Polynomial Structures in Code-Based Cryptography

Abstract

Abstract. In this article we discus a probability problem applied in the code based cryptography. It is related to the shape of the polynomials with exactly t different roots. We will show that the structure is very dense and the probability that this type of polynomials has at least one coefficient equal to zero is extremelly low. We treated this issue in our research of natural countermeasures to a timing attack against the polynomial evaluation.

Higher-Order Masking Scheme against DPA Attack in Practice: McEliece Cryptosystem Based on QD-MDPC Code

“…If the work factor (WF), namely, the time complexity of successfully attacking a cryptosystem, is greater, the public key cryptosystem based on the error-correcting code has more security. For the LDPC code, the WF of the DCA is heavily correlated with the line weight of ′ , and does not have a significant correlation with code length n [20]. To allow the QD code to resist a DCA, we use the MDPC code to improve the QD code, and the QD-MDPC code then generates a parity-check matrix H that has a larger density.…”

“…If = 0, attackers can easy obtain the plaintext = ′−1 . Therefore, the WF of ISDA is associated with the error correcting vector e, and increases when the density of the error correcting vector e is added [20]. The MDPC code is compared with the LDPC code, and has a higher error correction capability.…”

“…The BF Decoding Algorithm -Gallager proposed a BF decoding algorithm for the LDPC code [19]. According to the parity-check matrix of the QD-MDPC code, which has a larger density, the decision threshold is set as − , where is a small integer [20]. The number of iterations is reduced through an increase in the number of flipping bits per iteration, and thus the decoding complexity is reduced.…”

Improved Timing Attacks against the Secret Permutation in the McEliece PKC

“…Each time a coefficient equals one or zero it will be store in a special table used afterwards for multiplication or addition. The case where a coefficient equals zero is rare and its probability has been studied in [4]. Nevertheless, each time there's a coefficient equal to zero we will no longer multiply it by the corresponding element as the multiplication equals zero.…”

“…This problem was treated in [4] and the results show that the probability can be bounded. For the classical parameters of the McEliece PKC, i.e.…”