On the Relations between the Attacks on Symmetric Homomorphic Encryption over the Residue Ring

• Main Author: Alina V. Trepacheva
• Format: Article
• Language: English
• Published: Moscow Engineering Physics Institute 2017-06-01
• Series: Bezopasnostʹ Informacionnyh Tehnologij
• Subjects: ring-homomorphic encryption; fully homomorphic encryption; RSA problem; number factorization problem; ciphertexts only attack; known plaintext attack
• Online Access: https://bit.mephi.ru/index.php/bit/article/view/108
• ISSN: 2074-7128; 2074-7136

The paper considers the security of symmetric homomorphic cryptosystems (HC) over the residue ring. The main task is to establish an equivalence between ciphertexts only attack (COA) and known plaintexts attack (KPA) for HC. The notion of reducibility between attacks and sufficient condition of reducibility from COA to KPA are given for this purpose. The main idea is: to prove reducibility from COA to KPA we need to find a function over residue ring being efficiently computable and having a small image size comparing with the size of residue ring. The study of reducibility existence is important since it allows to understand better the security level of symmetric HC proposed in literature. A vulnerability against KPA has been already found for the majority of these HC. Thus the reducibility presence can demonstrate that cryptosystems under the study are not secure even against COA, and therefore they are totally insecure and shouldn’t be used in practice. We give an example of reducibility from COA to KPA for residue ring being a simple field. Based on this example we show an efficient COA on one symmetric HC for small field. Also we separately consider the case of residue ring composed using number n being hard-to-factor. For such n an efficient algorithm to construct an efficiently computable function with small image is unknown so far. So further work related to cryptanalysis of existing symmetric HC will be directed into study of functions properties over residue rings modulo numbers hard for factorization.