A Trend-following Trading Indicator on Homomorphically Encrypted

Data

Haotian Weng

and Artem Lenskiy

Research School of Computer Science, The Australian National University, Canberra, Australia

Keywords:

Homomorphic Encryption, Quantitative Finance, Algorithmic Trading.

Abstract:

Algorithmic trading has dominated the area of quantitative ﬁnance for already over a decade. The decisions

are made without human intervention using the data provided by brokerage ﬁrms and exchanges. An emerging

intermediate layer of ﬁnancial players that are placed in between a broker and algorithmic traders has recently

been introduced. The role of this layer is to aggregate market decisions from the algorithmic traders and send

a ﬁnal market order to a broker. In return, the quantitative analysts receive incentives proportional to the cor-

rectness of their predictions. In such a setup, the intermediate player — an aggregator — does not provide the

market data in plaintext but encrypts it. Encrypting market data prevents quantitative analysts from trading on

their own, as well as keeps valuable ﬁnancial data private. This paper proposes an implementation of a popular

trend-following indicator with two different homomorphic encryption libraries — SEAL and HEAAN — and

compares it to the trading indicator implemented for plaintext. Then, an attempt to implement a trading strat-

egy is presented and analysed. The trading indicator implemented with SEAL and HEAAN is almost identical

to that implemented on the plaintext, with the percentage error of 0.14916% and 0.00020% respectively. De-

spite many limitations that homomorphic encryption imposes on this algorithm’s implementation, quantitative

ﬁnance has a potential of beneﬁting from the methods of homomorphic encryption.

1 INTRODUCTION

The most prominent approach that provides the means

for data analysis and also keeps the data private is

based on homomorphic encryption (HE). The idea of

HE that allows computation on encrypted data was

ﬁrstly proposed in the 1970s. However, no practi-

cal implementation existed until Craig Gentry pro-

posed one in his PhD thesis in 2009 (Gentry and

Boneh, 2009). Modern cryptosystems are capable of

performing arbitrary computation on encrypted data

- ciphertexts, facilitating the implementation of var-

ious data analysis tools (Aslett et al., 2015). As an

active area of research, there is a multitude of HE

schemes that have been proposed and implemented

as open-source libraries. CKKS (Cheon et al., 2017)

and BFV (Brakerski and Vaikuntanathan, 2014) are

among the most popular HE schemes. Simple En-

crypted Arithmetic Library (SEAL) developed by Mi-

crosoft Cryptography Research (SEAL, 2019) imple-

ments both BFV and CKKS scheme. Homomorphic

https://orcid.org/0000-0002-3993-7621

https://orcid.org/0000-0002-4745-6756

Encryption for Arithmetic of Approximate Numbers

(HEAAN) is the original name of CKKS scheme de-

veloped by the Seoul National University CryptoLab

which only supports CKKS scheme. A particular area

of HE that has not receive well-deserved attention

is the privacy-preserving analysis of time-dependent

data. Such topic attracts ﬁnancial industry’s attention

where time-series analysis is widely applied.

The ﬁnancial industry is known to be extremely

cautious about privacy aspects of storing, analysing

and distributing data. The solutions for secure data

storage have been available for decades and are al-

ready provided by numerous cloud services. Secure

data distribution is nowadays an integral part of the

Internet, thanks to Secure Sockets Layer (SSL). How-

ever, data analysis that preserves privacy is still in its

infancy, even though, as mentioned, it has been ac-

tively developed, and adopted by the ﬁnancial indus-

try.

Numer.ai is a hedge fund powered by thousands

of independent quantitative analysts striving to out-

perform the market (Numer.ai, 2019). The quantita-

tive analysts compete with each other, and those with

accurate predictions are rewarded. Numer.ai does not

602

Weng, H. and Lenskiy, A.

A Trend-following Trading Indicator on Homomorphically Encrypted Data.

DOI: 10.5220/0009835706020607

In Proceedings of the 17th International Joint Conference on e-Business and Telecommunications (ICETE 2020) - SECRYPT, pages 602-607

ISBN: 978-989-758-446-6

provide the market data in plaintext but transforms it

in a form that makes it impossible to know what ﬁ-

nancial asset a particular time series represents. This,

in turn, prevents quantitative analysts from trading on

their own and keeps valuable ﬁnancial data private.

To the best of our knowledge, Numer.ai relies on pro-

prietary obfuscation methods.

