# Design and Application of Secret Codes for Learning Medical Data
## 2 Preliminaries  
In the distributed learning schemes depicted in Figure 1, the distribution of secret or private information is very important in order to retain secret properties. Linear block codes can be used as a mathematical tool to design such distribution strategies [17; 18]. In this section, we present some preliminary knowledge on finite fields and algebraic codes that are used in secret-sharing schemes. Minimal codes are a class of linear block codes, in which the support of a codeword is not included to that of another codeword. A binary linear block code C of length \(N\) is a subspace of vectors in the space \(\left\{0,1\right\}^{N}\). An element of a linear block code is called a codeword. A codeword \(x\) of length \(N\) in a binary linear block code can be expressed as  
\[x=\left\{x[n]\ :x[n]\in\left\{0,1\right\},\ n\in Z_{N}\right\}\]  
where \(n\) represent the index of each symbol, and \(Z_{N}\) is the set of integers modulo \(N\). Because a linear block code is a subspace, the vector addition of any two codewords in a code is included as another codeword in the code. The support of \(x\) is a subset of \(Z_{N}\), which is defined as  
\[\text{supp}(x)=\left\{n\ :x[n]=1,\ n\in Z_{N}\right\} \tag{1}\]  
The size of \(\text{supp}(x)\) is called the Hamming weight of \(x\). In secret-sharing schemes, a secret codeword is assigned to a user or a device. Information in the codeword should not  
Figure 1: Distributed learning model for medical data. be subordinate to any other codewords. A minimal code is defined as a linear block code in which the support of a codeword is not a subset of the support of any other codeword, as shown in Figure 2. Using this characteristic, secret information can be distributed to users, and the information of a user is not fully revealed to the other users. Due to this characteristic, minimal codes have been applied to secret-sharing schemes [15; 17]. However, in distributed learning, the secret-storing structure is important, as is the weight distribution of a code. Research on analytic approaches to the structure is ongoing. A secret-sharing scheme is, from a mathematical viewpoint, constructed on a finite field. A finite field is an algebraic structure in which addition, subtraction, multiplication, and division can be freely operated [24]. For a prime \(p\) and a positive integer \(m\), a finite field \(\mathrm{GF}(p^{m})\) consists of the additive identity \(0\) and \(1=\alpha^{0},\alpha^{1},\alpha^{2},\ldots,\ \alpha^{p^{m}-2}\), where \(\alpha\) is called a primitive element of \(\mathrm{GF}(p^{m})\). The finite field \(\mathrm{GF}(p^{m})\) is an Abelian group with respect to the addition modulo \(p\), and \(\mathrm{GF}(p^{m})\backslash\{0\}\) is a cyclic group with respect to the multiplication. Any element of a finite field can be represented not only in a multiplicative way using the power of the primitive element, but also in an additive way using vector representation on the basis \(\{1,\ \alpha^{1},\alpha^{2},\ldots,\alpha^{m-1}\}\). Note that \(\mathrm{GF}(p^{m})\) can be interpreted as a vector space of dimension \(m\) over \(\mathrm{GF}(p)\). Most of the well-known error-correction codes and pseudorandom sequences have been constructed based on the properties of finite fields [14]. In this paper, we assume that \(p=2\), which corresponds to the cases of binary codes. Because most machine learning algorithms are based on the binary operation, the assumption is appropriate. In a secret-sharing scheme defined on \(\mathrm{GF}(p^{m})\), a secret corresponds to an element of the finite field. Let \(U_{1},\ldots,U_{l}\) be users or devices engaged in the scheme. Pieces of the secret are assigned to each user using a secret code. This secret information must be encoded so that the whole information is not synthesized from any proper subset of information of users. Moreover, in a partial group of users, if a user can recover some part of the secret, the other user should also be able to recover the same information.