Identity-Based Encryption from Codes with Rank Metric

Abstract

Code-based cryptography has a long history, almost as long as the history of public-key encryption (PKE). While we can construct almost all primitives from codes such as PKE, signature, group signature etc, it is a long standing open problem to construct an identity-based encryption from codes. We solve this problem by relying on codes with rank metric. The concept of identity-based encryption (IBE), introduced by Shamir in 1984, allows the use of users' identifier information such as email as public key for encryption. There are two problems that makes the design of IBE extremely hard: the requirement that the public key can be an arbitrary string and the possibility to extract decryption keys from the public keys. In fact, it took nearly twenty years for the problem of designing an efficient method to implement an IBE to be solved. The known methods of designing IBE are based on different tools: from elliptic curve pairings by Sakai, Ohgishi and Kasahara and by Boneh and Franklin in 2000 and 2001 respectively; from the quadratic residue problem by Cocks in 2001; and finally from the Learning-with-Error problem by Gentry, Peikert, and Vaikuntanathan in 2008. Among all candidates for post-quantum cryptography, there only exist thus lattice-based IBE. In this paper, we propose a new method, based on the hardness of learning problems with rank metric, to design the first code-based IBE scheme. In order to overcome the two above problems in designing an IBE scheme, we first construct a rank-based PKE, called RankPKE, where the public key space is dense and thus can be obtained from a hash of any identity. We then extract a decryption key from any public key by constructing an trapdoor function which relies on RankSign-a signature scheme from PQCrypto 2014. In order to prove the security of our schemes, we introduced a new problem for rank metric: the Rank Support Learning problem (RSL). A high technical contribution of the paper is devoted to study in details the hardness of the RSL problem.

Interleaving Loidreau’s Rank-Metric Cryptosystem

“…Note that the Interleaved Search RSD problem is similar to the problem proposed in [29,Definition 7]. The only difference is that the rows of the matrix X in Interleaved RSD Distribution have the same row space whereas the rows of U ⊤ in [29,Definition 7] have the same column space.…”

“…Note that the Interleaved Search RSD problem is similar to the problem proposed in [29,Definition 7]. The only difference is that the rows of the matrix X in Interleaved RSD Distribution have the same row space whereas the rows of U ⊤ in [29,Definition 7] have the same column space. For a small interleaving order ℓ, the currently most efficient algorithm to solve both, the Interleaved Search RSD Problem and the problem given in [29,Definition 7], was presented in [3] and will be analyzed in Section IV.…”

“…The only difference is that the rows of the matrix X in Interleaved RSD Distribution have the same row space whereas the rows of U ⊤ in [29,Definition 7] have the same column space. For a small interleaving order ℓ, the currently most efficient algorithm to solve both, the Interleaved Search RSD Problem and the problem given in [29,Definition 7], was presented in [3] and will be analyzed in Section IV. For a high interleaving order ℓ ≥ w, the algorithm proposed in [30] is able to solve the Interleaved Search RSD Problem with high probability in polynomial time.…”

“…For a high interleaving order ℓ ≥ w, the algorithm proposed in [30] is able to solve the Interleaved Search RSD Problem with high probability in polynomial time. For an interleaving order greater than wk, the algorithm proposed in [31] is able to efficiently solve [29,Definition 7], see [31, Section 6.5].…”

Efficient Decoding of Interleaved Low-Rank Parity-Check Codes

“…The resulting code has length un and dimension uk, where n and k are the length and the dimension of the component code, respectively. This LRPC code construction is motivated by the difficult generic decoding problem stated in [14,Definition 7].…”