552d4c62d2940ea88b017e30ce8c8c9911011 | subsection | 4 | 60 | Background | This section provides a brief introduction to the main technologies
that underpin Raziel: blockchains, secure multi-party computation
and formal verification. | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 158,
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"raw": "Shafi Goldwasser. Multi-Party Computations: Past and Present, 1997. https://groups.csail.mit.edu/cis/pubs/shafi/1997-podc.pdf.",
"source_ref_id": "73f6f8f773df4d8e2c054cdcd845c21c7a323c70",
... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
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ebfe9afd75b6917c94b2f8eee0cd3f09561222e6 | subsection | 5 | 60 | Blockchains | A blockchain is a distributed ledger that stores a growing list of
unmodifiable records called blocks that are linked to previous blocks.
Blockchains can be used to make online secure transactions, authenticated
by the collaboration of the P2P nodes allowing participants to verify
and audit transactions. Blockchains ca... | {
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{
"arxiv_id": "",
"doi": "10.2139/ssrn.3440802",
"end": 576,
"openalex_id": "https://openalex.org/W4248175462",
"raw": "Satoshi Nakamoto. Bitcoin: A Peer-to-Peer Electronic Cash System, 2008. https://bitcoin.org/bitcoin.pdf.",
"source_ref_id": "9f02459b5e414ba... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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ea94aa1d6135ba7948f00759dc25da5e09e883b0 | subsection | 6 | 60 | Secure Multi-Party Computation | Protocols for secure multi-party computation (MPC) enable multiple
parties to jointly compute a function over inputs without disclosing
said inputs (i.e., secure distributed computation). MPC protocols
usually aim to at least satisfy the conditions of inputs privacy (i.e.,
the only information that can be inferred abou... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/359168.359176",
"end": 1185,
"openalex_id": "https://openalex.org/W2141420453",
"raw": "Adi Shamir. How to Share a Secret, 1979. https://cs.jhu.edu/~sdoshi/crypto/papers/shamirturing.pdf.",
"source_ref_id": "8f95a4706a947c85b6... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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cf3b55b7ee453ddff899098190a26f859e9dddc2 | subsection | 7 | 60 | Formal verification | Formal verification uses formal methods of mathematics on software
to prove or disprove its correctness with respect to certain formal
specifications. Deductive verification is the preferred approach:
smart contracts are annotated to generate proof obligations that are
proved using theorem provers or satisfiability mod... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 705,
"openalex_id": "",
"raw": "Phil Daian. Analysis of the DAO exploit. Hacking Distributed, 2016. http://hackingdistributed.com/2016/06/18/analysis-of-the-dao-exploit.",
"source_ref_id": "0930a560fee456dc4fc2f98e02a8360d01... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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a503446cf85c1f9632d7f4f374ad9461c0751048 | subsection | 8 | 60 | Model and Goals | Parties executing smart contracts need to protect their private financial
information and obtain formal guarantees regarding their execution.
The central goal for Raziel is to offer a programming framework to
facilitate the development of formally-verified, privacy-preserving
smart contracts with secure computation. | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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cc4b9a9e3cf078ab2cba72491d029cebefd00764 | subsection | 9 | 60 | Threat Model and Assumptions | A conservative threat model assumes that parties wish to execute smart
contracts but mutually distrust each another. Each party is potentially
malicious and the smart contract is developed by one of the parties
or externally developed. We assume that each party trusts its own
environment and the blockchain; the rest of... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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... | |
f48e5d00e28afcccda27c2a13a633383e0a1e496 | subsection | 10 | 60 | Goals | A secure smart contract system should operate as follows: a restricted
set of parties willing to execute a smart contract check the accompanying
proofs/certificates to verify it before its execution. Then these
parties send their private inputs to the nodes at the start of the
execution of the smart contract; after tha... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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0... | |
492f67a5d176d87f47667df730c287af7812018d | subsection | 11 | 60 | Private Smart Contracts | Enabling smart contracts with secure computation techniques (e.g.,
secure multi-party computation, homomorphic encryption,
indistinguishability obfuscation) is a key-step
to the global adoption of blockchain technologies: encrypted transactions
could be stored on the blockchain; secure computations could be carried
out... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 415,
"openalex_id": "https://openalex.org/W1568804695",
