Improved Security for a Ring-Based Fully Homomorphic Encryption Scheme

• Published in: Cryptography and Coding Vol. 8308; pp. 45 - 64
• Main Authors: Bos, Joppe W., Lauter, Kristin, Loftus, Jake, Naehrig, Michael
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2013; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Arithmetic Circuit; Homomorphic Encryption; Homomorphic Encryption Scheme; Message Space; Ring Element
• ISBN: 9783642452383; 3642452388
• ISSN: 0302-9743; 1611-3349; 1611-3349
• DOI: 10.1007/978-3-642-45239-0_4

In 1996, Hoffstein, Pipher and Silverman introduced an efficient lattice based encryption scheme dubbed NTRUEncrypt. Unfortunately, this scheme lacks a proof of security. However, in 2011, Stehlé and Steinfeld showed how to modify NTRUEncrypt to reduce security to standard problems in ideal lattices. In 2012, López-Alt, Tromer and Vaikuntanathan proposed a fully homomorphic scheme based on this modified system. However, to allow homomorphic operations and prove security, a non-standard assumption is required. In this paper, we show how to remove this non-standard assumption via techniques introduced by Brakerski and construct a new fully homomorphic encryption scheme from the Stehlé and Steinfeld version based on standard lattice assumptions and a circular security assumption. The scheme is scale-invariant and therefore avoids modulus switching and the size of ciphertexts is one ring element. Moreover, we present a practical variant of our scheme, which is secure under stronger assumptions, along with parameter recommendations and promising implementation results. Finally, we present an approach for encrypting larger input sizes by extending ciphertexts to several ring elements via the CRT on the message space.