There is a great adventure of watermarking usage in the context of conventional authentication since it does not require additional storage space for supplementary metadata. However, JPEG compression, being a conventional method to compress images, leads to exact authentication breaking. We discuss a semi-fragile watermarking system for digital images tolerant to JPEG/JPEG2000 compression. Recently we have published a selective authentication method based on Zernike moments. But unfortunately it has large computational complexity and not sufficiently good detection of small image modifications. In the current paper it is proposed (in contrast to Zernike moments approach) the usage of image finite differences and 3-bit quantization as the main technique. In order to embed watermark (WM) into the image, some areas of the Haar wavelet transform coefficients are used. Simulation results show a good resistance of this method to JPEG compression with CR≤30% (Compression Ratio), high probability of small image modification recognition, image quality assessments PSNR≥40 dB (Peak signal-to-noise ratio) and SSIM≥0.98 (Structural Similarity Index Measure) after embedding and lower computation complexity of WM embedding and extraction. All these properties qualify this approach as effective.