"raw": "R L Rivest, L Adleman, and M L Dertouzos. On Data Banks and Privacy Homomorphisms. Foundations of Secure Computation, Academia Press, pages 169–179, 1978. people.csai... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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1bbf0ae7ac74f0d76611050bebe16292127cb977 | subsection | 12 | 60 | Off-chain computation | Even when using the fastest available secure multi-party computation
techniques, the overhead would be very significant if secure computations
would have to be executed on every full node of a blockchain, as have
been previously proposed. In Ethereum, at
current prices (380 $/ETH, 21 Gwei/gas, 30/August/2017), multiply... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 238,
"openalex_id": "",
"raw": "Vitalik Buterin. Secret Sharing DAOs: The Other Crypto 2.0, 2014. https://blog.ethereum.org/2014/12/26/secret-sharing-daos-crypto-2-0/.",
"source_ref_id": "19c994aef0f5b6811d3e9dc06925317ccef0... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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... | |
73570ba9867e74ea158b37190c0954df7ce88a79 | subsection | 13 | 60 | Latency and its impact | As previously mentioned, MPC protocols can be roughly divided into
two classes: constant-round protocols, ideal for high latency settings;
and protocols with rounds dependent on the depth of the evaluated
circuit, usually faster but only on very low latency settings. The
present paper proposes the use of two protocol s... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/2976749.2978331",
"end": 581,
"openalex_id": "https://openalex.org/W2536058570",
"raw": "Toshinori Araki, Jun Furukawa, Yehuda Lindell, Ariel Nof, and Kazuma Ohara. High-throughput semi-honest secure three-party computation with an ... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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a901c3b2766659cd0ae378f4b879eaba318a4732 | subsection | 14 | 60 | Blockchain Solutions | The guarantees of privacy, correctness and verifiability are designed
for the more threatening setting of public permissionless blockchains,
although the smart contracts can also be used on private permissioned
blockchains: it's also preferred that private blockchains keep their
communications open to public blockchain... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 658,
"openalex_id": "",
"raw": "DEDIS. Cothority, 2019. https://github.com/dedis/cothority/.",
"source_ref_id": "5468a5654f0b1dfeb963b190ed1b4e71a3b1668a",
"start": 376
},
{
"arxiv_id": "",
"doi": "... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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863cce7d916744e738b8b3bda5f9f927c238ded5 | subsection | 15 | 60 | Alternative Protocols and Standards | Although this paper is focused on blockchains, it could be adopted
to other financial standards such as:Financial Information eXchange: messaging standard for
trade communication in the equity markets, with presence in the foreign
exchange, fixed income and derivatives market.
Financial Products Markup Language (FpML)... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 277,
"openalex_id": "",
"raw": "FIX Trading Community. FIX Protocol Application Layer, 2016. http://www.fixtradingcommunity.org/pg/structure/tech-specs/fix-protocol.",
"source_ref_id": "7f79df3e32ed2d2c84bc11e76421528694d00d... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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120813bcbe2d7b587c90f54c6c0d773c83b1364e | subsection | 16 | 60 | Functionality and Protocol | The functionalities and protocols of this sub-section and section
constitute an open framework
on which to instantiate different secure multi-party computation protocols,
thus benefiting from upcoming research advances.In this sub-section we present our secure protocol for private smart
contracts, consisting of the fo... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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0.0... | |
d5cd0633a474b7f3af6c5314b9ac45e1afd72bac | subsection | 17 | 60 | Functionality and Protocol | Output: results from the secure computation E_{1}:\overrightarrow{r_{1}},E_{2}:\overrightarrow{r_{2}},...,E_{N}:\overrightarrow{r_{N}}.The security of the protocol is proved on REF . | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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e3d8d5d91b4737d1d02a9ffc7b3c43b5b2bd7781 | subsection | 18 | 60 | Experimental Results | Table REF summarizes the execution cost for
several example applications, chosen for their economic significance:Millionaire's Problem (i.e., determining who's got the bigger number
without revealing anything else)
Second-price auction: sealed-bid auction not revealing the bids between
the participants and without an ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1111/j.1540-6261.1961.tb02789.x",
"end": 403,
"openalex_id": "https://openalex.org/W2015007620",
"raw": "William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. The Journal of Finance, 16(1):8–37, 1961. http://dx.d... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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-0.... | |
