A Note on the Chi-square Method : A Tool for Proving Cryptographic Security

In CRYPTO 2017, Dai, Hoang, and Tessaro introduced the {\em Chi-square method} ($\chi^2$ method) which can be applied to obtain an upper bound on the statistical distance between two joint probability distributions. The authors applied this method to prove the {\em pseudorandom function security} (PRF-security) of sum of two random permutations. In this work, we revisit their proof and find a non-trivial gap in the proof and describe how to plug this gap as well; this has already been done by Dai {\em et al.} in the revised version of their CRYPTO 2017 paper. A complete, correct, and transparent proof of the full security of the sum of two random permutations construction is much desirable, especially due to its importance and two decades old legacy. The proposed $\chi^2$ method seems to have potential for application to similar problems, where a similar gap may creep into a proof. These considerations motivate us to communicate our observation in a formal way.\par On the positive side, we provide a very simple proof of the PRF-security of the {\em truncated random permutation} construction (a method to construct PRF from a random permutation) using the $\chi^2$ method. We note that a proof of the PRF-security due to Stam is already known for this construction in a purely statistical context. However, the use of the $\chi^2$ method makes the proof much simpler

Sequential Indifferentiability of Confusion-Diffusion Networks

A large proportion of modern symmetric cryptographic building blocks are designed using the Substitution-Permutation Networks (SPNs), or more generally, Shannon\u27s confusion-diffusion paradigm. To justify its theoretical soundness, Dodis et al. (EUROCRYPT 2016) recently introduced the theoretical model of confusion-diffusion networks, which may be viewed as keyless SPNs using random permutations as S-boxes and combinatorial primitives as permutation layers, and established provable security in the plain indifferentiability framework of Maurer, Renner, and Holenstein (TCC 2004). We extend this work and consider Non-Linear Confusion-Diffusion Networks (NLCDNs), i.e., networks using non-linear permutation layers, in weaker indifferentiability settings. As the main result, we prove that 3-round NLCDNs achieve the notion of sequential indifferentiability of Mandal et al. (TCC 2012). We also exhibit an attack against 2-round NLCDNs, which shows the tightness of our positive result on 3 rounds. It implies correlation intractability of 3-round NLCDNs, a notion strongly related to known-key security of block ciphers and secure hash functions. Our results provide additional insights on understanding the complexity for known-key security, as well as using confusion-diffusion paradigm for designing cryptographic hash functions

Cryptosystems Resilient to Both Continual Key Leakages and Leakages from Hash Functions

Yoneyama et al. introduced Leaky Random Oracle Model (LROM for short) at ProvSec2008 in order to discuss security (or insecurity) of cryptographic schemes which use hash functions as building blocks when leakages from pairs of input and output of hash functions occur. This kind of leakages occurs due to various attacks caused by sloppy usage or implementation. Their results showed that this kind of leakages may threaten the security of some cryptographic schemes. However, an important fact is that such attacks would leak not only pairs of input and output of hash functions, but also the secret key. Therefore, LROM is rather limited in the sense that it considers leakages from pairs of input and output of hash functions alone, instead of taking into consideration other possible leakages from the secret key simultaneously. On the other hand, many other leakage models mainly concentrate on leakages from the secret key and ignore leakages from hash functions for a cryptographic scheme exploiting hash functions in these leakage models. Some examples show that the above drawbacks of LROM and other leakage models may cause insecurity of some schemes which are secure in the two kinds of leakage model. In this paper, we present an augmented model of both LROM and some leakage models, which both the secret key and pairs of input and output of hash functions can be leaked. Furthermore, the secret key can be leaked continually during the whole life cycle of a cryptographic scheme. Hence, our new model is more universal and stronger than LROM and some leakage models (e.g. only computation leaks model and bounded memory leakage model). As an application example, we also present a public key encryption scheme which is provably IND-CCA secure in our new model

Security of Practical Cryptosystems Using Merkle-Damgard Hash Function in the Ideal Cipher Model

