Abstract
A block is an n-bit string, and a (possibly keyed) block-function is a non-linear mapping that maps one block to another, e.g., a block-cipher. In this paper, we consider various symmetric key primitives with
{\ell}
block inputs and raise the following question: what is the minimum number of block-function invocations required for a mode to be secure? We begin with encryption modes that generate
{\ell^{\prime}}
block outputs and show that at least
{(\ell+\ell^{\prime}-1)}
block-function invocations are necessary to achieve the PRF security. In presence of a nonce, the requirement of block-functions reduces to
{\ell^{\prime}}
blocks only. If
{\ell=\ell^{\prime}}
, in order to achieve SPRP security, the mode requires at least
{2\ell}
many block-function invocations. We next consider length preserving r-block (called chunk) online encryption modes and show that, to achieve online PRP security, each chunk should have at least
{2r-1}
many and overall at least
{2r\ell-1}
many block-functions for
{\ell}
many chunks. Moreover, we show that it can achieve online SPRP security if each chunk contains at least
{2r}
non-linear block-functions. We next analyze affine MAC modes and show that an integrity-secure affine MAC mode requires at least
{\ell}
many block-function invocations to process an
{\ell}
block message. Finally, we consider affine mode authenticated encryption and show that in order to achieve INT-RUP security or integrity security under a nonce-misuse scenario, either (i) the number of non-linear block-functions required to generate the ciphertext is more than
{\ell}
or (ii) the number of extra non-linear block-functions required to generate the tag depends on
{\ell}
.
New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure
