Secure floating point arithmetic and private satellite collision analysis
Abstract
In this paper, we show that it is possible and, indeed, feasible to use secure multiparty computation (SMC) for calculating the probability of a collision between two satellites. For this purpose, we first describe basic floating point arithmetic operators (addition and multiplication) for multiparty computations. The operators are implemented on the \({\textsc {Sharemind}}\) SMC engine. We discuss the implementation details, provide methods for evaluating example elementary functions (inverse, square root, exponentiation of \(e\), error function). Using these primitives, we implement a satellite conjunction analysis algorithm and give benchmark results for the primitives as well as the conjunction analysis itself.
Keywords
Secure multiparty computation Secure floating point arithmetic Satellite collision analysis Benchmark results1 Introduction
The Earth is orbited by nearly 7,000 spacecraft,^{1} orbital debris larger than 10 cm are routinely tracked and their number exceeds 21,000.^{2} It is understandable that countries do not want to reveal orbital information about their more strategic satellites. On the other hand, satellites are a big investment, and it is in every satellite owner’s interest to keep their property intact. Currently, approximate information about the whereabouts and orbits of satellites is available. These data can be analyzed to predict collisions and hopefully react to the more critical results.
However, in 2009, two communications satellites belonging to the US and Russia collided in orbit [23]. Based on the publicly available orbital information, the satellites were not supposed to come closer than half a kilometer to each other at the of the collision. As there are many orbital objects and satellite operators, an alltoall collaboration between parties is infeasible. However, public data are precise enough to perform a prefilter and exclude all pairs of objects that cannot collide at least within a given period of time (e.g., the next 24 h). Once the satellite pairs with a sufficiently high collision risk have been found, the satellite operators should exchange more detailed information and determine if a collision is imminent and decide if the trajectory of either object should be modified. This is similar to today’s process, where the Space Data Center performs collision predictions on 300 satellite pairs twice a day [16].
We show that secure multiparty computation (SMC) can be used as a possible solution for this problem. We propose a secure method that several satellite operators can jointly use for determining if the orbital objects operated by them will collide with each other. SMC allows satellite operators to secret share data about their satellites so that no party can see the individual values, but collision analysis can still be conducted. The solution provides cryptographic guarantees for the confidentiality of the input trajectories.
To be able to implement the collision analysis algorithm in a privacypreserving way, we first provide floating point arithmetic for general multiparty computations and present an implementation of floating point operations in an SMC setting based on secret sharing. We build the basic arithmetic primitives (addition and multiplication), develop example elementary functions (inverse, square root, exponentiation of \(e\), error function), and present benchmarks for the implementations. As a concrete instantiation of the underlying SMC engine, we have chosen Sharemind [7], as it provides both excellent performance and convenient development tools [17]. However, all the algorithms developed are applicable for general SMC platforms in which the necessary technical routines are implemented. Using these available resources, we finally implemented one possible collision probability computation algorithm, and we give the benchmark results for this implementation.
2 Preliminaries
2.1 Secure multiparty computation
In this subsection, we give an overview of SMC.
Secure multiparty computation (SMC) with \(n\) parties \(P_1,\ldots ,P_n\) is defined as the computation of a function \(f(x_1,\ldots ,x_n) = (y_1,\ldots ,y_n)\) so that the result is correct and the input values of all the parties are kept private. Every party \(P_i\) will provide an input value \(x_i\) and learn only the result value \(y_i\). For some functions \(f\) and some input values, it might be possible to deduce other parties’ inputs from the result, but in the case of data aggregation algorithms, it is generally not possible to learn the inputs of other parties from the result.
In this paper, we concentrate on SMC methods based on secret sharing—also called share computing techniques. Share computing uses secret sharing for the storage of data.
Definition 1
 1.
Correctness: \(s\) is uniquely determined by any \(k\) shares from \(\{s_1, \dots , s_n\}\), and there exists an algorithm \(\mathbf {S^\prime }\) that efficiently computes \(s\) from these \(k\) shares.
 2.
Privacy: Having access to any \(k  1\) shares from \(\{s_1, \dots , s_n\}\) gives no information about the value of \(s\), i.e., the probability distribution of \(k  1\) shares is independent of \(s\).
We use \([\![s]\!]\) to denote the shared value, i.e., the tuple \((s_1,\ldots ,s_n)\).
After the data have been stored, the SMC nodes can perform computations on the shared data. Notice that none of the SMC nodes can reconstruct the input values thanks to the properties of secret sharing. We need to preserve this security guarantee during computations. This is achieved by using SMC protocols that specify which messages the SMC nodes should exchange in order to compute new shares of a value that corresponds to the result of an operation with the input data.
Several SMC frameworks have implementations [14, 21, 28, 29]. While the majority of implementations are research oriented, it is expected that more practical systems will be created as the technology matures.
2.2 The Sharemind platform
\({\textsc {Sharemind}}\) [7] is an implementation of privacypreserving computation technology based on SMC. In the most recent version of \({\textsc {Sharemind}}\), the users can choose which underlying SMC method suits them best. For each available method, some basic operations (e.g., addition, multiplication) have been implemented. More complex operations (e.g., division) often rely on these basic operations. If the basic operations are universally composable, it is possible to use them to implement the complex operations without additional effort. Knowing this, we can design protocols for these complex operations without having a specific SMC method in mind. Hence, the operations can be used with any kind of underlying method as long as it supports the basic operations that we need for the protocol. Some complex protocols, on the other hand, require us to know what the underlying schemas are and must, therefore, be redesigned for each underlying SMC method.
