Re: [Cfrg] Structure in the S-box of the Russian algorithms (RFC 6986, RFC 7801)

[Paul Lambert <[e-mail]>]

Hi Leo,


> On Feb 10, 2019, at 1:49 PM, Leo Perrin <[e-mail]> wrote:
> 
> Dear CFRG Participants,
> 
> My name is Léo Perrin, I am a post-doc in symmetric cryptography at Inria, and I would like to bring recent results of mine to your attention. They deal with the last two Russian standards in symmetric crypto, namely RFC 7801 (Kuznyechik, a block cipher) and RFC 6986 (Streebog, a hash function). My conclusion is that their designers purposefully used (and did not disclose) a very specific structure to build their S-box. The knowledge of this structure demands a renewed analysis of their algorithms in its light. While I do not have an attack at the moment, these results lead me to urge caution about using these algorithms.
> 
> Let me summarize my results.
> 
> Both algorithms use the same 8-bit S-box, pi, which is only specified via its lookup table. The designers never disclosed their rationale for their choice and never disclosed the structure they used. I have managed to identify what I claim to be the structure purposefully used by its designers to construct pi. The corresponding paper was accepted at ToSC and is already on eprint: https://eprint.iacr.org/2019/092 <https://urldefense.proofpoint.com/v2/url?u=https-3A__eprint.iacr.org_2019_092&d=DwMFAw&c=qgVugHHq3rzouXkEXdxBNQ&r=oIg4FfS8P761BlhMPJ2ys3IvSyH4XQ12Mbj_mXrCAJs&m=S0H637y57cjgJ_W4QG8e2TIQ0lLL3cDdmc5Lf8_MxfI&s=uVblnQv8g31qEeLvM80bundiG3ewh95591JrcjZKvGQ&e=>
> 
> With my then colleagues from the university of Luxembourg, we previously found two different structures in this component and published them a couple years ago [1,2]. However, we were not satisfied with these results as the structures we found were bulky and just plain weird. The one I just found is much simpler and has both previous decompositions as side effects---in fact, we conjectured the existence of such a nicer structure in [2]. Much more importantly, this new decomposition highlights some very specific (and, in my opinion, worrying) properties of pi that were not known before.
> 
> In a nutshell, pi is actually defined over the finite field GF(2^8) in such a way as to map multiplicative cosets of GF(2^4) to additive cosets of GF(2^4). Furthermore, the restriction of the permutation to each multiplicative coset is always the same. Also, the linear layer of Streebog---specified via a 64x64 binary matrix by its designers, including in RFC 6986---is in fact an 8x8 matrix defined over GF(2^8) using the same irreducible polynomial as in the S-box. Thus, at least in the case of Streebog, both the linear layer and the S-box interact in a highly structured way with two partitions of GF(2^8) and one of those is its partition into additive cosets of the subfield (this will be important later).
> 
> This situation is unlike anything else in the literature. For example, while the inverse in GF(2^8) preserves the partition into multiplicative cosets of GF(2^8), the AES designers composed it with an affine mapping breaking the GF(2^8) structure. It is not the case here. On the other hand, Arnaud Bannier proved in his PhD (see also [3]) that an S-box preserving a partition of the space into additive cosets in such a way that it interacts with the linear layer was necessary to build some specific backdoors.
> 
> Still, at the moment, I don't know of any attack leveraging my new decomposition as the partition in the input is the partition in multiplicative cosets (and not additive ones). Nevertheless, I can't think of a good reason for the designers of these algorithms to use this structure and, worse, to keep this fact secret; especially since the presence of such properties demands a specific analysis to ensure that the algorithms are safe.
> 
> I felt I had to bring these results to the attention of the CFRG. If you have any questions I'd be happy to answer them!

Interesting work … looking at the walsh function based non-linearity of Streebog, it is non-optimal (compared to AES and SMS4):
	AES non-linearity  (min, max) =  (112.0, 112.0)
	sms4 non-linearity (min, max) =  (112.0, 112.0)
	Streebog non-linearity  (min, max) =  (102.0, 110.0)

This was using: https://github.com/nymble/cryptopy/blob/master/analysis/sbox_nonlinearity.py 

Paul


> 
> Best regards,
> /Léo Perrin
> 
> [1] Alex Biryukov, Léo Perrin, Aleksei Udovenko. "Reverse-Engineering the S-Box of Streebog, Kuznyechik and STRIBOBr1". Eurocrypt'16, available online: https://eprint.iacr.org/2016/071
> [2] Léo Perrin, Aleksei Udovenko. "Exponential S-Boxes: a Link Between the S-Boxes of BelT and Kuznyechik/Streebog ". ToSC'16. Available online: https://tosc.iacr.org/index.php/ToSC/article/view/567
> [3] Arnaud Bannier, Nicolas Bodin, Éric Filiol. "Partition-based trapdoor ciphers". https://eprint.iacr.org/2016/493
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