b870e44dca6e84c6a9f630bccc79787b032a67b5 | subsection | 19 | 60 | Experimental Results | Suppose two risky currencies with different interest rates but constant
exchange rate: we calculate the calls and puts with the following
equations,
Call & = & S_{0}e^{-\rho t}N\left(d_{1}\right)-X^{-rt}N\left(d_{2}\right)\\
Put & = & Xe^{-rt}N\left(-d_{2}\right)-S_{0}e^{-\rho t}N\left(-d_{1}\right)\\
d_{1} & = & \fra... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1949,
"openalex_id": "https://openalex.org/W2400524050",
"raw": "Charanjit S. Jutla. Upending stock market structure using secure multi-party computation. Cryptology ePrint Archive, Report 2015/550, 2015. http://eprint.iacr.org/20... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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f177892b5c006971eb7acf623ae98285bb9e1f78 | subsection | 20 | 60 | Modes of Interaction | Secure computation can be carried on the nodes of the blockchain or
on the parties themselves. Additionally, said secure computations
can be outsourced to the cloud (see REF ) or use
mined pre-processing data for secure multi-party computation (see
REF ). | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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-0.02023562416434288,... | |
c96fdef5009d1f93b56760bcb695577abe7d1d3b | subsection | 21 | 60 | Outsourcing Secure Computations for Cloud-based Blockchains | The following scheme makes use of a multi-party non-interactive key
exchange, to establish
a shared secret between the computing parties and the executing nodes
of the blockchain: replacing the NIKE protocol would require a PKI
infrastructure between the computing parties and the executing nodes
that will be used to es... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-662-44371-2_27",
"end": 447,
"openalex_id": "https://openalex.org/W288529411",
"raw": "Dan Boneh and Mark Zhandry. Multiparty key exchange, efficient traitor tracing, and more from indistinguishability obfuscation. Cryptology ... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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0.03991... | |
dba1980fa3c26b919dfa201e4ae1ee129afbdc14 | subsection | 22 | 60 | Outsourcing Secure Computations for Cloud-based Blockchains | The following protocol uses a pivot table
during the secure computation, allowing the evaluator node to obliviously
map the encoded inputs by the parties to the encoding expected by
the circuit created by the garbling node. To implement Functionality
B.1 (Outsourcing for Cloud-based Blockchains), the following protocol... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.05538938567042351,
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... | |
e6d5651108b8c164ba4f8a43d5da0e1d510293cd | subsection | 23 | 60 | Outsourcing Secure Computations for Cloud-based Blockchains | Compute and save garbled inputs: using the pivot keys, encrypt Enc_{s_{jl}^{0}}\left(w_{jl}^{0}\right)
and Enc_{s_{jl}^{1}}\left(w_{jl}^{1}\right); then save them into
pivot table P_{q}\left[j,l\right] in random order.
Secure computation at node N_{E}: for every bit of x_{j} and
using the encoding X_{jl}^{x_{j}\left[... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.03316422179341316,
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b61529a7a7fe25e07acf08f09d7ceeb398aa7922 | subsection | 24 | 60 | Outsourcing Secure Computations for Cloud-based Blockchains | Then, the
simulator computes one entry of the pivot tables by encrypting random
values and the other entry by encrypting each garbled value with the
random values generated in the SendPrivateParameters phase:
A simulator of garbled circuits generates a simulated garbled circuit
GC_{SC} and garbled values w_{jl} for ea... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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0.00... | |
69070450b2eb7816189ce93e716f1d11b206134f | subsection | 25 | 60 | Outsourcing Secure Computations for Cloud-based Blockchains | In this hybrid, the call SECCOMP by party E_{i}
computes the pivot table P_{q}\left[j,l\right]=Enc_{s_{jl}^{x_{j}[l]}}\left(0\right),Enc_{s_{jl}^{x_{j}[l]}}\left(w_{jl}^{x_{j}[l]}\right),
that is, only one garbled value per wire is encrypted: therefore,
a contradiction will be reached since an adversary distinguishing
... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
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0.01836000196635723,
-0.001877207076177001,
-... | |
4817c352e44cea4c1057787eff21e9076fe79bba | subsection | 26 | 60 | Outsourcing Secure Computations for Cloud-based Blockchains | Therefore, the indistinguishability
of the view of E_{i} is due to the secure channels: the case for
more corrupted parties trivially follows from this argument. | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.010024523362517357,
-0.0036047089379280806,
-0.0009111902909353375,
0.013503354042768478,
0.00025819442817009985,
-0.012145389802753925,
0.0007891260902397335,
0.041379768401384354,
0.0014437908539548516,
0.032774243503808975,
0.016844861209392548,
-0.024870583787560463,
-0.02438232675194... | |
79fa277acf6fc29d63e0d08243abeaeb1ee38580 | subsection | 27 | 60 | Mining pre-processing data for Secure Multi-Party Computation | The Proof-of-Work of crypto-currencies consumes great amounts of computational
