Julien Lavauzelle
email: julien.lavauzelle@inria.fr

Françoise Levy-Dit-Vehel

Generic constructions of PoRs from codes and instantiations

In this paper, we show how to construct, from any linear code, a Proof of Retrievability (PoR) which features very low computation complexity on both the client (Verifier) and server (Prover) side, as well as small client storage (typically 512 bits). We adapt the security model initiated by Juels and Kaliski [JK07] to fit into the framework of Paterson et al. [PSU13], from which our construction evolves. We thus provide a rigorous treatment of the security of our generic design; more precisely, we sharply bound the extraction failure of our protocol according to this security model. Next, we instantiate our formal construction with codes built from tensor-products as well as with Reed-Muller codes and lifted codes [GKS13], yielding PoRs with moderate communication complexity and (server) storage overhead, in addition to the aforementioned features.

Introduction

Motivation

Cloud computing and storage has evolved quite spectacularly over the past decade. Especially, data outsourcing allows users and companies to lighten their storage burden and maintenance cost. Though, it raises several issues: for example, how can someone check efficiently that he can retrieve without any loss a massive file that he had uploaded on a distant server and erased from his personal system? Proofs of retrievability (PoRs) address this issue. They are cryptographic protocols involving two parts: a client (or a verifier) and a server (or a prover). PoRs usually consist in the following phases. First, a key generation process creates secret material related to the file, meant to be kept by the client only. Then the file is initialised, that is, it is encoded and/or encrypted according to the secret data held by the client. This processed file is uploaded to the server. In order to check retrievability, the client can run a verification procedure, which is the core of the PoR. Finally, if the client is convinced that the server still holds his file, the client can proceed at any time to the extraction of the file.

Several parameters must be taken into account. Plainly, the verification process has to feature a low communication complexity, as the main goal is to avoid downloading a large part of the file to only check its extractability. Second, the storage overhead induced by the protocol must be low, as large server overhead would imply high fees for the customer. Third, the computation cost of the verification procedure must be low, both for the client (which is likely to own a lightweight device) and the server (whose computation work could also be expensive for the client).

Notice that proofs of data possession (PDP) represent protocols close to what is needed in PoRs. However, in PDPs one does not require the client to be able to extract the file from the server. Instances of PDPs are given by Ateniese et [START_REF] Naor | The Complexity of Online Memory Checking[END_REF] are very often seen as precursors for PoRs. For instance, the work of Naor and Rothblum [START_REF] Naor | The Complexity of Online Memory Checking[END_REF] considers a setting in which the client directly accesses the file stored by the prover/server (while the actual PoR definition uses "an arbitrary program as opposed to a simple memory layout and this program may answer these questions in an arbitrary manner" [START_REF] Shacham | Compact Proofs of Retrievability[END_REF]).

Previous work

Juels and Kaliski [START_REF] Juels | PORs: Proofs of Retrievability for Large Files[END_REF] gave the first formal definition of PoRs. They also proposed a first construction based on so-called sentinels (namely, random parts of the file to be checked during the verification step) the client keeps secretly on his device. Additionally, an erasure code ensures the integrity of the file to be extracted. This seminal work also raised several interesting points. On the one hand, it revealed that (i) the client must store secret data to be used in the verification step and (ii) coding is needed in order to retrieve the file without erasures or errors. On the other hand, in JK's construction the verification step can only be performed a finite number of times, since sentinels cannot be reused endlessly.

As a consequence, Shacham and Waters proposed to consider unbounded-use PoRs in [START_REF] Shacham | Compact Proofs of Retrievability[END_REF], where they built two kinds of PoRs. The first one is based on linear combinations of authenticators produced via pseudo-random functions; its security was proved using cryptographic tools such as unforgeable MAC scheme, semantically secure symmetric encryption and secure PRFs. The second one is a publicly verifiable scheme based on the Diffie-Hellman problem in bilinear groups.

Bowers, Juels and Oprea [BJO09] adopted a coding-theoretic approach (inner code, outer code) to compare variants of SW and JK schemes. They focused on the efficiency of the schemes, and proved that, despite bounded-use, new variants of JK construction are highly competitive compared to other existing schemes.