In our scenario, we distinguish three independent

players: a broker, a decision aggregator and algo-

rithmic traders. A broker provides access to the ex-

changes that is a mediator between sellers and buyers.

In particular, it provides such market data as order

books, recent trades and price quotes, so the market

participants are able to draw a trading decision. The

decision aggregator (analogous to Numer.ai) receives

market data M in plaintext and returns market orders

O. The data received by the aggregator is encrypted

as c

= E

} and is sent to an algorithmic trader

. Algorithmic traders operate over the encrypted

data c

and the decision o

∗

drawn by the traders are

also encrypted and unknown to the traders (ﬁg. 1).

These decisions are then decrypted by the aggrega-

tor o

= Da

∗

} and transmitted to the broker in the

form of market orders.

In this paper, we focus on making a very ﬁrst step

in the direction of applying HE in developing an algo-

rithm that operates on encrypted time-dependent data.

On the example of a popular trend following trading

strategy based on Moving Average Convergence Di-

vergence (MACD) indicator, we demonstrate how an

algorithmic trader could employ methods of homo-

morphic encryption to make trading decisions.

2 BACKGROUND

2.1 Moving Average Convergence

Divergence

MACD is a momentum indicator which uses the dif-

ference between fast and slow moving averages to

indicate market trend (Appel, 1979). MACD was a

valuable tool for traders during the 1980s. In this pa-

per, we implement a modiﬁed MACD algorithm that

operates on encrypted stock price.

2.2 Homomorphic Encryption

In the past decade, several homomorphic encryption

schemes have been introduced. One of the most pop-

ular schemes is CKKS that implements approximate

arithmetic of complex numbers. The scheme supports

https://github.com/woonhulktin/HETSA

addition, subtraction and multiplication (Cheon et al.,

2017).

Every operation in CKKS, especially multiplica-

tion, adds a certain amount of noise, which limits the

number of operations allowed before the accumulated

noise grows to the point making the ﬁnal result in-

accurate. Levelled schemes limit the maximal quan-

tity of sequential homomorphic multiplications before

the noise becomes intolerable (Brakerski and Vaikun-

tanathan, 2014).

To ﬁght the noise, it introduces rescaling as well as

bootstrapping. The rescaling is a scale-invariant tech-

nique that scales down the size of ciphertext modulus

to reduce the noise and preserve the precision (Cheon

et al., 2017). The bootstrapping operation in theory

eliminates the noise accumulated throughout homo-

morphic computations by refreshing the noise in a ci-

phertext. The signiﬁcant disadvantages of the boot-

strapping method are the dramatic increase in com-

putational time and signiﬁcant memory consumption

(Cheon et al., 2018).

3 RELATED WORK

Previous studies have shown the practicality of

privacy-preserving analysis of time-series data. An

additive homomorphic encryption scheme was pro-

posed to aggregate time-series data without sacriﬁc-

ing privacy (Shi et al., 2011). The ciphertexts are en-

crypted under different users’ secret keys respectively

to achieve secure multi-party computation. Addition-

ally, Paillier encryption scheme as a partial homo-

morphic encryption was applied to privacy-preserving

similarity evaluation of time-series data (Zhu et al.,

2014). Pallier scheme is adequate for computing

the square of Euclidean distance as it supports ho-

momorphic addition of ciphertexts and homomor-

phic multiplication to plaintexts. Another partially

homomorphic-encryption-based access control con-

struction (HEAC) was introduced to support both ac-

cess control and aggregation-based computations on

encrypted data (Burkhalter et al., 2020). However,

these studies rely on additive homomorphic encryp-

tion schemes and thus the types of computations al-

lowed are limited. Our approach focuses on ﬁnancial

time-series data and applies both homomorphic addi-

tion and homomorphic multiplication of ciphertexts

to generate the trading decisions.

A Trend-following Trading Indicator on Homomorphically Encrypted Data

603

Decision Aggregator

Brokerage

…

𝑀

𝑂

𝒟

𝑜

∗

ℰ

𝑚

…

𝒟

𝑜

∗

ℰ

𝑚

𝑜

∗

= 𝑇

𝑐

…

𝑜

∗

𝑜

∗

= 𝑇

𝑐

𝑜

∗

Intranet

Internet

𝑇

𝑖th trader (strategy)

𝑀

market data (orderbooks, quotes,…)

𝑚

𝑘th part of 𝑀 (e.g. AMD price quote)

𝑐

encrypted

𝑚

𝑜

trading order by

𝑡

𝑜

∗

encrypted

𝑜

𝑂

aggregated orders

ℰ

encryption with

𝑖th key

𝒟

decryption with

𝑖th key

Exchanges

Figure 1: An intermediate layer collects encrypted decision, decrypts them and makes a ﬁnal decision on whether to buy or

to sell.