The default protocol suite in \({\textsc {Sharemind}}\) relies on the threeparty additive secret sharing scheme working on \(n\)bit integers, \(n \in \{8, 16, 32, 64\}\), i.e., each value \(x\) is split into three shares \(x_1,x_2,x_3\) so that \(x_1+x_2+x_3=x\,\hbox {mod}\, 2^{n}\) and \([\![x]\!]=(x_1,x_2,x_3)\). In this article, this is the underlying scheme we use for testing the protocols and for benchmarking results. This is also the scheme for which we have implemented targeted optimizations that are dependent on the format in which the data are stored.
The default protocol suite of \({\textsc {Sharemind}}\) has formal security proofs in the passive security model allowing one corrupt miner. This ensures that the security of inputs is guaranteed as long as the miner servers do not deviate from the protocols and do not disclose the contents of their private database. The protocols are also proven to be universally composable, meaning that we can run them sequentially or in parallel without losing the privacy guarantees. For more details on the security model of \({\textsc {Sharemind}}\), please refer to the PhD thesis [6]. Note that since then, the primitive protocols of \({\textsc {Sharemind}}\) have been updated to be more efficient [8].
3 Collaborative satellite collision probability analysis
3.1 Motivation and state of the art
The first confirmed and wellstudied collision between a satellite and another orbital object was the collision of the French reconnaissance satellite Cerise with a fragment of an Ariane rocket body in 1996 [25].
The first accidental collision between two intact satellites happened on February 10th, 2009 when the US communications satellite Iridium 33 collided with a decommissioned Russian communications satellite Kosmos 2251 [23]. The two orbital planes intersected at a nearly right angle, resulting in a collision velocity of more than 11 km/s. Not only did the US lose their working satellite, the collision left two distinct debris clouds floating in Earth’s orbit. Even though the number of tracked debris is already high, an even larger amount is expected in case of a bodytobody hit. Preliminary assessments indicate that the orbital lifetimes of many of the debris are likely to be tens of years, and as the debris gradually form a shell about the Earth, further collisions with new satellites may happen.
This event was the fourth known accidental collision between two cataloged objects and produced more than 1,000 pieces of debris [18]. The previous collisions were between a spacecraft and a smaller piece of debris and did not produce more than four pieces of new debris each.
The US Space Surveillance Network (part of the US Strategic Command) is maintaining a catalog of orbital objects, like satellites and space debris, since the launch of early satellites like the Sputnik. Some of the information is available on the SpaceTrack.org^{3} web page for registered users. However, publicly available orbital data are often imprecise. For example, an analysis based on the publicly available orbital data determined that the Iridium 33 and Kosmos 2251 would come no closer to each other than half a kilometer at the time of the collision. This was not among the top predicted close approaches for that report nor was it the top predicted close approach for any of the Iridium satellites for the coming week [18].
This, however, was not a fault in the analysis software but rather the quality of the data. As a result of how the orbital data are generated, they are of low fidelity, and are, therefore, of limited value during analysis. If more precise data were acquired directly from satellite operators and governments, and the collision analysis process were agreed upon, these kinds of collisions could be avoided [19].
3.2 Underlying math
Algorithm 1 describes how to compute the probability of collision between two spherical objects and is based on [2, 3, 16]. As input, the satellite operators give the velocity vectors \(\mathbf{v} \in \mathbb {R}^{3}\), covariance matrices \({\mathbf {C}} \in \mathbb {R}^{3 \times 3}\), position vectors \(\mathbf{p} \in \mathbb {R}^{3}\) and radii \(R \in \mathbb {R}\) of two space objects. First, on line 2, we define an orthogonal coordinate system in terms of the velocity vectors and their difference. The resulting \(\mathbf{j}\)\(\mathbf{k}\) plane is called the conjunction plane. We consider space objects as spheres but as all the uncertainty in the problem is restricted to the conjunction plane, we can project this sphere onto the conjunction plane (line 4) and treat the problem as 2dimensional [2].
Next, we diagonalize the matrix \(C\) to choose a basis for the conjunction plane where the basis vectors are aligned with the principal axes of the ellipse (lines 5–9). We go on to project the center of the hardbody (\(\mathbf{p}_\mathbf{b}\mathbf{p}_\mathbf{a}\)) onto the encounter plane and then put it in the eigenvector basis (line 10) to gain the coordinates \((x_m, y_m)\). Finally, the probability of collision is computed on line 12 [3].
3.2.1 Vector and matrix operations
To implement the analysis given in Algorithm 1, we need to compute the unit vector, dot product and the cross product for vectors of length \(3\). Furthermore, we need matrix multiplication for twodimensional matrices and, also, the computation of the eigensystem for \(2 \times 2\) symmetric matrices. We only need to consider the eigensystem calculation for symmetric \(2 \times 2\) matrices, so specializing the general method [22] for \(2\) dimensions is sufficient for our needs. These operations require multiplication, addition, division and the computation of square root.