In [START_REF] Paterson | A Coding Theory Foundation for the Analysis of General Unconditionally Secure Proof-of-Retrievability Schemes for Cloud Storage[END_REF], Paterson et al. provide a general framework for PoRs in the unconditional security model. They show that retrievability of the file can be expressed as error-correction of a so-called response code. That allows them to precisely quantify the extraction success as a function of the success probability of a proving algorithm: indeed, in this setting, extraction can be naturally seen as nearest-neighbour decoding in the response code. They notably apply their framework to prove the security of a modified version of SW scheme. Also notice that, prior to [START_REF] Paterson | A Coding Theory Foundation for the Analysis of General Unconditionally Secure Proof-of-Retrievability Schemes for Cloud Storage[END_REF], Dodis, Vahan and Wichs [START_REF] Dodis | Proofs of Retrievability via Hardness Amplification[END_REF] proposed another coding-theoretic model for PoRs that allowed them to build efficient bounded-use and unbounded-use PoR schemes.

With practicality in mind, other features have been deployed on PoRs. For instance, Wang et al. [WWR + 11] presented a PoR construction based on Merkle hash trees, which allows efficient file updates on the server. Their scheme is provably secure under cryptographic assumptions (hardness of Diffie-Hellman in bilinear groups, unforgeable signatures, etc.), and has been improved by Mo, Zhou and Chen [START_REF] Mo | A Dynamic Proof of Retrievability (PoR) Scheme with O(log n) Complexity[END_REF] in order to prevent unbalanced trees. More recently, other features have been proposed for PoRs, such as multi-prover PoRs (see [PSU18]) or public verifiability (for instance in [START_REF] Sengupta | Efficient Proofs of Retrievability with Public Verifiability for Dynamic Cloud Storage[END_REF]).

Our approach

As we remarked before, most PoR schemes rely on two techniques: (i) the client locally stores secret data in order to check the integrity of the file and (ii) the client encodes the file in order to repair a small number of erasures and errors that could have been missed during the verification step.

In this work, we propose to build PoR schemes using codes that fulfil the two previous goals, when equipped with a suitable family of efficiently computable random permutations. More precisely, our idea is the following. Given a file F , a code C and a family of random permutations σ K , the client sends to the server an encoded and scrambled version σ K (C(F )) of his file. Then, the verification step consists in checking "short" relations among descrambled symbols of w = C(F ), which come for instance from low-weight parity-check equations for C. Moreover, during the extraction step, the code C provides the redundancy necessary to repair erasures and potential unnoticed errors.

In the present work we develop a seminal idea that appeared in [START_REF] Lavauzelle | New Proofs of Retrievability using Locally Decodable Codes[END_REF] where the authors proposed a construction of PoRs based on lifted codes. We here provide a more generic construction, and give a deeper analysis of its security.

While our scheme does not feature updatability nor public verifiability, we emphasize the genericity of our construction, which is based on well-studied algebraic and combinatorial structures, namely codes and their parity-check equations. Moreover, since the code C is public, the client must only store the secret material associated to the random permutations σ K , which consist in a few bytes. Besides, an honest server simply needs to read pieces of w during the verification step, and therefore has very low computational burden compared to many other PoR schemes.

Organisation

Section 2 is devoted to the definition and security model of proofs of retrievability. Despite the great disparity of models in PoR literature, we try to keep close to the definitions given in [START_REF] Juels | PORs: Proofs of Retrievability for Large Files[END_REF][START_REF] Paterson | A Coding Theory Foundation for the Analysis of General Unconditionally Secure Proof-of-Retrievability Schemes for Cloud Storage[END_REF], for the sake of uniformity.

Section 3 presents our construction of PoR. Precisely, in Subsection 3.1, we introduce objects called verification structures for a code C that will be used in the definition of our PoR scheme (Subsection 3.2). A rigorous analysis of our scheme is the purpose of the remainder of that section.

The performance of our generic construction is given in Section 4. We then provide several instances in Section 5, proving the practicality of our PoR schemes for some classes of codes.

Proofs of retrievability

Definition of underlying protocols

We recall that in proofs of retrievability, a user wants to estimate if a message m can be retrieved from a encoded version w of the message stored on a server. In all what follows, the user will be known as the Verifier (wants to verify the retrievability of the message) while the server is the Prover (aims at proving the retrievability). The message space is denoted by M while W, the (server) file space, is the set of encoded versions of the messages. We also denote by K the set of secret values (or keys) kept by the Verifier, and by R the space of responses to challenges.

Throughout the paper, symbols ← R and ← respectively denote the output of randomised and deterministic algorithms.

