Volume 8, Issue 3

We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the parties to construct a shared commutative square despite the non-commutativity of the endomorphism ring. We give a precise formulation of the necessary computational assumptions along with a discussion of their validity, and prove the security of our protocols under these assumptions. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves. This paper is an extended version of [Lecture Notes in Comput. Sci. 7071, Springer (2011), 19–34]. We add a new zero-knowledge identification scheme and detailed security proofs for the protocols. We also present a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data.

Hardware devices can be protected against side-channel attacks by introducing one random mask per sensitive variable. The computation throughout is unaltered if the shares (masked variable and mask) are processed concomitantly, in two distinct registers. Nonetheless, this setup can still be attacked if the side-channel is squared, because this operation causes an interference between the two shares. This more sophisticated analysis is referred to as a zero-offset second-order correlation power analysis (CPA) attack. When the device leaks in Hamming distance, the countermeasure can be improved by the “leakage squeezing”. It consists in manipulating the mask through a bijection, aimed at reducing the dependency between the shares' leakage. Thus d th-order zero-offset attacks, that consist in applying CPA on the d th power of the centered side-channel traces, can be thwarted for d ≥ 2 at no extra cost. We denote by n the size in bits of the shares and call F the transformation function, that is, a bijection of 𝔽2n$\mathbb {F}_2^n$. In this paper, we explore the functions F that thwart zero-offset high-order CPA (HO-CPA) of maximal order d . We mathematically demonstrate that optimal choices for F relate to optimal binary codes (in the sense of communication theory). First, we exhibit optimal linear F functions. They are suitable for masking schemes where only one mask is used throughout the algorithm. Second, we note that for values of n for which non-linear codes exist with better parameters than linear ones, better protection levels can be obtained. This applies to implementations in which each mask is randomly cast independently of the previous ones. These results are exemplified in the case n = 8, where the optimal F can be identified: it is derived from the optimal rate 1/2 binary code of size 2n$2n$, namely the Nordstrom–Robinson (16,256,6)$(16, 256, 6)$ code. This example provides explicitly with the optimal protection that limits to one mask of byte-oriented algorithms such as AES or AES-based SHA-3 candidates. It protects against all zero-offset HO-CPA attacks of order d≤5$d \le 5$. Eventually, the countermeasure is shown to be resilient to imperfect leakage models, where the registers leak differently than the sum of their toggling bits.

The Modified Rivest Scheme (MRS) is an additive homomorphic scheme recently used in many applications which demands third party processing of encrypted data. The present study carries out a comprehensive security analysis of MRS. We work out an attack from the category of known plaintext, chosen plaintext, chosen ciphertext where the adversary is having with him the pair of plaintext and its corresponding ciphertext. It is demonstrated that in such a scenario the adversary can compute the private key of the legitimate node causing threat to the security of the entire system. The novelty of the present study lies in the fact that any attack from the above mentioned category could be mounted on MRS (which is not being attacked so far), irrespective of the fact whether the modulus of the underlying MRS is kept private or made public.

In this paper, we revisit the fully homomorphic encryption (FHE) scheme implemented by Gentry and Halevi, which is just an instantiation of Gentry's original scheme based on ideal lattices. Their FHE scheme starts from a somewhat homomorphic encryption (SHE) scheme, and its decryption range is deeply related with the FHE construction. Gentry and Halevi gave an experimental evaluation of the decryption range, but theoretical evaluations have not been given so far. Moreover, we give a theoretical upper bound, and reconsider suitable parameters for theoretically obtaining an FHE scheme. In particular, while Gentry and Halevi use the Euclidean norm evaluation in the noise management of ciphertexts, our theoretical bound enables us to use the ∞-norm evaluation, and hence it helps to lower the difficulty of controlling the noise density of ciphertexts.