3.2.2 Computing integrals
3.3 Privacypreserving solution using secure multiparty computation
The locations and orbits of communications satellites are sensitive information and governments or private companies might not be willing to reveal these data. One possible solution is to use a trusted third party who will gather all the data and perform the analysis. This, however, still requires the satellite owners to reveal their information to an outside party. While this could be done within one country, it is doubtful that different governments are willing to disclose the orbits of their satellites to a single third party that is inevitably connected with at least one country.
We need the entire trajectory of the satellite to compute the collision probability. So, for example, sharing only the difference between the publicly available trajectory and the precise one does not reduce the amount of work we have to perform, in fact it is likely to increase it as we would need to combine the imprecise public data with the precise secure difference and only then start the computation.
4 Secure floating point operations
4.1 Related work
Integer data domains are not well suited for scientific and engineering computations, where rational and real numbers allow achieving a much greater precision. Catrina et al. [10, 11, 12] took the first steps in adapting real number arithmetic to secure multiparty computation by developing secure fixed point arithmetic and applying it to linear programming. In 2011, Franz and Katzenbeisser [15] proposed a solution for the implementation of floating point arithmetic for secure signal processing. However, their approach relies on twoparty computations working over garbled circuits, and they do not provide the actual implementation nor any benchmarking results. In 2012, Aliasgari et al. [4] designed and evaluated floating point computation techniques and protocols for square root, logarithm and exponentiation in a standard linear secret sharing framework.
4.2 Representation of floating point numbers
The most widely used standard for floating point arithmetic is IEEE 754 [1]. To reach our goal of implementing the arithmetic, we will also make use of Donald Knuth’s more generic presentation given in Sect. 4.2 of the book [20] and ideas used in FPGA implementations [24].

Radix (base) \(b\) (usually \(2\) or \(10\)).

Sign \(s\) (e.g., having the value \(0\) or \(1\) corresponding to signs plus and minus, respectively).

Significand \(f\) representing the significant digits of \(N\). This number is assumed to belong to a fixed interval like \([1,2)\) or \([1/b,1)\).

Exponent \(e\) showing the number of places the radix point of the significand needs to be shifted in order to produce the number \(N\).

Bias (excess) \(q\) is a fixed number used to make the representation of \(e\) nonnegative; hence, one would actually use \(E\) such that \(e=Eq\) for storing the exponent. For example, for IEEE 754 doubleprecision arithmetic, it is defined that \(q=1{,}023\), \(e\in [1{,}022,1{,}023]\) and \(E\in [1,2{,}046]\). The value \(E=0\) is reserved for representing \(N=0\) and some other very small values. The value \(E=2{,}047\) is reserved for special quantities like infinity and Not a Number (NaN) occurring as a result of illegal operations (e.g., \(0/0\) or \(\sqrt{1}\)).
In case of a secretshared representation; however, access to the bits is nontrivial and involves a lot of computation for bit extraction [7, 8]. An alternative would be storing the values shared bitwise, but this would render the underlying processor arithmetic unusable, and we would need to reimplement all the basic arithmetic operations on these shares. At this point, we do not rule this possibility out completely, but this approach needs good justification before actual implementation.
In order to save time on bit extraction in the course of packing and unpacking, we will store the crucial parts of the floating point numbers (sign, significand and exponent) in separately shared values \(s\), \(e\) and \(f\) that have \(8\), \(16\) and \(32\) bits, respectively. We also allow doubleprecision floating point values, where the sign and exponent are the same as for \(32\)bit floating point values, but the significand is stored using \(64\) bits.
The (unsigned) significand \(f\) will represent the real number \(\overline{f}=f/2^{32}\). In other words, the highest bit of \(f\) corresponds to \(1/2\), the second one to \(1/4\), etc. In order to obtain real floating point behavior, we require that the representation is normalized, i.e., that the highest bit of \(f\) is always \(1\) (unless the number to be represented is zero, in which case \(f=0\), \(s\) corresponds to the sign of the value, and the biased exponent \(e = 0\)). Hence, for all nonzero floats, we can assume that \(f\in [1/2,1)\). With our singleprecision floating point values, we achieve \(32\)bit significand precision, which is somewhere in between IEEE single and double precision. The doubleprecision floating point values achieve \(64\)bit significand precision, which is more than the IEEE double precision.
The sign \(s\) can have two values: \(1\) and \(0\) denoting plus and minus, respectively. Note that we are using a convention opposite to the standard one to implement specific optimizations. As we are using a \(16\)bit shared variable \(e\) to store the exponent, we do not need to limit the exponent to \(11\) bits like in the standard representation. Instead we will use \(15\) bits (we do not use all \(16\) because we want to keep the highest bit free), and we introduce a bias of \(2^{14}  1\).
In a typical implementation of floating point arithmetic, care is taken to deal with exponent under and overflows. In general, all kinds of exceptional cases (including under and overflows, division by zero, \(\sqrt{1}\)) are very problematic to handle with secretshared computations, as issuing any kind of error message as a part of control flow leaks information (e.g., the division by zero error reveals the divisor).