Definition 2.1. A keyed proof of retrievability (PoR) is a tuple of algorithms (KeyGen, Init, Verify, Extract) running as follows:

1. The key generation algorithm KeyGen generates uniformly at random a key κ ← R K. The key κ is secretly kept by the Verifier. 2. The initialisation algorithm Init is a deterministic algorithm which takes as input a message m ∈ M and a key κ ∈ K, and outputs a file w ∈ W. Init is run by the Verifier which initially holds the message m. After the process, the file w is sent to the Prover and the message m is erased on Verifier's side. Upon receipt of w, the Prover sets a deterministic algorithm P (w) that will be run during the verification procedure. 3. The verification algorithm Verify is a randomised algorithm initiated by the Verifier which needs a secret key κ ∈ K and interacts with the Prover. Verify is depicted in Figure 1 and works as follows:

(i) the Verifier runs a random query generator that outputs a challenge u ← R Q (the set Q being the so-called query set); (ii) the challenge u is sent to the Prover; (iii) the Prover outputs a response r u ← P (w) (u) ∈ R; (iv) the Verifier checks the validity of r u according to u and κ: the algorithm Verify finally outputs the boolean value Check(u, r u , κ). 4. The extraction algorithm Extract is run by the Verifier. It takes as input κ and r = (r u : u ∈ Q) ∈ R Q , and outputs either a message m ∈ M, or a failure symbol ⊥. We say that extraction succeeds if Extract(r, κ) = m.

The vector r = (r u ← P (w) (u)) u∈Q ∈ R Q is called the response word associated to P (w) .

Verifier Prover

κ w Pick u ← R Q at random r u ← P (w) (u)
Output Check(u, r u , κ) u r u

Figure 1: Definition of algorithm Verify

Note that, in assuming that the response algorithm P (w) is deterministic and non-adaptive1 we follow the work of Paterson et al. [START_REF] Paterson | A Coding Theory Foundation for the Analysis of General Unconditionally Secure Proof-of-Retrievability Schemes for Cloud Storage[END_REF]. The authors justify determinism of response algorithms by the fact that any probabilistic prover can be replaced by a deterministic prover whose success probability is at least as good as the probabilistic one.

In Definition 2.1, we can see that a deterministic algorithm P (w) can be represented by the vector of its outputs r = (P (w) (u)) u∈Q , called the response word of P (w) . Therefore, we can assume that before the verification step, the Prover produces a word r (w) ∈ R Q related to the file w he holds. In other words, we model provers as algorithms P which, given as input w, return a word r ∈ R Q .

Following [START_REF] Paterson | A Coding Theory Foundation for the Analysis of General Unconditionally Secure Proof-of-Retrievability Schemes for Cloud Storage[END_REF], we also assume in this chapter that the extraction algorithm Extract is deterministic, though in general it can be randomised. Finally, notice that proofs of retrievability aim at proving the extractability of a file. The extraction algorithm is therefore a tool to retrieve the whole file. Hence, its computational efficiency is not a crucial feature.

The following table summarises the information held by each entity after the initialisation step:

Verifier Prover κ w

Let us also report the inputs and outputs of the algorithms involved in a PoR:

algorithm

Security models

One should first notice that, despite many efforts, proofs of retrievability lack a general agreement on the definition of their security model. Nevertheless, our definitions remain very close to the ones given in the original work of Juels and Kaliski [START_REF] Juels | PORs: Proofs of Retrievability for Large Files[END_REF].

For a response word r ∈ R Q given by the Prover and a key κ ∈ K kept by the Verifier, we first define the success of r according to κ as:

succ(r, κ) := Pr u (Check(u, r u , κ) = True) ,
where the probability is taken over the internal randomness of Verify. A first security model can be defined as follows.

Definition 2.2 (security model, strong version). Let ε, τ ∈ [0, 1]. A proof of retrievability (KeyGen, Init, Verify, Extract) is strongly (ε, τ )-sound if, for every initial file m ∈ M, every uploaded file w ∈ W and every prover P : W → R Q we have:

Pr   Extract(r, κ) = m and succ(r, κ) ≥ 1 -ε κ ← R KeyGen(1 λ ) w ← Init(m, κ) r ← P(w)   ≤ τ , (1)
the probability being taken over the internal randomness of KeyGen under the constraint that w = Init(m, κ).

A remark concerning parameters ε and τ . In proofs of retrievability, we aim at making the extraction of the desired file m as sure as possible when the audit succeeds. Hence, it is desirable to have τ small. On the other hand, the parameter ε measures the rate of unsuccessful audits which leads the Verifier to believe the extraction will fail. Therefore, one does not necessarily need to look for large values of ε, though in practice, large ε afford more flexibility, for instance if communication errors occur between the Prover and the Verifier during the verification procedure.

Definition 2.2 provides a strong security model, in the sense that (i) it does not require any bound on the response algorithms given by the Prover (ii) the probability in (1) is taken over fixed messages m (informally, it means the Prover knows m).