power and electricity: 14TWh for Bitcoin
and 4.25TWh for Ethereum just calculating
hash functions. Miners could create pre-processing data for secure
multi-party computation and be incentivised with crypto-tokens: 50-80%
of total execution t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 177,
"openalex_id": "",
"raw": "Digiconomist. Bitcoin Energy Consumption Index, 2017. http://digiconomist.net/bitcoin-energy-consumption.",
"source_ref_id": "c52e0f71e28d17c0f93745e5c4efd52bcfd43372",
"start": 0
},... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.010177689604461193,
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0.054993096739053726,
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0.030105819925665855,
-0.023513972759246826,
-0.0014944190625101328,
0.009681775234639645,
... | |
1047707c13e2b515c4775b124cf15a385cad342d | subsection | 28 | 60 | Mining pre-processing data for Secure Multi-Party Computation | Specifically, a value
x\in \mathbb {\mathbb {F}}_{q} is secret shared among parties P
by sampling \left(x_{i}\right)_{i\in P}\leftarrow \mathbb {F}_{q}^{\left|P\right|}
subject to x=\sum _{i\in P}x_{i} with a party i holding the value
x_{i}; the MAC is obtained by sampling \left(\gamma \left(x\right)_{i}\right)_{i\in P... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1224,
"openalex_id": "",
"raw": "Peter Scholl, Nigel P. Smart, and Tim Wood. When It's All Just Too Much: Outsourcing MPC-Preprocessing. Cryptology ePrint Archive, Report 2017/262, 2017. http://eprint.iacr.org/2017/262.pdf.",
... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.02863943949341774,
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0.018370743840932846,
-0.033140577375888824,
0.015654802322387695,
0.01003219559788704,
0.... | |
1502df430f5aac001da4ffb977602c99f22af8c5 | subsection | 29 | 60 | Security setting | Let E denote the set of n_{E} parties who are to run the online
phase and O the set of n_{O} outsourcing parties that run the
pre-preprocessing for the executing parties E (respectively Q
and R in ). Adversaries can corrupt
a majority of parties in E and in O, but not all parties in
E nor all parties in O: that is, eac... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 199,
"openalex_id": "",
"raw": "Peter Scholl, Nigel P. Smart, and Tim Wood. When It's All Just Too Much: Outsourcing MPC-Preprocessing. Cryptology ePrint Archive, Report 2017/262, 2017. http://eprint.iacr.org/2017/262.pdf.",
... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.05864162743091583,
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0.007757354062050581,
0.019954020157456398,
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0.051227521151304245,
-0.008756580762565136,
0.019145485013723373,
0.008909134194254875,
0.... | |
741a38b300b543402de4237fa2f2dbdd1c2d172d | subsection | 30 | 60 | Preventing Sybil attacks | In the context of blockchains, Sybil attacks in which an attacker
creates a large number of pseudonymous identities can easily be prevented:
before running secure computations, any party could be required to
deposit some arbitrarily high amount of money on a smart contract
that would be confiscated in case any abnormal... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.021860580891370773,
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0.01760440319776535,
0.0024045875761657953,
0.0009100629831664264,
-0.0271693617105484,
-... | |
98bf936b2de688ab7ac2ac36f984948f13639e0f | subsection | 31 | 60 | Other Applications | Applications of this technology can be found on many commercial/financial
settings. Some of the most noteworthy are as follows:the most immediate application of secure multi-party computation is
the removal of third parties and the financial industry has plenty
of them: market makers, escrows, custodians, brokers, even... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1257/aer.102.3.65",
"end": 1853,
"openalex_id": "https://openalex.org/W3123665471",
"raw": "Andrew W. Lo Emmanuel A. Abbe, Amir Khandani. Privacy-Preserving Methods for Sharing Financial Risk Exposures, 2011. https://papers.ssrn.com/sol3... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.025890355929732323,
0.028377173468470573,
-0.03258797898888588,
0.0526961050927639,
-0.031184377148747444,
0.010977083817124367,
0.024395214393734932,
0.006346722599118948,
0.05416073277592659,
0.016705917194485664,
-0.016385529190301895,
0.028346659615635872,
-0.005435144063085318,
-0.... | |
15b1270babd2b4a1e4a83959a089a1245d2b4073 | subsection | 32 | 60 | Verifiable Smart Contracts | After the DAO attack, there has been some work (, , , , )
to include formal methods in the development of smart contracts: unfortunately,
current solutions are very complex and cumbersome, , , ,
almost equivalent to formally verifying assembly code. Only very high-level
languages should be used to write smart contracts... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 249,
"openalex_id": "",
"raw": "Yoichi Hirai. A Next-Generation Smart Contract and Decentralized Application Platform, 2017. https://yoichihirai.com/malta-paper.pdf.",