We argue, that in many practical cases, it is possible to analyze the format of the data, based on knowledge about their contents. This allows the analysts to know that the data fall within certain limits and construct their algorithms accordingly. This makes the control flow of protocols independent of the inputs and preserves privacy in all possible cases. For example, consider the scenario, where the data are gathered as \(32\)bit integers and then the values are cast to floating point numbers. Thus, the exponent cannot exceed \(32\), which means that we can multiply \(2^{9}\) of such values without overflowing the maximal possible exponent \(2^{14}\). This approach to exceptions is basically the same as Strategy 3 used in the article [15].
4.3 Multiplication
For the casting on line 3, keep in mind that as Sharemind uses modular arithmetic, then while casting the significand up, we need to account for the overflow error generated by the increase in the modulus. For instance, if we are dealing with \(32\)bit floats, the significand is \(32\) bits. When we cast it to \(64\) bits, we might get an overflow of up to 2 bits. To deal with this problem, we check obliviously whether our resulting \(64\) bit float is larger than \(2^{32}\) and \(2^{33}\) and adjust the larger value accordingly.
Truncation on line 4 can be performed by deleting the lower \(n\) bits of each share. This way we will lose the possible overflow that comes from adding the lower bits. This will result in a rounding error, so we check whether the lower \(n\) bits actually generate overflow. We do this similarly to the upcasting process—by checking whether the sum of the three shares of the lower \(n\) bits is larger than \(2^{32}\) and \(2^{33}\). Then, we truncate the product of the significands and obliviously add \(0\), \(1\) or \(2\) depending on the overflow we received as the result of the comparison.
For nonzero input values, we have \(f,f'\in [1/2,1)\), and hence \(f''=f\cdot f'\in [1/4,1)\) (note that precise truncation also prevents us from falling out of this interval). If the product is in the interval \([1/4,1/2)\), we need to normalize it. We detect the need for normalization by the highest bit of \(f''\) which is \(0\) exactly if \(f\in [1/4,1/2)\). If so, we need to shift the significand one position left (line 7) and reduce the exponent by \(1\) (line 8).
4.4 Addition
The overall structure of the addition routine is presented in Algorithm 4. In order to add two numbers, their decimal points need to be aligned. The size of the required shift is determined by the difference of the exponents of the inputs. We perform an oblivious switch of the inputs based on the bit \([\![e']\!]\mathop {\ge }\limits ^{?}[\![e]\!]\), so that in the following, we can assume that \([\![e']\!]\ge [\![e]\!]\) (line 1). Our reference architecture Sharemind uses \(n\)bit integers to store the significands, hence if \([\![e']\!][\![e]\!]\ge n\), adding \(N\) is efficiently as good as adding \(0\), and we can just output \([\![N']\!]\) (line 2). Unfortunately, in the implementation, we cannot halt the computation here, as the control flow must be data independent, but this check is needed for the following routines to work correctly. Instead of returning, the result of the private comparison is stored, and an oblivious choice is performed at the end of the protocol to make sure that the correct value is returned.
On line 5, we shift one of the significands to align the decimal points and adjust the exponent. Moving on to the ifelse statement on lines 6–22, the operation performed on the significands depends on the sign bits. If they are equal (lines 6–13), we need to add the significands, and otherwise subtract them. When adding the significands, the result may exceed \(1\), so we may need to shift it back by one position. In order to achieve this functionality, we use temporary casting to \(2n\) bits similarly to the process in Algorithm 2 for multiplication.
Normalization of the significand and the exponent on line 22 of Algorithm 4 can be performed by using Algorithm 6. It works with a similar principle as Algorithm 5 and performs an oblivious selection from a vector of elements containing all the possible output values. The selection vector is computed using the \({\mathsf {MSNZB}}\) (most significant nonzero bit) routine from [8] which outputs \(n\) additively shared bits, at most one being equal to \(1\) in the position corresponding to the highest nonzero bit of the input. The vectors from which we obliviously select consist of all the possible shifts of the significands (lines 2 and 3) and the corresponding adjustments of the exponents (line 4).
5 Elementary functions
In this section, we present algorithms for four elementary functions, namely inversion, square root, exponentiation of \(e\) and error function. All of them are built using Taylor series expansion, and we also give a version using the Chebyshev polynomial for inversion and the exponentiation of \(e\). Polynomial evaluations are sufficiently robust for our purposes. They allow us to compute the functions using only additions and multiplications, and do so in a dataindependent manner. Taylor series expansion also gives us a convenient way to have several tradeoffs between speed and precision.
5.1 Inversion
As Sharemind works fastest with parallelised computations, we do as much of the Taylor series evaluation in parallel as possible. This parallelization is described in Algorithm 8. Variations of this algorithm can be adopted for the specific polynomial evaluations where needed, but this algorithm gives a general idea of how the process works. We use Algorithm 8 to parallelise polynomial evaluation in all elementary functions we implement.