However, keyed proofs of retrievability are usually insecure according to the security model given in Definition 2.2. For instance, in A proof of retrievability (KeyGen, Init, Verify, Extract) is weakly (ε, τ )-sound (or simply (ε, τ )-sound) if, for every polynomial-time prover P : W → R Q and every uploaded file w ∈ W, we have:

Pr     Extract(r, κ) = m and succ(r, κ) ≥ 1 -ε m ← R M κ ← R KeyGen(1 λ ) w ← Init(m, κ) r ← P(w)     ≤ τ . (2)
In Equation ( 2), the randomness comes from pairs (m, κ) ∈ M × K picked uniformly at random among those satisfying w = Init(m, κ).

Since we deal with values of τ very close to 0, we also say that a strongly (ε, τ )-sound PoR admits λ = -log 2 (τ ) bits of security against ε-adversaries.

Informally, saying that a PoR is not weakly sound amounts to finding a polynomial-time deterministic algorithm P which

• takes as input a file w ∈ W and outputs a response word r ∈ R Q ,

• makes the extraction fail with non-negligible probability (over messages m and keys κ such that the corresponding response words are successfully audited).

Our generic construction

Schematically, in the initialisation phase of our construction, the Verifier (i) encodes his file according to a code C;

(ii) scrambles the resulting codeword using a tuple of permutations over the base field;

(iii) uploads the result to the Prover.

As we explained in the introduction, the verification step then consists in checking that the server is still able to give answers that, once descrambled, satisfy low-weight parity-check equations for C.

For this purpose, we next introduce objects called verification structures for codes, which will be used in the definition of our generic PoR scheme.

Verification structures: a tool for our PoR scheme

We here consider F q , the finite field with q elements. From well-known coding theory terminology, the support of a word w ∈ F n q is supp(w) := {i ∈ [1, n], w i = 0}, and its weight is wt(w) := | supp(w)|.

In this work, we need to consider codes whose alphabets are finite dimensional spaces R over

F q , typically R = F s q . Precisely, a code C of length n over R is a subset of R n . A code C ⊆ R n is F q -linear
if C is a vector space over F q . When R = F q , we get the usual definition of linear codes over finite fields. Unless stated otherwise, we only consider F q -linear codes, that we will refer to as codes.

We usually denote by k the dimension over F q of a code C. Its minimum distance d min (C) is the smallest Hamming distance between two distinct codewords. If n is the length of C, then

d min (C)/n ∈ [0, 1] is the relative minimum distance of the code C, while k/n represents its rate. If C ⊆ F n q , its dual code C ⊥ is defined as {h ∈ F n q , n i=1 h i c i = 0, ∀c ∈ C}.
Codewords in C ⊥ are also called parity-check equations for C. Definition 3.1 (Verification structure). Let 1 ≤ ≤ n and C ⊆ F n q be a code. Let also Q be a non-empty set of -subsets of [1, n]. Set R = F q . We define the restriction map R associated to Q as: R :

Q × F n q → R (u, w) → w |u Given an integer s ≥ 1 and a map V : Q × R → F s q , we say that (Q, V ) is a verification structure for C if the following holds: 1. for all i ∈ [1, n], there exists u ∈ Q such that i ∈ u; 2. for all u ∈ Q, the map F n q → F s q given by a → V (u, R(u, a)) is surjective and vanishes on the code C. Explicitly, ∀c ∈ C, V (u, R(u, c)) = 0 .
The map V is then called a verification map for C, and the set Q a query set for C. By convention, for w ∈ F n q and r ∈ R Q , we define

R(w) := (R(u, w) : u ∈ Q) ∈ R Q , V (r) := (V (u, r u ) : u ∈ Q) ∈ (F s q ) Q . Finally, the code R(C) := {R(c), c ∈ C} is called the response code of C.
Example 3.2 (Fundamental example). Let C be a code, and let H be a set of parity-check equations for C of Hamming weight , whose supports are pairwise distinct. Define the query set Q = {supp(h), h ∈ H}, and for any u ∈ Q, h(u) to be the unique parity-check equation in H whose support is u. Finally, we define a map V by:

V : Q × R → F q (u, r) → i=1 h(u) u i r i
Notice that we set s = 1 here. By construction, it is clear that (Q, V ) is a verification structure for C.

