Paper 2026/383

HCTR$^{++}$ : A Beyond Birthday Bound Secure HCTR2 Variant

Onur Koçak, TUBITAK BILGEM

Oğuz Yayla, Middle East Technical University

Abstract

Current industry-standard block cipher modes of operation, such as CBC and GCM, are fundamentally limited by the birthday bound $O(2^{n/2})$, a constraint that has evolved from a theoretical concern into a practical security bottleneck in contemporary high-throughput, high-data-volume environments. To address this, the cryptographic community and NIST are prioritizing Beyond Birthday Bound (BBB) security to extend the operational security margin toward the full block size $O(2^n)$. Achieving BBB security requires a departure from traditional constructions, primarily utilizing three methodologies: XOR of Permutations (XORP), Tweakable Block Ciphers (TBCs), and Fresh Re-keying. While none of these innovative BBB modes have been formally standardized, NIST has initiated the Accordion Mode project, defining a new primitive class: the Tweakable Variable-Input-Length Strong Pseudorandom Permutation (VIL-SPRP). This primitive treats the entire message as a single, indivisible block and expects the submission of BBB-secure variants. To contribute to this standardization effort, we propose a simple BBB-secure variant of the HCTR2 algorithm. We first explain the core BBB methodologies, then discuss the operational mechanism of HCTR2, and finally present our proposed BBB-secure construction.

@misc{cryptoeprint:2026/383,
      author = {Gülnihal Öztürk and Onur Koçak and Oğuz Yayla},
      title = {{HCTR}$^{++}$ : A Beyond Birthday Bound Secure {HCTR2} Variant},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/383},
      year = {2026},
      url = {https://eprint.iacr.org/2026/383}
}