4 METHODS

4.1 Data Encoding

Both SEAL and HEAAN libraries process data in

batches. This, in turn, speeds up the processing since

all components of the vector are processed simultane-

ously (Chen et al., 2017). However, in our scenario,

the server sends encrypted price one at a time, and the

remaining components are padded with zeros. This

process repeats every time a new price quote is avail-

able. Therefore, only the ﬁrst slot of the plaintext

vector is used to store the asset price. This results in

higher memory requirement and slows computation.

Once receiving the latest encrypted price quote,

the algorithm appends it to the vector of previously

received ciphertexts, and a MACD signal value is pro-

duced, that is either used to generate a trading deci-

sion or returned on its own.

As the share prices are rational numbers and BFV

is slower than CKKS when performing multiplication

followed by rescaling (Chen et al., 2019), the frac-

tional encoder is chosen over the integer one.

To test our algorithm on encrypted data, we se-

lected Apple’s daily stock price (NASDAQ: AAPL)

on the interval from 06/01/2015 to 21/10/2015.

4.2 Weighted Moving Average

A moving average ﬁlter is a low pass ﬁlter with a lin-

ear phase shift. In the context of ﬁnancial time-series,

it is used to determine a trend of an asset price and

is the foundation of the MACD indicator and corre-

sponding trading strategy. The emphasis is put on

recent price quotes by assigning different weighting

factors to the asset prices. The original MACD indica-

tor employs the exponential moving average (EMA),

and as a ﬁrst-order autoregressive ﬁlter, it requires re-

cursion. Both SEAL and HEAAN libraries are lim-

ited by the noise that accumulates with every arith-

metic operation. Without an efﬁcient bootstrapping

method, EMA is infeasible as it requires a signiﬁcant

number of multiplicative depths (in theory inﬁnitely

many). One solution is to replace the EMA by a non-

recursive ﬁlter. One such ﬁlter is the weighted mov-

ing average (WMA). The WMA has a ﬁnite impulse

response and does not require inﬁnite multiplications.

Instead, it limits the multiplicative depth by the or-

der n that deﬁnes the number of multiplications and is

SECRYPT 2020 - 17th International Conference on Security and Cryptography

604

equal to the window size. The weighting coefﬁcients

of the WMA are chosen as follows:

w[i] =

2(i + 1)

n(n + 1)

(1)

for i ∈ [0, n), where w

w[i] is the weight and n is the

window size.

Algorithm 1: Weighted Moving Average (WMA).

Input: a vector of ciphertexts c

c, window size n

Output: a vector of encrypted weighted moving av-

erages a

w = zeros(n)

for i in range [0, n) do

w[i] = FHE.encode(2(i+1)/(n(n+1)))

end for

a = zeros(c

c.size − n)

for i in range [0, c

c.size − n) do

a[i] = FHE.encrypt(0)

for j in range [0, n) do

r = FHE.multiplyConstant(c

i:i+n

[ j], w

w[ j])

a[i] = FHE.add(a

a[i], r)

end for

return a

4.3 Moving Average Convergence

Divergence

One of the popular indicators for detecting a market

turning point is the MACD indicator (Appel, 2003).

The original MACD indicator computes two mov-

ing averages: the 12-period EMA and the 26-period

EMA. Both EMAs are replaced by the WMAs for the

reasons explained above.

The trading signals are triggered by the MACD

signal line crossing the x-axis. First, a 12-period

WMA α

α and a 26-period WMA β

β are computed.

Then, the differences between β

β and α

α are calculated.

Finally, a 9-period WMA γ

γ is applied to the differ-

ences to produce the MACD signal line m

m. Algorithm

2 illustrates a library-independent MACD implemen-

tation using homomorphic encryption method.

When the MACD signal line crosses the x-axis

from below, it indicates a buy signal and when the

signal line crosses the x-axis from above, it triggers a

sell signal.