We need to compute powers of \([\![x]\!]\) from \([\![x^{1}]\!]\) to \([\![x^{p}]\!]\). To do this in parallel, we fill vector \([\![\mathbf{u}]\!]\) with all the powers we have already computed \([\![x^{1}]\!], \ldots , [\![x^{h}]\!]\) and the second vector \([\![\mathbf{v}]\!]\) with the highest power of \([\![x]\!]\) we have computed \([\![x^{h}]\!]\). When we multiply these vectors elementwise, we get powers of \([\![x]\!]\) from \([\![x^{h+1}]\!]\) to \([\![x^{2h}]\!]\) (lines 6–16). If the given precision \(p\) is not a power of \(2\); however, we also need to do one extra parallel multiplication to get the rest of the powers from \([\![x^{h}]\!]\) to \([\![x^{p}]\!]\). This is done on lines 18–29. If the precision is \(<1.5\) times larger than the highest power, it is not necessary to add elements to the vector \([\![\mathbf{u}]\!]\) (checked on line 20).
The ifelse statements in Algorithm 8 are based on public values, so there is no need to use oblivious choice to hide which branch is taken. The summation of vector elements on line 35 is done in parallel as much as possible.
5.2 Square root
5.3 Exponentiation of \(e\)
Unfortunately, the larger \([\![N]\!]\) is, the more we have to increase precision \(p\) to keep the result of the Taylor series evaluation from deviating from the real result too much. This, however, increases the working time of the algorithm significantly. To solve this problem, we implemented another algorithm using the separation of the integer and fractional parts of the power \([\![N]\!]\) and using the Chebyshev polynomial to approximate the result [26]. This solution is described in Algorithm 12.
In this case, we exponentiate \(e\) using the knowledge that \(e^{x} = (2^{\log _2 e})^{x} = 2^{(\log _2 e)\cdot x}\). Knowing this, we can first compute \(y = (\log _2 e)\cdot x \approx 1.44269504 \cdot x\) (line 1) and then \(2^{y}\). Notice, that \(2^{y} = 2^{[y] + \{y\}} = 2^{[y]} \cdot 2^{\{y\}}\), where \([y]\) denotes the integer part and \(\{y\}\) the fractional part of \(y\).
First, we need to separate the integer part from the shared value \([\![y]\!] = ([\![s]\!], [\![f]\!], [\![e]\!])\). Let \(n\) be the number of bits of \(f\). In this case, \([\![[y]]\!] = [\![f \gg (n  e)]\!]\) (line 3) is true if and only if \([\![e]\!] \le n\). On the other hand, if \([\![e]\!] > n\), then \([\![y]\!] \ge 2^{[\![e]\!]  1} > 2^{n  1}\) would also hold, and we would be trying to compute a value that is not smaller than \(2^{2^{n  1}}\). Hence, we consider the requirement \([\![e]\!] \le n\) to be reasonable.
Note, however, that if we want to use the Chebyshev polynomial to compute \(2^{\{y\}}\), we need to use two different polynomials for the positive and negative case. We would like to avoid this if possible for optimization, hence, we want to keep \({\{y\}}\) always positive. This is easily done if \(y\) is positive, but causes problems if \(y\) is negative. So if \(y\) is negative and we want to compute the integer part of \(y\), we compute it in the normal way and add \(1\) (line 3) making its absolute value larger than that of \(y\). Now, when we subtract the gained integer to gain the fraction on line 4, we always get a positive value. This allows us to only evaluate one Chebyshev polynomial, thus giving us a slight performance improvement.
We construct the floating point value \(2^{[\![[y]]\!]}\) by setting the corresponding significand to be \(100\ldots 0\) and the exponent to be \([\![[y]]\!]+1\). As \([\![\{y\}]\!] \in [0,1]\) holds for the fractional part of \(y\), and we have made sure that the fraction is always positive, we can compute \(2^{[\![\{y\}]\!]}\) using the Chebyshev polynomial on line 7. In this domain, these polynomials will give a result that is accurate to the fifth decimal place. Finally, we multiply \(2^{[\![[y]]\!]} \cdot 2^{[\![\{y\}]\!]}\) to gain the result.
5.4 Error function
Note that, in addition, we check whether the exponent is larger or smaller than \(3\), which shows us whether \(x \in [2, 2]\) and based on the private comparison, we obliviously set the result as \(\pm 1\) according to the original sign \([\![s]\!]\) (line 2). We perform this approximation because at larger values, more cycles are needed for a sufficiently accurate result.
6 Performance of floating point operations
6.1 Benchmark results
We implemented the secure floating point operations described in this paper on the \({\textsc {Sharemind}}\) 3 secure computation system. We built the implementation using the threeparty protocol suite already implemented in the \({\textsc {Sharemind}}\) system. Several lowlevel protocols were developed and optimized to simplify the use and conversion of shared values with different bit sizes. Secure floating point operations are programmed as compositions of lowlevel integer protocols.
To measure the performance of the floating point operations, we deployed the developed software on three servers connected with fast network connections. More specifically, each of the servers used contains two Intel X5670 2.93 GHz CPUs with 6 cores and 48 GB of memory. The network connections are pointtopoint 1 Gb/s connections. We note that during the experiments, peak resource usage for each machine did not exceed 2 cores and 1 GB of RAM. The peak network usage varied during the experiments, but did not exceed 50 Mb/s. Therefore, we believe that the performance can be improved with further optimizations.
