Sudoku refers to a two-dimensional (2D) number-placement puzzle with simple constraints that there are no repeated digits in each row, each column, or each block. Motivated by this Sudoku configuration, we introduce a number of Sudoku associated matrix element representations besides the conventional representation using matrix row-column pair. Specifically, they are representations via Sudoku matrix row-digit pair, digit-row pair, column-digit pair, digit-column pair, block-digit pair, and digit-block pair. This means we can secretly represent matrix elements via a Sudoku matrix, and furthermore develop new Sudoku associated 2D parametric bijections. To demonstrate the effectiveness and randomness of bijections, we introduce a simple but effective Sudoku Associated Image Scrambler only using 2D Sudoku associated bijections for image scrambling without bandwidth expansion. Simulations and comparisons demonstrate that the proposed method outperforms several state-of-the-art methods.