4.4 Trading Decision

Unfortunately due to the theoretical limitations of

CKKS as well as other popular HE schemes, only ba-

sic arithmetic operations are provided. None of

Algorithm 2 : Moving Average Convergence Divergence

Function (MACD).

Input: a vector of ciphertexts of asset prices d

Output: a vector of ciphertexts of MACD signals m

α = wma(d

d, 12)

β = wma(d

d, 26)

θ = zeros(β

β.size)

for i in range[0, β

β.size) do

θ[i] = FHE.sub(α

14:α

α.size

[i], β

β[i])

end for

γ = wma(θ

θ, 9)

m = zeros(γ

γ.size)

for i in range[0, γ

γ.size) do

m[i] = FHE.sub(θ

9:θ

θ.size

[i], γ

γ[i])

end for

return m

widespread used HE libraries is equipped with logic

and relational operators on numbers (Acar et al.,

2018). Hence, there is no direct method to compare

two values and deduce where the MACD signal line

is greater than, less than, or equal to zero which in

turn determines the time of buying, selling or doing

nothing. Nevertheless, the decision of sell, hold, or

buy could be associated with a decision function o

that produces either -1, 0 or 1 that correspond to a

sell, hold or buy order. To deﬁne a decision function

o, we ﬁrst deﬁne a sign function that return the sign

of a number. Then we deﬁne a vector of differences

of adjacent MACD values δ

δ as δ

δ[i] = m

m[i − 1] − m

m[i]

with δ

δ[0] = 0, and the product of consecutive MACD

values π

π as π

π[i] = m

m[i − 1]m

m[i] for i ∈ [1, m

m.size) with

π[0] = 0. Then the trading decision is deﬁned as:

m, i) =

sign(δ

δ[i]) · (sign(π

π[i]) − 1) (2)

where i ∈ [0, m

m.size).

The range of the function above is {−1, 0, 1}. The

downside of the function is its dependence on the sign

function that is not implementable using available HE

operations. The ﬁrst sign function determines the

trend change and the second is the moment of cross-

ing x-axis.

4.5 Polynomial Approximation of the

ReLU Function

Given that HE schemes are limited to arithmetic op-

erations, only polynomial functions can be imple-

mented homomorphically. Additionally, due to the

noise accumulation discussed earlier, there is a lim-

itation on the order of a polynomial function. From

A Trend-following Trading Indicator on Homomorphically Encrypted Data

605

this perspective, due to discontinuity of the sign func-

tion, polynomials of a lower order do not approximate

it well. A better function in terms of polynomial ap-

proximation is the ReLU function that is deﬁned as

follows:

r(x) =

(

x x > 0

0 x ≤ 0

(3)

Then an equivalent to o

decision function could

be implemented using ReLU as follows:

m, i) = −sign(δ

δ[i] · r(−π

π[i])) (4)

where i ∈ [0, m

m.size).

We employed polynomial regression to approxi-

mate the ReLU function to estimate the polynomial

coefﬁcients:

ˆr(x) = −0.0001x

− 0.0003x

+ 0.0025x

+ 0.009x

− 0.0253x

− 0.0984x

+ 0.0882x

+ 0.5173x

+ 0.4475x + 0.0753 (5)

Then using the ˆr we deﬁne an approximation of

the decision function as follows:

ˆo

m, i) = −δ

δ[i] · ˆr(−π

π[i]) (6)

where i ∈ [0, m

m.size).

We apply both o

and ˆo

on plaintext MACD sig-

nals and employ only ˆo

on encrypted MACD sig-

nals. The percentage errors between plaintext and

ciphertext implementations of ˆo

are then compared

and presented in Section 4.

4.6 Multiplicative Depth

Multiplicative depth is the maximal number of se-

quential homomorphic multiplications allowed, while

the multiplication level represents the number of se-

quential multiplications performed on the ciphertext

(Brakerski and Vaikuntanathan, 2014). Namely, a

polynomial’s multiplicative depth depends on its de-

gree. Multiplicative depth is deﬁned by the param-

eters of the HE library. Every time a multiplication

operation is performed on a ciphertext, its multiplica-

tion level goes one level deeper. The number of se-

quential multiplications on the ciphertext are limited

by the multiplicative depth. The most substantial part

of the trading strategy in terms of multiplicative depth

is the implementation of ˆr function. A multiplicative

depth of 3 is required to generate m

m but 7 more lev-

els are needed to produce δ

δ. Currently, ˆr is of degree

8 and consumes 4 multiplication levels. The trading

decision ˆo

is of degree 9. Therefore the maximum

multiplication level of our approach is 9.

