# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021021

## Designing tweakable enciphering schemes using public permutations

 1 Indian Statistical Institute, 203, B.T. Road, Kolkata, India 700108 2 Institute for Advancing Intelligence, TCG-CREST, Sector V, Salt Lake, Kolkata, India 700091

* Corresponding author: Debrup Chakraborty

Received  February 2021 Revised  April 2021 Published  June 2021

A tweakable enciphering scheme (TES) is a length preserving (tweakable) encryption scheme that provides (tweakable) strong pseudorandom permutation security on arbitrarily long messages. TES is traditionally built using block ciphers and the security of the mode depends on the strong pseudorandom permutation security of the underlying block cipher. In this paper, we construct TESs using public random permutations. Public random permutations are being considered as a replacement of block cipher in several cryptographic schemes including AEs, MACs, etc. However, to our knowledge, a systematic study of constructing TES using public random permutations is missing. In this paper, we give a generic construction of a TES which uses a public random permutation, a length expanding public permutation based PRF and a hash function which is both almost xor universal and almost regular. Further, we propose a concrete length expanding public permutation based PRF construction. We also propose a single keyed TES using a public random permutation and an AXU and almost regular hash function.

Citation: Debrup Chakraborty, Avijit Dutta, Samir Kundu. Designing tweakable enciphering schemes using public permutations. Advances in Mathematics of Communications, doi: 10.3934/amc.2021021
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##### References:
$\textsf{HCTR}$ construction based on an $n$-bit block cipher $\textsf{E}_k$ and an $n$-bit Polyhash function. Left part of the algorithm is the encryption function and right part is the decryption function
$\textsf{HCTR}$ construction with tweak $T$ and message $M_1 \| M_2 \| \ldots \| M_{l}$ and the corresponding ciphertext $C_1 \| C_2 \| \ldots \| C_{l}$. $\textsf{Poly}_{K_h}$ is the polynomial hash function with hash key $K_h$. $\textsf{Ctr}_{ \textsf{E}_K}$ is the block cipher based counter mode of encryption
$\textsf{ppTES}$ based on an $n$-bit public random permutations $\pi_1$, an AXUAR hash function $\textsf{H}_{k_h}$ and a public permutation based length expanding PRF $\textsf{F}^{\pi_2}_k$. $M \in \{0,1\}^{\geq n}$ is the input message and $T \in \{0,1\}^{\tt tw}$ is the tweak. Left part of the algorithm is the encryption function and right part is the decryption function
Algorithm corresponding to a length expanding random function. $\mathbb{T}[x]_{1, \ldots, b}$ denotes the first $b$ many blocks stored at the $x$-th entry of table $\mathbb{T}$
$\textsf{ppCTR}$ construction with an $n$-bit input $z$ and an integer $b = 3$ and corresponding output $S_1\|S_2\|S_3$. $\pi$ is the public random permutation, $k$ is the key and $\gamma$ is the root of a primitive polynomial of $\mathrm{GF}(2^n)$
$\textsf{ppHCTR+}$ based on an $n$-bit public random permutation $\pi$ and an $n$-bit random hash key $k_h$. Left part is the encryption algorithm and right part is its decryption algorithm
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