Abstract:
This thesis is a compilation of various message authentication codes having
beyond the birthday bound (BBB) security. Kicking off with preliminary
development in chapter 1, it proceeds to introduce the nEHtM (nonce-based
Enhanced Hash-then-Mask) MAC in chapter 2, which is BBB-secure when
nonce misuse occurs, through the concept of faulty nonces. The construction
is based on a single block cipher, used on the inputs after they undergo a
domain-separation. Next, chapter 3 tackles the security and cryptanalysis of
MAC constructions that use pseudorandom permutations as primitives by
introducing the construction PDMMAC (Permutation-based Davies-Meyer
MAC) and its variants. The work on obtaining pseudorandom functions from
PRPs by [53] lead to our exploration of PRP-based MACs, and one of our
constructions was inspired by the DWCDM of [62]. This was instrumental
in the search for an inverse-free permutation-based MAC with a single
instance of permutation. This is addressed in chapter 4 through the p-EDM
(permutation-based Encrypted Davies-Meyer), which follows the trend of
constructing n-bit to n-bit PRFs by summing smaller constructions such
as the Even-Mansour and the Davies-Meyer, like the SoEM and SoKAC
constructions of [53] and the PDMMAC and variant constructions of [47]
before it. The BBB security is again tight.
Two interesting treatments of the DbHtS construction [61] can be found in
chapters 5 and 6. A permutation-based version, dubbed p-DbHtS (permutation-
based Double-block Hash-then-Sum) is proven to possess BBB security and
a matching attack provided. Finally, a block cipher-based version of the
original construction is shown to have BBB security in the multi-user setting
for underlying hash functions that are constructed without the use of block Ciphers. Furthermore, each chapter extends Patarin’s Mirror Theory to provide
partial bounds for solutions to a system of affine bivariate equations and
non-equations satisfying certain conditions.
