IST Austria Thesis

In this thesis we discuss the exact security of message authentications codes HMAC , NMAC , and PMAC . NMAC is a mode of operation which turns a fixed input-length keyed hash function f into a variable input-length function. A practical single-key variant of NMAC called HMAC is a very popular and widely deployed message authentication code (MAC). PMAC is a block-cipher based mode of operation, which also happens to be the most famous fully parallel MAC. NMAC was introduced by Bellare, Canetti and Krawczyk Crypto’96, who proved it to be a secure pseudorandom function (PRF), and thus also a MAC, under two assumptions. Unfortunately, for many instantiations of HMAC one of them has been found to be wrong. To restore the provable guarantees for NMAC , Bellare [Crypto’06] showed its security without this assumption. PMAC was introduced by Black and Rogaway at Eurocrypt 2002. If instantiated with a pseudorandom permutation over n -bit strings, PMAC constitutes a provably secure variable input-length PRF. For adversaries making q queries, each of length at most ` (in n -bit blocks), and of total length σ ≤ q` , the original paper proves an upper bound on the distinguishing advantage of O ( σ 2 / 2 n ), while the currently best bound is O ( qσ/ 2 n ). In this work we show that this bound is tight by giving an attack with advantage Ω( q 2 `/ 2 n ). In the PMAC construction one initially XORs a mask to every message block, where the mask for the i th block is computed as τ i := γ i · L , where L is a (secret) random value, and γ i is the i -th codeword of the Gray code. Our attack applies more generally to any sequence of γ i ’s which contains a large coset of a subgroup of GF (2 n ). As for NMAC , our first contribution is a simpler and uniform proof: If f is an ε -secure PRF (against q queries) and a δ - non-adaptively secure PRF (against q queries), then NMAC f is an ( ε + `qδ )-secure PRF against q queries of length at most ` blocks each. We also show that this ε + `qδ bound is basically tight by constructing an f for which an attack with advantage `qδ exists. Moreover, we analyze the PRF-security of a modification of NMAC called NI by An and Bellare that avoids the constant rekeying on multi-block messages in NMAC and allows for an information-theoretic analysis. We carry out such an analysis, obtaining a tight `q 2 / 2 c bound for this step, improving over the trivial bound of ` 2 q 2 / 2 c . Finally, we investigate, if the security of PMAC can be further improved by using τ i ’s that are k -wise independent, for k > 1 (the original has k = 1). We observe that the security of PMAC will not increase in general if k = 2, and then prove that the security increases to O ( q 2 / 2 n ), if the k = 4. Due to simple extension attacks, this is the best bound one can hope for, using any distribution on the masks. Whether k = 3 is already sufficient to get this level of security is left as an open problem. Keywords: Message authentication codes, Pseudorandom functions, HMAC, PMAC

LNCS

NMAC is a mode of operation which turns a fixed input-length keyed hash function f into a variable input-length function. A practical single-key variant of NMAC called HMAC is a very popular and widely deployed message authentication code (MAC). Security proofs and attacks for NMAC can typically be lifted to HMAC. NMAC was introduced by Bellare, Canetti and Krawczyk [Crypto'96], who proved it to be a secure pseudorandom function (PRF), and thus also a MAC, assuming that (1) f is a PRF and (2) the function we get when cascading f is weakly collision-resistant. Unfortunately, HMAC is typically instantiated with cryptographic hash functions like MD5 or SHA-1 for which (2) has been found to be wrong. To restore the provable guarantees for NMAC, Bellare [Crypto'06] showed its security based solely on the assumption that f is a PRF, albeit via a non-uniform reduction. - Our first contribution is a simpler and uniform proof for this fact: If f is an ε-secure PRF (against q queries) and a δ-non-adaptively secure PRF (against q queries), then NMAC f is an (ε+ℓqδ)-secure PRF against q queries of length at most ℓ blocks each. - We then show that this ε+ℓqδ bound is basically tight. For the most interesting case where ℓqδ ≥ ε we prove this by constructing an f for which an attack with advantage ℓqδ exists. This also violates the bound O(ℓε) on the PRF-security of NMAC recently claimed by Koblitz and Menezes. - Finally, we analyze the PRF-security of a modification of NMAC called NI [An and Bellare, Crypto'99] that differs mainly by using a compression function with an additional keying input. This avoids the constant rekeying on multi-block messages in NMAC and allows for a security proof starting by the standard switch from a PRF to a random function, followed by an information-theoretic analysis. We carry out such an analysis, obtaining a tight ℓq2/2 c bound for this step, improving over the trivial bound of ℓ2q2/2c. The proof borrows combinatorial techniques originally developed for proving the security of CBC-MAC [Bellare et al., Crypto'05]