4.7 Conﬁdentiality

CKKS scheme is an asymmetric cryptosystem with a

public and a private keys. The public key is shared

with both the decision aggregator and algorithmic

traders while the private key is shared with the de-

cision aggregator. Without knowing the private key,

algorithmic traders as well as the attackers who hi-

jack the communication are not able to decrypt the ci-

phertexts. Therefore the conﬁdentiality of our method

is well maintained by CKKS scheme. Moreover, ev-

ery trader is equipped with a unique public key, that

serves as a digital sign and prevents traders from im-

personating each other.

5 RESULTS

To compare the errors between the WMA-based ci-

phertext and plaintext implementations, we employ

the mean absolute percentage error deﬁned as:

e(x

x, y

y) =

∑

i=1

x[i] − y

y[i]|

y[i]

× 100% (7)

where i ∈ [0, x

x.size) and N = x

x.size = y

y.size. x

x is ei-

ther decrypted WMA, MACD signals or the trading

decisions and y

y is the vector of corresponding plain-

text WMA, MACD signals or trading decisions.

Table 2 presents the comparison results. As the er-

rors of WMA and MACD signals between encrypted

and plaintext analysis are insigniﬁcant, the WMA and

MACD signals generated with SEAL and HEAAN

are almost identical to those in plaintext. In terms of

trading decisions, we compare trading decision func-

tions over encrypted data with SEAL and plaintext

data. The peaks of the approximated trading deci-

sions generally correspond to the exact trading deci-

sions, but there are errors around the peaks. Addi-

tionally, the error increases as the multiplication level

gets deeper, and hence the percentage error of a trad-

ing decision function is larger than the that of MACD

and much larger than that of WMA. The reason for

the increasing error is the noise added by every arith-

metic operation.

Table 1: Errors between ciphertext and plaintext analysis.

Result Percentage Error

WMA-SEAL 0.00918%

WMA-HEAAN 0.00019%

MACD-SEAL 0.14916%

MACD-HEAAN 0.00020%

Decision-SEAL 3.19030%

Decision-HEAAN 0.03794%

SECRYPT 2020 - 17th International Conference on Security and Cryptography

606

The computation was conducted on the Intel(R)

Core(R) CPU i7-7820HQ @ 2.9GHz with 16GB

RAM. The task consisted of the following steps: (1)

200 AAPL share prices were encrypted, (2) MACD

analysis was performed, (3) encrypted trading deci-

sions were generated and (4) the decisions were de-

crypted. The time required to produce the MACD

signal as well as the trading decision for a single day

were measured and summarised in table 3. The trad-

ing indicator implemented with SEAL can also run on

1-second candles and HEAAN implementation can be

applied to 15-seconds candles. The total time con-

sumed per data unit is 0.99 and 12.34 seconds for

SEAL and HEAAN implementations respectively.

Table 2: Performance of MACD and decision analysis.

Method Computation Time

MACD-SEAL 0.73 sec

MACD-HEAAN 7.085 sec

Decision-SEAL 0.26 sec

Decision-HEAAN 5.255 sec

Total-SEAL 0.99 sec

Total-HEAAN 12.34 sec

At this point, there are two signiﬁcant limitations

in our implementation. Firstly, the lack of recursion

reduces the decision accuracy as the multiplicative

depth is constrained by the HE parameters. Although,

HEAAN supports bootstrapping, it introduces sub-

stantial noise and is also computationally intensive

(Acar et al., 2018). Therefore, we did not implement

bootstrapping in our approach. Secondly, operation

of bootstrapping is slow, due to the absence of logi-

cal and relational operators in the state-of-the-art HE

libraries. Only approximate decisions can be imple-

mented. However, the presented algorithm accurately

implements the MACD indicator and could be applied

in the real world applications.

6 CONCLUSIONS

We have implemented the MACD indicator on a stock

price time-series. To the best of our knowledge, this

is the ﬁrst time HE methods are applied to ﬁnan-

cial time-series analysis. The algorithm implemented

with SEAL is able to produce the trading indicator in

less than a second and could be applied to 1-second

candle data as well as to lower resolution data. For

the future work we plan to implement linear systems

and corresponding recessive AR and MA based ﬁl-

ters, including exponential moving average.

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