Since the Merkle-Damgård (MD) type hash functions are differentiable from ROs even when compression functions are modeled by ideal primitives, there is no guarantee as to the security of cryptosystems when ROs are instantiated with structural hash functions. In this paper, we study the security of the instantiated cryptosystems whereas the hash functions have the well known structure of Merkle-Damgård construction with Stam\u27s type-II compression function (denoted MD-TypeII) in the Ideal Cipher Model (ICM). Note that since the Type-II scheme includes the Davies-Meyer compression function, SHA-256 and SHA-1 have the MD-TypeII structure. We show that OAEP, RSA-KEM, PSEC-KEM, ECIES-KEM and many other encryption schemes are secure when using the MD-TypeII hash function. In order to show this, we customize the indifferentiability framework of Maurer, Renner and Holenstein. We call the customized framework ``indifferentiability with condition\u27\u27. In this framework, for some condition $\alpha$ that cryptosystem $C$ satisfies, if hash function $H$ is indifferentiable from RO under condition $\alpha$, $C$ is secure when RO is instantiated with $H$. We note the condition of ``prefix-free\u27\u27 that the above schemes satisfy. We show that the MD-TypeII hash function is indifferentiable from RO under this condition. When the output length of RO is incompatible with that of the hash function, the output size is expanded by Key Derivation Functions (KDFs). Since a KDF is specified as MGF1 in RSA\u27s PKCS $\#$1 V2.1, its security discussion is important in practice. We show that, KDFs using the MD-TypeII hash function (KDF-MD-TypeII) are indifferentiable from ROs under this condition of ``prefix-free\u27\u27. Therefore, we can conclude that the above practical encryption schemes are secure even when ROs are instantiated with (KDF-)MD-TypeII hash functions. Dodis, Ristenpart and Shrimpton showed that FDH, PSS, Fiat-Shamir, and so on are secure when RO is instantiated with the MD-TypeII hash function in the ICM, their analyses use the different approach from our approach called indifferentiability from public-use RO (pub-RO). They showed that the above cryptosystems are secure in the pub-RO model and the MD-TypeII hash function is indifferentiable from pub-RO. Since their analyses did not consider the structure of KDFs, there might exist some attack using a KDF\u27s structure. We show that KDFs using pub-RO (KDF-pub-RO) is differentiable from pub-RO. Thus, we cannot trivially extend the result of Dodis et al to the indifferentiability for KDF-MD-TypeII hash functions. We propose a new oracle called private interface leak RO (privleak-RO). We show that KDF-pub-ROs are indifferentiable from privleak-ROs and the above cryptosystems are secure in the privleak-RO model. Therefore, by combining the result of Dodis et al. with our result, we can conclude that the above cryptosystems are secure when ROs are instantiated with KDF-MD-TypeII hash functions. Since OAEP, RSA-KEM, PSEC-KEM, ECIES-KEM and many other encryption schemes are insecure in the pub-RO (privleak-RO) model, we cannot confirm the security of these encryption schemes from the approach of Dodis et al. Therefore, the result of Dodis et al can be supplemented with our result. Consequently, from the two results we can confirm the security of almost practical cryptosystems when ROs are instantiated with (KDF-)MD-TypeII hash functions

Salvaging Merkle-Damgard for Practical Applications

Many cryptographic applications of hash functions are analyzed in the random oracle model. Unfortunately, most concrete hash functions, including the SHA family, use the iterative (strengthened) Merkle-Damgard transform applied to a corresponding compression function. Moreover, it is well known that the resulting ``structured\u27\u27 hash function cannot be generically used as a random oracle, even if the compression function is assumed to be ideal. This leaves a large disconnect between theory and practice: although no attack is known for many concrete applications utilizing existing (Merkle-Damgard-based) hash functions, there is no security guarantee either, even by idealizing the compression function. Motivated by this question, we initiate a rigorous and modular study of finding kinds of (still idealized) hash functions which would be (a) elegant and interesting in their own right; (b) still enough to argue security of important applications; and (c) provably instantiable by the (strengthened) Merkle-Damgard transform, applied to a strong enough compression function. We develop two such notions which we believe are natural and interesting in their own right: preimage awareness and being indifferentiable from a public-use random oracle

IMPROVING THE ROUND COMPLEXITY OF IDEAL-CIPHER CONSTRUCTIONS

Block ciphers are an essential ingredient of modern cryptography. They are widely used as building blocks in many cryptographic constructions such as encryption schemes, hash functions etc. The security of block ciphers is not currently known to reduce to well-studied, easily formulated, computational problems. Nevertheless, modern block-cipher constructions are far from ad-hoc, and a strong theory for their design has been developed. Two classical paradigms for block cipher design are the Feistel network and the key-alternating cipher (which is encompassed by the popular substitution-permutation network). Both of these paradigms that are iterated structures that involve applications of random-looking functions/permutations over many rounds. An important area of research is to understand the provable security guarantees offered by these classical design paradigms for block cipher constructions. This can be done using a security notion called indifferentiability which formalizes what it means for a block cipher to be ideal. In particular, this notion allows us to assert the structural robustness of a block cipher design. In this thesis, we apply the indifferentiability notion to the two classical paradigms mentioned above and improve upon the previously known round complexity in both cases. Specifically, we make the following two contributions: (1) We show that a 10-round Feistel network behaves as an ideal block cipher when the keyed round functions are built using a random oracle. (2) We show that a 5-round key-alternating cipher (also known as the iterated Even-Mansour construction) with identical round keys behaves as an ideal block cipher when the round permutations are independent, public random permutations