Integer operations on \({\textsc {Sharemind}}\) perform more efficiently on multiple inputs [8]. Therefore, we also implemented floating point operations as vector operations. Unary operations take a single input vector and binary operations take two input vectors of an equal size. Both kinds of operations produce a vector result. We performed the benchmarks on vectors of various sizes to estimate how much the size of the vector affects the processing efficiency.
For each operation and input size, we performed 10 iterations of the operation and computed the average. The only exceptions to this rule were operations based on Taylor series or Chebyshev polynomials running on the largest input vector. Their running time was extensively long so we performed only a few tests for each of them. In the presentation, such results are marked with an asterisk.
Performance of singleprecision floating point operations
Input size in elements  

1  10  100  1,000  10,000  
Private \(x\)  Add  0.73 ops  7.3 ops  69 ops  540 ops  610 ops 
Private \(y\)  Multiply  2.3 ops  23 ops  225 ops  1,780 ops  6,413 ops 
Divide (TS)  0.08 ops  0.81 ops  6.8 ops  25.5 ops  38 ops\(^{*}\)  
Divide (CP)  \(0.16\) ops  \(1.6\) ops  \(14\) ops  \(59\) ops  \(79\) ops\(^{*}\)  
Private \(x\)  Multiply  2.3 ops  23 ops  220 ops  1,736 ops  6,321 ops 
Public \(y\)  Multiply by \(2^{k}\)  \(5.9\times 10^{4}\) ops  \(6.1\times 10^{5}\) ops  \(4.2\times 10^{6}\) ops  \(1.1\times 10^{7}\) ops  \(1.4\times 10^{7}\) ops 
Divide  2.3 ops  23 ops  221 ops  1,727 ops  6,313 ops  
Divide by \(2^{k}\)  \(6.4\times 10^{4}\) ops  \(6.1\times 10^{5}\) ops  \(4.1\times 10^{6}\) ops  \(1.2\times 10^{7}\) ops  \(1.4\times 10^{7}\) ops  
Private \(x\)  Find \(e^{x}\) (TS)  0.09 ops  0.9 ops  7.3 ops  23 ops  30 ops\(^{*}\) 
Find \(e^{x}\) (CP)  \(0.12\) ops  \(1.2\) ops  \(11\) ops  \(71\) ops  \(114\) ops  
Invert (TS)  0.09 ops  0.84 ops  6.8 ops  14.7 ops  35.7 ops\(^{*}\)  
Invert (CP)  \(0.17\) ops  \(1.7\) ops  \(15.3\) ops  \(55.2\) ops  \(66.4\) ops  
Square root  0.09 ops  0.85 ops  7 ops  24 ops  32 ops\(^{*}\)  
Error function  \(0.1\) ops  0.97 ops  8.4 ops  30 ops  39 ops 
Performance of doubleprecision floating point operations
Input size in elements  

1  10  100  1,000  10,000  
Private \(x\)  Add  0.68 ops  6.8 ops  60 ops  208 ops  244 ops 
Private \(y\)  Multiply  2.1 ops  21 ops  195 ops  1,106 ops  2,858 ops 
Divide (TS)  0.08 ops  0.7 ops  3 ops  5.2 ops  12.4 ops\(^{*}\)  
Divide (CP)  \(0.15\) ops  \(1.5\) ops  \(12\) ops  \(23\) ops  \(52\) ops\(^{*}\)  
Private \(x\)  Multiply  2.1 ops  21 ops  193 ops  1263 ops  2667 ops 
Public \(y\)  Multiply by \(2^{k}\)  \(3.3\times 10^{4}\) ops  \(3.2\times 10^{5}\) ops  \(2.1\times 10^{6}\) ops  \(8.5\times 10^{6}\) ops  \(1.3\times 10^{7}\) ops 
Divide  2.1 ops  21 ops  194 ops  1223 ops  2573 ops  
Divide by \(2^{k}\)  \(6.4\times 10^{4}\) ops  \(6.1\times 10^{5}\) ops  \(4\times 10^{6}\) ops  \(1\times 10^{7}\) ops  \(1.2\times 10^{7}\) ops  
Private \(x\)  Find \(e^{x}\) (TS)  0.09 ops  0.8 ops  3.1 ops  4.5 ops  6.2 ops\(^{*}\) 
Find \(e^{x}\) (CP)  \(0.11\) ops  \(1.1\) ops  \(9.7\) ops  \(42\) ops  \(50\) ops  
Invert (TS)  0.08 ops  0.75 ops  3.6 ops  4.8 ops  10.7 ops\(^{*}\)  
Invert (CP)  \(0.16\) ops  \(1.5\) ops  \(11.1\) ops  \(29.5\) ops  \(47.2\) ops  
Square root  0.08 ops  0.76 ops  4.6 ops  9.7 ops  10.4 ops\(^{*}\)  
Error function  \(0.09\) ops  0.89 ops  5.8 ops  16 ops  21 ops\(^{*}\) 
Addition and multiplication are the primitive operations with addition being about five times slower than multiplication. With singleprecision floating point numbers, both can achieve the speed of a thousand operations per second if the input vectors are sufficiently large. Based on the multiplication primitive, we also implemented multiplication and division by a public constant. For multiplication, there was no gain in efficiency, because we currently simply classify the public factor and use the private multiplication protocol.
