## 1 Introduction  
Visual cryptography (VC) was firstly designed by [PERSON] and [PERSON] in 1994 to share an image secret in visual pattern instead of computational pattern as the traditional secret sharing [1]. It is a new type of secret sharing scheme that encodes an image secret into \(n\) seemingly random shadow images and printed them on \(n\) transparencies. When \(k\) or more of them are stacked together, the secret image will be recognized by the human visual system. This visual decoding method attracted much attention and was used in many applications such as visual image encoding [2; 3; 4], visual authentication [5; 6], and steganography [7; 8; 9; 10], as well as an important solution for secure multimedia content protection on cloud-based media hosting [11; 12; 13]. In a \((k,n)\)-VC scheme, each pixel in the secret image is encoded into \(m\) subpixels on each shadow image. \(m\) is called the pixel expansion. Generally speaking, to minimize the pixel expansion, contrast takes the value \(1/m\). Since the pioneering work of [1], much effort has been devoted to putting forward the optimization of VC construction. Most of the VC schemes inherited two drawbacks that are high pixel expansion and low contrast, which results in lower visual quality of the recovered image. Although some methods, including linear coding [1, 14], supergraph [15; 16], integer linear programming method [17; 18; 19], and linear algebraic technique [20; 21], formulated the construction of VC schemes as the mathematical model to decrease the pixel expansion, the resulting pixel expansion \(m\) is still large located in \([C_{n}^{k}\cdot 2^{k-2}+1,C_{n-1}^{k-1}\cdot 2^{k-1}+1]\). The resulting contrast is quite small with a value \(1/m\). A good strategy was proposed to improve the contrast, which introduced the \"XOR\" operation in recovering the secret [11; 19; 22; 23; 24; 25]. It is investigated that the contrast for a \((k,n)\)-XVC scheme is \(2^{k-1}\) times the contrast for the \((k,n)\)-OVC scheme if the the basis matrices of the latter scheme are adopted [26]. The \"XOR\" operation can be easily run by the special lightweight devices such as smart phone or iPad. It just requires a little computation overload, but it can greatly enhance the contrast compared with the \"OR\" operation. With the \"XOR\" operation, some special schemes can greatly improve the contrast. The \((2,n)\)-XVC scheme can even reach contract being \(1/2\) with no pixel expansion [23]. The special \((n,n)\)-XVC scheme has perfect contrast and no pixel expansion [21; 23]. Some systematic methods by using linear coding [22], and integer linear programming [25], and linear algebraic [21] were proposed to improve the pixel expansion of a \((k,\ n)\)-XVC scheme. However, these XVC schemes need complex computation to construct the basis matrices and also suffer from severe pixel expansion problem. As proved by [26], the basis matrices in a \((k,\ n)\)-OVC scheme also satisfy the contrast and security conditions of a \((k,\ n)\)-XVC scheme. Meanwhile, the contrast is \(2^{k-1}\) times enhanced by \"XOR\" operation. This is also an effective construction method for a \((k,n)\)-XVC scheme. However, it has the same large pixel expansion as a \((k,\ n)\)-OVC scheme. We propose to use part columns of the basis matrices of \((k,\ n)\)-OVC scheme to construct a \((k,\ n)\)-XVC scheme with flexible pixel expansion, and it can reduce the pixel expansion from the least \(1\) pixel to the largest \(2^{k-1}-1\) pixels at the same time it can keep the same average contrast with the corresponding \((k,\ n)\)-XVC scheme. It is a deterministic scheme suitable for any secret image. The resulting pixel expansion is still optimal to the related \((k,\ n)\)-XVC schemes [21; 22; 25]. The remainder of the article is organized as follows. Section 2 briefly introduces the definitions of a \((k,\ n)\)-OVC scheme and a \((k,n)\)-XVC scheme, and the relationship between them.