"source_ref_id": "de5ec62b60803351a6b963b7ee04056fce042b... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.045505862683057785,
0.021074309945106506,
-0.0371127612888813,
-0.006916677113622427,
0.010430334135890007,
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0.059575747698545456,
0.009499463252723217,
-0.01727452501654625,
0.026690058410167694,
0.007122689858078957,
-0.015191511251032352,
0.008271018043160439,
-0... | |
da71c29cac7c4517907ea64f675bbfd540ae14b3 | subsection | 33 | 60 | Verifiable Smart Contracts | Smart contracts must be written with the specific purpose of
verification in mind, otherwise it becomes extremely complex to generate
complete proofs; regarding
the size, effort and duration of the verification process, there is
a strong linear relationship between effort and proof size
and a quadratic relationship bet... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00165-007-0060-5",
"end": 397,
"openalex_id": "https://openalex.org/W2009752377",
"raw": "Jim Woodcock, Susan Stepney, David Cooper, John Clark, and Jeremy Jacob. The Certification of the Mondex Electronic Purse to ITSEC Level E6. ... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.03955772519111633,
0.012491511180996895,
0.006963044870644808,
0.0020240521989762783,
0.01573457382619381,
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0.004120596218854189,
0.007180520799010992,
0.028523683547973633,
0.011110348626971245,
-0.008309869095683098,
0.019488895311951637,
0.005635296925902367,
0.0... | |
29b21efb91bd60ab463d72bae1dc891fc5c83302 | subsection | 34 | 60 | Case Study | [H]
class Crowdfunding {int minimum = 1000;// requires 0 < n// ensures \result >= minimumpublic int crowdfund(int n, int[] inputs) {int sum = 0;// invariant 0 <= i && i <= nfor (int i = 0; i < n; i++) {sum += inputs[i];}return sum;}}Crowdfund example processed with PCC toolchainProofs are written in Coq: the following ... | {
"cite_spans": []
} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.05558697134256363,
0.011707738973200321,
-0.031134195625782013,
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0.019922224804759026,
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0.008450928144156933,
0.041552938520908356,
... | |
0a7dca9175b00b56452dd853b208dbf306a57ea8 | subsection | 35 | 60 | On the Size of Certifiable Certificates | Since the invention of Proof Carrying Code,
the question of how to efficiently represent and validate proofs
has been a focus of much research. Different techniques are considered
in this work to reduce the size of certificates:The reflection technique. This technique
decreases the size of the proof using the reduction... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/263699.263712",
"end": 143,
"openalex_id": "https://openalex.org/W2034711041",
"raw": "George C. Necula. Proof-carrying Code. In Proceedings of the 24th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '97, ... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.024872956797480583,
0.004970776382833719,
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0.005001295357942581,
-0.010735046118497849,
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0.031098823994398117,
-0.024644063785672188,
-0.0005660314345732331,
-0.009842365980148315... | |
c6af913a09cee8648cceddde9e3adbea4f7fd975 | subsection | 36 | 60 | Zero-Knowledge Proofs of Proofs | Although it's possible to obfuscate certifiable certificates in such
a way that de-obfuscation wouldn't be any easier
while keeping the certifiable certificate sound and complete, this
level of security isn't acceptable in a formal cryptographic model.
When smart contracts are fully encrypted with homomorphic encryptio... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 252,
"openalex_id": "",
"raw": "François Dupressoir. Code and Proof Obfuscation, 2008. http://fdupress.net/files/m2-material/report.pdf.",
"source_ref_id": "9d08e5a8dca2b6d3b473ba4f478c34d7d5a26987",
"start": 0
},
... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.045255597680807114,
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0.054257892072200775,
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0.002759813331067562,... | |
f27764ddd1cae8a5b9674c6f1ba4dde9e9ae57e9 | subsection | 37 | 60 | Proof-Carrying Data | A conceptually related technique to Proof-Carrying Code is Proof-Carrying
Data: in a distributed computation setting, PCD allows
messages to be accompanied by proofs that said messages and the history
leading to them follow a compliance predicate, in such a way that
verifiers can be convinced that the compliance predic... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 399,
"openalex_id": "https://openalex.org/W1504441476",
"raw": "Alessandro Chiesa and Eran Tromer. Proof-Carrying Data and Hearsay Arguments from Signature Cards, 2010. https://people.eecs.berkeley.edu/~alexch/docs/CT10.pdf.",
... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
"David Cerezo Sánchez"
] | [
"cs.CR"
] | 2,018 | en | Computer Science | [
-0.043723512440919876,
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0.03902468457818031,
-0.04314378648996353,
-0.022502504289150238,
-0.01437109149992466,
... |
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