However, for division by a public constant, we can have serious performance gains as we just invert the divisor publicly and then perform a private multiplication. Furthermore, as our secret floating point representation is based on powers of two, we were able to make a really efficient protocol for multiplying and dividing with public powers of two by simply modifying the exponent. As this is a local operation, it requires no communication, and is, therefore, very fast.
Although we isolated the network during benchmarking, there was enough fluctuation caused by the flow control to slightly vary the speeds of similar operations. Although private multiplication, multiplication and division by constant are performed using the same protocol, their speeds vary due to effects caused by networking.
The unary operations are implemented using either Taylor series or Chebyshev polynomials. When available, we benchmarked both cases. In all such cases, Chebyshev polynomials provide better performance, because smaller polynomials are evaluated to compute the function. In general, all these operations have similar performance.
As mentioned in Sect. 5.1, division between two private floating point values is implemented by inverting the second parameter and performing a private multiplication. As inversion is significantly slower than multiplication, the running time of division is mostly dictated by the running time of inversion.
6.2 Difference in the precision of Taylor series and Chebyshev polynomials
In Sect. 5, we give two algorithms for computing inverse and powers of \(e\) based on Taylor series and Chebyshev polynomials. It is easy to see from Tables 1 and 2 that using the Chebyshev polynomial versions to approximate the results gives us better performance.
Panel (b) shows similar reasoning for exponentiation with the different polynomials. As \(32\) elements of Taylor series for exponentiation will not be enough to produce a precise result after \(x = 13\) and will produce an absolute error of more than \(2\times 10^{6}\) at \(x = 20\), we give this plot in the interval \([0, 15]\) and show the relative error instead of the absolute error. As can be seen from the figure, the relative error produced by evaluation of 5 terms of the Chebyshev polynomial always stays between \(2.9\times 10^{6}\) and \(3.4\times 10^{6}\). Taylor series evaluation is more precise until \(x = 13\) after which the relative error increases. This point can be pushed further by adding more terms to the Taylor series evaluation at the cost of performance.
As expected, Fig. 5 indicates that using the Taylor series evaluation gives us a more precise result but, in larger algorithms, it can be a bottleneck. We propose that the choice of the algorithm be done based on the requirements of the problem being solved—if precision is important, Taylor series evaluation should be used, if speed is important, Chebyshev polynomials are the wiser choice. If the problem requires both speed and precision, it is possible to add terms to the Chebyshev polynomial to make it more precise. In addition, as the error in exponentiation using Taylor series evaluation becomes significant when \(x\) grows, Chebyshev polynomials should be used if it is expected that \(x\) will exceed \(10\). Another option is to raise the number of terms in Taylor series but this will, in turn, increase the running time of the algorithm.
6.3 Benefits of parallelization
Note that the running time of the secure computation protocols grows very little until a certain point, when it starts growing nearly linearly in the size of the inputs. At this point, the secure computation protocols saturate the network link and are working at their maximum efficiency. Addition is a more complex operation with more communication and; therefore, we can perform less additions before the network link becomes saturated. Addition takes even more communication with doubleprecision numbers so it is predictable that the saturation point occurs earlier than for singleprecision values. Multiplication uses significantly less network communication, so its behavior is roughly similar with both single and doubleprecision numbers.
Sequential and parallel performance of the matrix product function
Number of matrix multiplications  

1 (s)  10 (s)  100 (s)  1,000 (s)  
\(3\times 3\)  Sequential  3.2  31.9  318.5  3,212.3 
Matrix  Parallel  3.2  3.3  5.2  45.7 
\(5\times 5\)  Sequential  4.6  46.3  466  4,641.4 
Matrix  Parallel  4.6  5.1  20.4  91.9 
\(10\times 10\)  Sequential  6.4  65.9  674.6  6,624.2 
matrix  Parallel  6.6  15.7  152  1,189.2 
We see that, for a single input, there is no significant difference between the running time of the sequential and parallel version. This is logical, as there is nothing to vectorize. However, as the number of multiplications grows, parallel execution becomes dramatically more efficient. However, when the network saturates, the performance of the parallel case also become more linear in the number of inputs, as illustrated by the \(10\times 10\) case. Even then, the parallel case is much faster than the sequential case. This justifies our decision to use vectorization to the highest possible degree. We note that this increase in efficiency comes at the cost of needing more memory. However, we can control the size of our inputs and balance vectorization and memory use.
7 Implementation of collision analysis
7.1 Reusable components
We implemented the collision analysis described in Algorithm 1 for Sharemind in its language SecreC. This allows us to easily do parallel operations on vectors as it has syntax for elementwise operations on vectors. We first implemented the algorithm and its component methods for one satellite pair using as much parallelization as possible and then went on to parallelise the solution further for \(n\) satellite pairs.
One of the most important values of our code is its reusability. The vector and matrix operations we have implemented can later be used in other algorithms similarly to the modular design of software solutions that is the norm in the industry. This makes it convenient to implement other algorithms that make use of vector and matrix operations. In addition, the operations have already been parallelised so future developers will not have to put time into this step of the coding process in SecreC.