Generic Universal Forgery Attack on Iterative Hash-based MACs

In this article, we study the security of iterative hash-based MACs, such as HMAC or NMAC, with regards to universal forgery attacks. Leveraging recent advances in the analysis of functional graphs built from the iteration of HMAC or NMAC, we exhibit the very first generic universal forgery attack against hash-based MACs. In particular, our work implies that the universal forgery resistance of an n-bit output HMAC construction is not 2^n queries as long believed by the community. The techniques we introduce extend the previous functional graphs-based attacks that only took in account the cycle structure or the collision probability: we show that one can extract much more meaningful secret information by also analyzing the distance of a node from the cycle of its component in the functional graph

Cryptanalysis of HMAC/NMAC-Whirlpool

In this paper, we present universal forgery and key recovery attacks on the most popular hash-based MAC constructions, e.g., HMAC and NMAC, instantiated with an AES-like hash function Whirlpool. These attacks work with Whirlpool reduced to 6 out of 10 rounds in single-key setting. To the best of our knowledge, this is the first result on ``original\u27\u27 key recovery for HMAC (previous works only succeeded in recovering the equivalent keys). Interestingly, the number of attacked rounds is comparable with that for collision and preimage attacks on Whirlpool hash function itself. Lastly, we present a distinguishing-H attack against the full HMAC- and NMAC-Whirlpool

The Exact PRF-Security of NMAC and HMAC

NMAC is a mode of operation which turns a fixed input-length keyed hash function f into a variable input-length function. A~practical single-key variant of NMAC called HMAC is a very popular and widely deployed message authentication code (MAC). Security proofs and attacks for NMAC can typically be lifted to HMAC. NMAC was introduced by Bellare, Canetti and Krawczyk [Crypto\u2796], who proved it to be a secure pseudorandom function (PRF), and thus also a MAC, assuming that (1) f is a PRF and (2) the function we get when cascading f is weakly collision-resistant. Unfortunately, HMAC is typically instantiated with cryptographic hash functions like MD5 or SHA-1 for which (2) has been found to be wrong. To restore the provable guarantees for NMAC, Bellare [Crypto\u2706] showed its security based solely on the assumption that f is a PRF, albeit via a non-uniform reduction. Our first contribution is a simpler and uniform proof: If f is an \eps-secure PRF (against q queries) and a \delta-non-adaptively secure PRF (against q queries), then NMAC^f is an (\eps+lq\delta)-secure PRF against q queries of length at most l blocks each. We then show that this \eps+lq\delta bound is basically tight. For the most interesting case where lq\delta>=\eps we prove this by constructing an f for which an attack with advantage lq\delta exists. This also violates the bound O(l\eps) on the PRF-security of NMAC recently claimed by Koblitz and Menezes. Finally, we analyze the PRF-security of a modification of NMAC called NI [An and Bellare, Crypto\u2799] that differs mainly by using a compression function with an additional keying input. This avoids the constant rekeying on multi-block messages in NMAC and allows for a security proof starting by the standard switch from a PRF to a random function, followed by an information-theoretic analysis. We carry out such an analysis, obtaining a tight lq^2/2^c bound for this step, improving over the trivial bound of l^2q^2/2^c. The proof borrows combinatorial techniques originally developed for proving the security of CBC-MAC [Bellare et al., Crypto\u2705]. We also analyze a variant of NI that does not include the message length in the last call to the compression function, proving a l^{1+o(1)}q^2/2^c bound in this case

Equivalent Key Recovery Attacks against HMAC and NMAC with Whirlpool Reduced to 7 Rounds

