On formal verification of arithmetic-based cryptographic primitives

Cryptographic primitives are fundamental for information security: they are used as basic components for cryptographic protocols or public-key cryptosystems. In many cases, their security proofs consist in showing that they are reducible to computationally hard problems. Those reductions can be subtle and tedious, and thus not easily checkable. On top of the proof assistant Coq, we had implemented in previous work a toolbox for writing and checking game-based security proofs of cryptographic primitives. In this paper we describe its extension with number-theoretic capabilities so that it is now possible to write and check arithmetic-based cryptographic primitives in our toolbox. We illustrate our work by machine checking the game-based proofs of unpredictability of the pseudo-random bit generator of Blum, Blum and Shub, and semantic security of the public-key cryptographic scheme of Goldwasser and Micali.Comment: 13 page

Quantifying Shannon's Work Function for Cryptanalytic Attacks

Attacks on cryptographic systems are limited by the available computational resources. A theoretical understanding of these resource limitations is needed to evaluate the security of cryptographic primitives and procedures. This study uses an Attacker versus Environment game formalism based on computability logic to quantify Shannon's work function and evaluate resource use in cryptanalysis. A simple cost function is defined which allows to quantify a wide range of theoretical and real computational resources. With this approach the use of custom hardware, e.g., FPGA boards, in cryptanalysis can be analyzed. Applied to real cryptanalytic problems, it raises, for instance, the expectation that the computer time needed to break some simple 90 bit strong cryptographic primitives might theoretically be less than two years.Comment: 19 page

Replacing Probability Distributions in Security Games via Hellinger Distance

Security of cryptographic primitives is usually proved by assuming "ideal" probability distributions. We need to replace them with approximated "real" distributions in the real-world systems without losing the security level. We demonstrate that the Hellinger distance is useful for this problem, while the statistical distance is mainly used in the cryptographic literature. First, we show that for preserving ?-bit security of a given security game, the closeness of 2^{-?/2} to the ideal distribution is sufficient for the Hellinger distance, whereas 2^{-?} is generally required for the statistical distance. The result can be applied to both search and decision primitives through the bit security framework of Micciancio and Walter (Eurocrypt 2018). We also show that the Hellinger distance gives a tighter evaluation of closeness than the max-log distance when the distance is small. Finally, we show that the leftover hash lemma can be strengthened to the Hellinger distance. Namely, a universal family of hash functions gives a strong randomness extractor with optimal entropy loss for the Hellinger distance. Based on the results, a ?-bit entropy loss in randomness extractors is sufficient for preserving ?-bit security. The current understanding based on the statistical distance is that a 2?-bit entropy loss is necessary

Forward-Security in Private-Key Cryptography

This paper provides a comprehensive treatment of forward-security in the context of sharedkey based cryptographic primitives, as a practical means to mitigate the damage caused by key-exposure. We provide definitions of security, practical proven-secure constructions, and applications for the main primitives in this area. We identify forward-secure pseudorandom bit generators as the central primitive, providing several constructions and then showing how forward-secure message authentication schemes and symmetric encryption schemes can be built based on standard schemes for these problems coupled with forward-secure pseudorandom bit generators. We then apply forward-secure message authentication schemes to the problem of maintaining secure access logs in the presence of break-ins

Simple, near-optimal quantum protocols for die-rolling

Die-rolling is the cryptographic task where two mistrustful, remote parties wish to generate a random $D$-sided die-roll over a communication channel. Optimal quantum protocols for this task have been given by Aharon and Silman (New Journal of Physics, 2010) but are based on optimal weak coin-flipping protocols which are currently very complicated and not very well understood. In this paper, we first present very simple classical protocols for die-rolling which have decent (and sometimes optimal) security which is in stark contrast to coin-flipping, bit-commitment, oblivious transfer, and many other two-party cryptographic primitives. We also present quantum protocols based on integer-commitment, a generalization of bit-commitment, where one wishes to commit to an integer. We analyze these protocols using semidefinite programming and finally give protocols which are very close to Kitaev's lower bound for any $D \geq 3$. Lastly, we briefly discuss an application of this work to the quantum state discrimination problem.Comment: v2. Updated titl

Using Simon's Algorithm to Attack Symmetric-Key Cryptographic Primitives

We present new connections between quantum information and the field of classical cryptography. In particular, we provide examples where Simon's algorithm can be used to show insecurity of commonly used cryptographic symmetric-key primitives. Specifically, these examples consist of a quantum distinguisher for the 3-round Feistel network and a forgery attack on CBC-MAC which forges a tag for a chosen-prefix message querying only other messages (of the same length). We assume that an adversary has quantum-oracle access to the respective classical primitives. Similar results have been achieved recently in independent work by Kaplan et al. Our findings shed new light on the post-quantum security of cryptographic schemes and underline that classical security proofs of cryptographic constructions need to be revisited in light of quantum attackers.Comment: 14 pages, 2 figures. v3: final polished version, more formal definitions adde