7.2 Vector and matrix operations

Vector length, unit vector and dot product for vectors of any size;

Cross product for vectors of size \(3\);

Matrix product for twodimensional matrices; and

Eigensystem computation for \(2 \times 2\) matrices.
Similarly, we implemented simple and parallel methods for matrix multiplication for twodimensional matrices. We also provide a slight optimization for the case of diagonal matrices as the covariance matrices we work with are diagonal. This does not provide a very large speedup in the case of one satellite pair but if many pairs are analyzed in parallel, then the benefit is worth the implementation of such a conditional method. We also implemented the eigensystem computation for \(2 \times 2\) matrices.
7.2.1 Dot product
We start the parallel dot product computation similarly to the simple version—we multiply the matrices together elementwise. The SecreC language allows us to do this easily by using the multiplication sign on two arrays of the same dimensions. As the result, we receive the matrix \([\![{\mathbf {product}}]\!]\) that has the same dimensions as the input matrices. To use the function sum that allows us to sum together vector elements, we first reshape the matrix \([\![{\mathbf {product}}]\!]\) to a vector. This is another array manipulation method which SecreC provides us. Finally, we use the method that lets us sum vector elements together. However, in the parallel case, we use a different version of the method sum that lets us sum elements together by groups, giving us \(xShape[0] = m\) sums as a result. The second parameter of the method sum must be a factor of the size of the vector, the elements of which are being summed.
The benefit in using the parallel version of the method is in the multiplication and summing sub methods. If we computed the dot product for each satellite pair as a cycle, we would get the products within one pair \([\![{\mathbf{x}_\mathbf{1}} \cdot {\mathbf{y}_\mathbf{1}}]\!], [\![{\mathbf{x}_\mathbf{2}} \cdot {\mathbf{y}_\mathbf{2}}]\!], [\![{\mathbf{x}_\mathbf{3}} \cdot {\mathbf{y}_\mathbf{3}}]\!]\) in parallel but would not be able to use the parallelization for computing many pairs at once. As the multiplication and summation of floating point values is quite timeconsuming, parallelization is essential.
7.3 Benchmark results
Finally, we measured the performance of the full privacypreserving collision analysis implementation. We performed two kinds of experiments. First, we measured the running time of the algorithm on various input sizes to know if it is feasible in practice and if there are efficiency gains in evaluating many satellite pairs in parallel. Second, we profiled the execution of a single instance of collision analysis to study, what the bottleneck operations in its execution are.
Collision analysis performance
Number of satellite pairs  

1  2  4  8  16  
Single  TS  258 (258) s  350 (175) s  488 (122) s  792 (99) s  1,348 (84) s 
Precision  CP  204 (204) s  255 (128) s  355 (89) s  557 (70) s  967 (60) s 
Double  TS  337 (337) s  447 (223) s  724 (181) s  1127 (141) s  2,064 (129) s 
Precision  CP  246 (246) s  340 (170) s  450 (113) s  724 (91) s  1,272 (80) s 
The running time for a single collision probability computation is 4–5 min and it is reduced 1–2 min with parallel executions. Consider the scenario reported in [16], where the Space Data Center computes the collision probabilities twice a day for 300 satellite pairs. By extrapolating our performance measurements (an average of 90 s per satellite pair), we see that a secure run would be shorter than 8 h. Also, given that we can balance the computation load to multiple secure computation clusters, we can scale the system to process a much larger number of satellites. Therefore, today’s performance is suitable for practical use and there is room for scalability through further optimizations.
When we compute collision probabilities using operations based on Taylor series, we see that division and square root are the most expensive operations, followed by the exponent function and addition. Using Chebyshev polynomials greatly reduces the importance of division and the exponent function, leaving square root as the most timeconsuming operation in the algorithm.
8 Conclusions and further work
In this paper, we showed how to perform satellite collision analysis in a secure multiparty setting. For this purpose, we first presented routines for implementing floating point arithmetic operations on top of a SMC engine. The general approach is generic, but in order to obtain the efficient real implementation, platformspecific design decisions have been made. In the current paper, we based our decisions on the architecture of \({\textsc {Sharemind}}\) which served as a basic platform for our benchmarks. We also showed how to implement several elementary functions and benchmarked their performance. However, by replacing the specific optimizations with generic algorithms or other specific optimizations, these floating point algorithms can be ported to other secure computation systems. We concluded by implementing the algorithm for computing the probability of satellite collision and benchmarked the performance of this implementation.
The largest impact of this work is the conclusion that it is possible and feasible to perform satellite collision analysis in a privacypreserving manner.
Footnotes
 1.
NSSDC Master Catalog, NASA. http://nssdc.gsfc.nasa.gov/nmc/, last accessed May 16, 2013.
 2.
Orbital Debris Program Office, NASA. http://orbitaldebris.jsc.nasa.gov/faqs.html, last accessed May 16, 2013.
 3.
SpaceTrack.org, https://www.spacetrack.org, last accessed May 16, 2013.
Notes
Acknowledgments
The authors would like to thank Galois Inc for the use of their prototype that works on public data. The authors also thank Nigel Smart for suggesting the use of Chebyshev polynomials, Dan Bogdanov for his help with benchmarking and Sven Laur for coming up with a neat trick for slightly speeding up exponentiation.
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