A Trend-following Trading Indicator on Homomorphically Encrypted Data

MACD is a momentum indicator which uses the difference between fast and slow moving averages to indicate market trend [Appel, 1979]. MACD was a valuable tool for traders during the 1980s. In this paper, we implement a modified MACD algorithm that operates on encrypted stock price.

In the past decade, several homomorphic encryption schemes have been introduced. One of the most popular schemes is CKKS that implements approximate arithmetic of complex numbers. The scheme supports addition, subtraction and multiplication [Cheon et al., 2017].

Both SEAL and HEAAN libraries process data in batches. This, in turn, speeds up the processing since all components of the vector are processed simultaneously [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 available. 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.

A moving average filter is a low pass filter with a linear 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 corresponding trading strategy. The emphasis is put on recent price quotes by assigning different weighting factors to the asset prices. The original MACD indicator employs the exponential moving average (EMA), and as a first-order autoregressive filter, it requires recursion. Both SEAL and HEAAN libraries are limited by the noise that accumulates with every arithmetic 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 moving average (WMA). The WMA has a finite impulse response and does not require infinite multiplications. Instead, it limits the multiplicative depth by the order $n$ that defines the number of multiplications and is equal to the window size. The weighting coefficients of the WMA are chosen as follows:

for $i\in[0,n)$, where $\bm{w}[i]$ is the weight and $n$ is the window size.

One of the popular indicators for detecting a market turning point is the MACD indicator [Appel, 2003]. The original MACD indicator computes two moving averages: the 12-period EMA and the 26-period EMA. Both EMAs are replaced by the WMAs for the reasons explained above.

Unfortunately due to the theoretical limitations of CKKS as well as other popular HE schemes, only basic arithmetic operations are provided. None of 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 $\bm{\delta}$ as $\bm{\delta}[i]=\bm{m}[i-1]-\bm{m}[i]$ with $\bm{\delta}[0]=0$, and the product of consecutive MACD values $\bm{\pi}$ as $\bm{\pi}[i]=\bm{m}[i-1]\bm{m}[i]$ for $i\in[1,\bm{m}.size)$ with $\bm{\pi}[0]$ = 0. Then the trading decision is defined as:

where $i\in[0,\bm{m}.size)$.

Given that HE schemes are limited to arithmetic operations, only polynomial functions can be implemented homomorphically. Additionally, due to the noise accumulation discussed earlier, there is a limitation on the order of a polynomial function. From this perspective, due to discontinuity of the $sign$ function, polynomials of a lower order do not approximate it well. A better function in terms of polynomial approximation is the ReLU function that is defined as follows:

where $i\in[0,\bm{m}.size)$.

where $i\in[0,\bm{m}.size)$.

Multiplicative depth is the maximal number of sequential homomorphic multiplications allowed, while the multiplication level represents the number of sequential multiplications performed on the ciphertext [Brakerski and Vaikuntanathan, 2014]. Namely, a polynomial’s multiplicative depth depends on its degree. Multiplicative depth is defined by the parameters of the HE library. Every time a multiplication operation is performed on a ciphertext, its multiplication level goes one level deeper. The number of sequential 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 $\hat{r}$ function. A multiplicative depth of 3 is required to generate $\bm{m}$ but 7 more levels are needed to produce $\bm{\delta}$. Currently, $\hat{r}$ is of degree 8 and consumes 4 multiplication levels. The trading decision $\hat{o}_{2}$ is of degree 9. Therefore the maximum multiplication level of our approach is 9.

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 decision aggregator. Without knowing the private key, algorithmic traders as well as the attackers who hijack the communication are not able to decrypt the ciphertexts. Therefore the confidentiality of our method is well maintained by CKKS scheme. Moreover, every trader is equipped with a unique public key, that serves as a digital sign and prevents traders from impersonating each other.