Processing convolution layers remains a huge bottleneck for private deep convolutional neural network (CNN) inference for large datasets. To solve this issue, this paper presents a novel homomorphic convolution algorithm that provides speedup, communication cost, and storage saving. We first note that padded convolution provides the advantage of model storage saving, but it does not support channel packing, thereby increasing the amount of computation and communication. We address this limitation by proposing a novel plaintext multiplication algorithm using the Walsh-Hadamard matrix. Furthermore, we propose the optimization techniques to significantly reduce the latency of the proposed convolution by selecting the optimal encryption parameters and applying lazy reduction. It achieves 1.6-3.8x speedup and reduces the weight storage by 2000-8000x compared to the conventional convolution. When the proposed convolution is employed for CNNs like VGG-16, ResNet-20, and MobileNetV1 on ImageNet, it reduces the end-to-end latency by 1.3-2.6x, the memory usage by 2.1-7.9x and communication cost by 1.7-2.0x compared to conventional method.

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