The square root law does not require a linear key

• Authors: Andrew D. Ker
• Venue: Proceedings of the 12th ACM workshop on Multimedia and security
• Year: 2010

Square root laws are theorems about imperfect steganography, embedding which fails to preserve all statistical properties of covers. They show that, in various situations, capacity of covers grows only with the square root of the available cover size. In a paper given at this conference last year [14], we showed an important caveat: when the sender's and recipient's shared embedding key determines the embedding path, its length must be at least linear in the size of the hidden payload to avoid their enemy exhausting over all possible sets of locations. It was left open to show that a linear key is sufficient. There is no necessity, however, for the recipient to know exactly which locations were changed during the embedding process. In this paper we remove that condition, allowing the embedder to combine more than one cover location to convey one bit of payload. As long as the embedder lives beneath the classic square root law bound, we can do more than prove the sufficiency of a linear key: we can even show that asymptotically perfect steganographic security is possible with no key at all. Furthermore, by computing Steganographic Fisher Information, we can show that the keyless embedding tends to perfect security at least as fast as the "ideal" embedding, which requires an unfeasibly large key to spread payload uniformly at random over the cover. Finally, we show asymptotic perfect security of a simple matrix embedding, which allows payload capacity of order √ n log n to be achieved.