A main contribution of this paper is an improved analysis against HMAC instantiating with reduced Whirlpool. It recovers equivalent keys, which are often denoted as Kin and Kout, of HMAC with 7-round Whirlpool, while the previous best attack can work only for 6 rounds. Our approach is applying the meet-in-the-middle (MITM) attack on AES to recover MAC keys of Whirlpool. Several techniques are proposed to bypass different attack scenarios between a block cipher and a MAC, e.g., the chosen plaintext model of the MITM attacks on AES cannot be used for HMAC-Whirlpool. Besides, a larger state size and different key schedule designs of Whirlpool leave us a lot of room to study. As a result, equivalent keys of HMAC with 7-round Whirlpool are recovered with a complexity of (Data, Time, Memory) = (2^481.7, 2^482.3, 2^481)

One-Key Compression Function Based MAC with Security beyond Birthday Bound

Ga{\v z}i et al. [CRYPTO 2014] analyzed the NI-MAC construction proposed by An and Bellare [CRYPTO 1999] and gave a tight birthday-bound of $O(\ell q^{2}/2^{n})$, as an improvement over the previous bound of $O(\ell^{2}q^{2}/2^{n})$. In this paper, we design a simple extension of NI-MAC, called NI$^+$-MAC, and prove that it has security bound beyond birthday (BBB) of order $O(q^2\ell^2 / 2^{2n})$ provided $\ell \leq 2^{n/4}$. Our construction not only lifts the security of NI-MAC beyond birthday, it also reduces the number of keys from 2 (NI uses 2 independent keys) to 1. Before this work, Yasuda had proposed [FSE 2008] a single fixed-keyed compression function based BBB-secure MAC with security bound $O(\ell q^2/2^{2n})$ that uses an extra mask, requires a storage space to store the mask. However, our proposed construction NI$^+$ does not require any extra mask and thereby has reduced the state size compared to Yasuda\u27s proposal [FSE 2008] with providing the same order of security bound for light-weight application

Distinguishing and Forgery Attacks on Alred and Its AES-based Instance Alpha-MAC

In this paper, we present new distinguishers of the MAC construction \textsc{Alred} and its specific instance \textsc{Alpha}-MAC based on AES, which is proposed by Daemen and Rijmen in 2005. For the \textsc{Alred} construction, we describe a general distinguishing attack which leads to a forgery attack directly. The complexity is $2^{64.5}$ chosen messages and $2^{64.5}$ queries with success probability 0.63. We also use a two-round collision differential path for \textsc{Alpha}-MAC, to construct a new distinguisher with about $2^{65.5}$ queries. The most important is that the new distinguisher can be used to recover the internal state, which is an equivalent secret subkey, and leads to a second preimage attack. Moreover, the distinguisher on \textsc{Alred} construction is also applicable to the MACs based on CBC and CFB encryption mode

On the Security of the COPA and Marble Authenticated Encryption Algorithms against (Almost) Universal Forgery Attack

COPA is a block-cipher-based authenticated encryption mode with a provable birthday-bound security under the assumption that the underlying block cipher is a strong pseudorandom permutation, and its instantiation with the AES block cipher is called AES-COPA. Marble is an AES-based COPA-like authenticated encryption algorithm with a full security. In this paper, we analyse the security of COPA and Marble against universal forgery attacks. We present beyond-birthday-bound (almost) universal forgery attacks on the COPA when used with constant or variable associate data, and present (almost) universal forgery attacks on the Marble when used without associated data or with (variable) associate data. Our attacks on the COPA with variable associate data have a complexity very near the birthday bound, and their applications to AES-COPA show that the security claim of AES-COPA against tag guessing may be not correct; and our attacks on the (newest as well as initial version of) Marble with associate data show that Marble does not provide a full security that the designer claimed. Like many recently published cryptanalytic results on message authentication algorithms with a provable birthday-bound security, our attacks on COPA do not violate its security proofs, but provide a comprehensive understanding of its security against universal forgery attack, show that the success probability of a universal forgery on the COPA is larger than the ideal bound $2^{-n}$ of the standard forgery-resistance, and boil down to an existing open question: Should a message authentication algorithm with a weaker security claim than the standard forgery-resistance be regarded as a sound design