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A new fractional difference equation two‐dimensional triangle function combination discrete chaotic map (2D‐TFCDM) based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent, and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect. [ABSTRACT FROM AUTHOR]