, Equation (4) turns into a proper inclusion as long as q is not a prime. Besides, by definition of lifted codes
, The restriction of f along L i can be interpolated as a univariate polynomial f |L i (T ) of degree at most q ? 2, since (f (Q)) Q?L i lies in the Reed-Solomon code RS q (q ? 1), by definition of lifted codes, Therefore f |L i (T ) = 0, and f vanishes on L i. Repeating arguments in the proof of Lemma 5.5, we get a ? deg f ? 2q ? 2, and d min
, Similarly to Proposition 5.6, we can then prove that practical PoRs can be constructed with the family of lifted codes Lift
, and (Q, V ) its associated verification structure. If every ? w is sufficiently uniform, then the PoR scheme associated to C and (Q, V ) is (?, ? )-sound for every ? < 1 and ? = O( 1 (1??)q 2 )
, The crucial improvement is that lifted codes potentially have much higher dimension than ReedMuller codes
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, We here confirm our heuristic on the fact that ? w is sufficiently uniform, by providing experimental estimates of ?
, as presented in Section 5.2.1. We also fix the word w ? F n q uploaded on the server during the initialisation step. Remark that, for varying w, all ? w are equivalently distributed. Indeed, if ? ? S(F q ) n satisfies ?(w) = w , then the distribution of permutations picked from ? w can be obtained by applying ? to permutations picked from ? w. Hence, without loss of generality we assume that w = 0. Proposition 3.16 claims that in this context, ? should be O(1/q) since ? = 2 and ? q. For convenience, Setup. We consider PoR schemes using Reed-Muller codes C = RM q
, ? Test 1. We sample N challenges u, and for each sample, we fix t ? and r u in {x ? F q , |Z u | = t}. Then, we estimate p ? by running M trials and computing the average
, ? Test 2. A challenge u is fixed, and for several values of t, we pick N responses r u randomly in {x ? F q , |Z u | = t}. For every r u , we estimate p ? with M samples. We collect the maximum value of ? M
, A challenge u is fixed, as well as a response r u to this challenge which satisfies |Z u | = t for several values of t ? [2, We then run M trials and collect ? M (p ? ), vol.3
, At the end of the document, Figures 5, 6 and 7 confirm that, for fixed N and q, and for any Test i we use