QARMA

QARMA overview (an overbar indicates an inverse transformation)

QARMA is an Even–Mansour cipher using three stages, with whitening keys w⁰ and w¹ XORed in between:

1. permutation F is using core key k⁰ and parameterized by a tweak T. It has r rounds inside (r = 7 for QARMA-64, r = 11 for QARMA-128);
2. "central" permutation C is using key k¹ and is designed to be reversible via a simple key transformation (contains two central rounds);
3. the third permutation is an inverse of the first (r more rounds).

All keys are derived from the master encryption key K using specialisation:

• K is partitioned into halves as w⁰ Concatenation k⁰, each will have halfsize bits;
• for encryption w¹ = (w⁰ >>> 1) + (w⁰ >> (halfsize-1));
• for encryption k¹ = k⁰;
• for decryption, the same design can be used as long as k⁰+α is used as a core key, k¹ = Q•k⁰, w¹ and w⁰ are swapped. α here is a special constant and Q a special involutary matrix. This construct is similar to the alpha reflection in PRINCE.

QARMA details. Rounds of ${\displaystyle \digamma }$ are at the top, rounds of ${\displaystyle {\overline {\digamma }}}$ are at the bottom, ${\displaystyle C}$ is on the right. Inner path describes the transformation of the internal state, outer path corresponds to the tweak update. c_i are round constants.

The data is split into 16 cells (4-bit nibbles for QARMA-64, 8-bit bytes for QARMA-128). Internal state also contains 16 cells, arranged in a 4x4 matrix, and is initialized by plaintext (XORed with w⁰). In each round of ${\displaystyle \digamma }$, the state is transformed via operations ${\displaystyle \tau ,M,S}$:

• ${\displaystyle \tau }$ is ShuffleCells, a MIDORI permutation of cells ([ 0, 11, 6, 13, 10, 1, 12, 7, 5, 14, 3, 8, 15, 4, 9, 2]);
• ${\displaystyle M}$ is MixColumns: each column is multiplied by a fixed matrix M;
• ${\displaystyle S}$ is SubCells: each cell is transformed using an S-box.

The tweak for each round is updated using ${\displaystyle h,\omega }$:

• ${\displaystyle h}$ is a cell permutation from MANTIS ([ 6, 5, 14, 15, 0, 1, 2, 3, 7, 12, 13, 4, 8, 9, 10, 11]);
• ${\displaystyle \omega }$ is an LFSR applied to each of the cells with numbers [0, 1, 3, 4, 8, 11, 13]. For QARMA-64, the LFSR is (b3, b2, b1, b0) ⇒ (b0 + b1, b3, b2, b1), for QARMA-128, (b7, b6, ..., b0) ⇒ (b0 + b2, b7, b6, ..., b1),

The rounds of ${\displaystyle {\overline {\digamma }}}$ consist of inverse operations ${\displaystyle {\overline {\tau }},{\overline {M}},{\overline {S}},{\overline {h}},{\overline {\omega }}}$. Central rounds, in addition to two rounds (${\displaystyle \tau ,M,S}$ and ${\displaystyle {\overline {\tau }},{\overline {M}},{\overline {S}}}$), include multiplication of the state by an involutary matrix Q.

• Avanzi, Roberto (2016). The QARMA Block Cipher Family (PDF). IACR Transactions on Symmetric Cryptology (ToSC). Vol. 17 (published 8 March 2017). pp. 4–44. doi:10.13154/tosc.v2017.i1.4-44. Archived from the original (PDF) on May 13, 2020.
• Zong, Rui; Dong, Xiaoyang (2016). "Meet-in-the-Middle Attack on QARMA Block Cipher" (PDF). iacr.org. IACR. Retrieved 10 June 2022.
• Kaur, Jasmin; Kermani, Mehran Mozaffari; Azarderakhsh, Reza (1 January 2022). "Hardware Constructions for Lightweight Cryptographic Block Cipher QARMA With Error Detection Mechanisms". IEEE Transactions on Emerging Topics in Computing. 10 (1): 514–519. doi:10.1109/TETC.2020.3027789. eISSN 2376-4562. S2CID 226665710.
• Li, Rongjia; Jin, Chenhui (4 May 2018). "Meet-in-the-Middle Attacks on Reduced-Round QARMA-64/128". The Computer Journal. 61 (8): 1158–1165. doi:10.1093/comjnl/bxy045. eISSN 1460-2067. ISSN 0010-4620.
• Yang, Dong; Qi, Wen-feng; Chen, Hua-jin (2018). "Impossible Differential Attack on QARMA Family of Block Ciphers". Cryptology ePrint Archive.

• Public-domain Python implementation of QARMA-64
• Open-source (MIT license) implementation of QARMA-64 in C