Example 3.3 (toy example). Let C ⊆ F 7 2 be a binary Hadamard code of length n = 7 and dimension k = 3. In other words, C is defined by a parity-check matrix

H =           1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 1           .
According to Example 3.2, we define Q to be the set of supports of rows of H. In other words,

Q = {{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 5, 6}, {2, 4, 7}, {3, 4, 6}, {3, 5, 7}} .
Then, the verification map V :

Q × F 3 2 → F 2 can be defined as follows. If u = {u 1 , u 2 , u 3 } ∈ Q and b ∈ F u
2 is indexed according to u, then we define

V (u, b) = 3 i=1 b u i . Now, let m = (m 1 , m 2 , m 3 ) ∈ F 3 2 . The message m can be encoded into c = (m 1 , m 2 , m 1 + m 2 , m 3 , m 1 + m 3 , m 1 + m 2 + m 3 , m 2 + m 3 ) ∈ C .
Hence, the word r = R(c) ∈ (F 3 2 ) 7 is:

r =     c 1 c 2 c 3   ,   c 1 c 4 c 5   ,   c 1 c 6 c 7   ,   c 2 c 5 c 6   ,   c 2 c 4 c 7   ,   c 3 c 4 c 6   ,   c 3 c 5 c 7     =     m 1 m 2 m 1 + m 2   ,   m 1 m 3 m 1 + m 3   ,   m 1 m 1 + m 2 + m 3 m 2 + m 3   ,   m 2 m 1 + m 3 m 1 + m 2 + m 3   ,   m 2 m 3 m 2 + m 3   ,   m 1 + m 2 m 3 m 1 + m 2 + m 3   ,   m 1 + m 2 m 1 + m 3 m 2 + m 3     For each vector-coordinate b ∈ F 3 2 of r = R(c)
, one can now check that j b j = 0. Hence, we get V (R(c)) = 0, as expected.

From now on, we denote by N = |Q| the length of the response code R(C) of a code C equipped with a verification structure (Q, V ).

Definition of our PoR scheme

Let (Q, V ) be a verification structure for C ⊆ F n q , and let σ ∈ S(F q ) n , where S(F q ) denotes the set of permutations over F q . Any n-tuple of permutations σ = (σ 1 , . . . , σ n ) ∈ S(F q ) n naturally acts on c ∈ F n q by: σ(c) → (σ 1 (c 1 ), . . . , σ n (c n )) , and we define σ(C) = {σ(c), c ∈ C}. Let finally

V σ : Q × F q → F s q (u, y) → V (u, σ -1 |u (y))
where σ -1 |u (y) = (σ -1 u 1 (y 1 ), . . . , σ -1 u (y )). The map V σ has been defined in order to satisfy

V σ (u, R(u, σ(c))) = V (u, R(u, c)) for every (c, u) ∈ C × Q.
Based on this, our PoR construction is given in Figure 2.

Analysis

Preliminary results

We first give results concerning verification structures and response codes. The following two lemmata are straightforward to prove.

Lemma 3.4. Let (Q, V ) be a verification structure for a code C ⊆ F n q . Then (Q, V σ ) is a verification structure for σ(C). Lemma 3.5. Let Q be any query-set for a code C ⊆ F n q whose elements have cardinality ≥ 1. Then its response code R(C) is an F q -linear code over the alphabet R F q .

The code C and the verification structure (Q, V ) for C are public parameters. We assume that C is linear, and set N = |Q|. We recall that R = F q and W = F n q .

• Key generation: The Verifier generates uniformly at random an n-tuple of permutations

(σ 1 , . . . , σ n ) = σ ← R S(F q ) n .
• Initialisation: The Verifier first encodes his file m ∈ F k q into a codeword c ∈ C with a systematic encoding algorithm for C. Then, the Verifier scrambles each coordinate c i using the permutation σ i :

w i = σ i (c i ), 1 ≤ i ≤ n .
Finally, w ∈ W is sent to the Prover, and m is erased by the Verifier. To sum up, the deterministic algorithm Init is defined by

Init(m, σ) := w = σ(C(m)) ∈ W .
Based on his knowledge of w and public parameters, the Prover produces a word r ← P(w), r ∈ R Q , which corresponds to the vector of outputs of the deterministic proving algorithm P on input w.

• Verification:

1. The Verifier picks uniformly at random u = (u 1 , . . . , u ) ← R Q. Then, the Verifier sends u to the Prover, meaning the Prover is asked to send back R(u, w) = w |u ∈ F q to the Verifier. 2. The Prover sends back the u-th coordinate r u ∈ R of his response word r to the Verifier. 3. On input r u ∈ R, the Verifier runs V σ (u, r u ) and outputs the result. Here we mean that:

Check(u, r u , σ) := True if V σ (u, r u ) = 0 False otherwise.
• Extraction: The Verifier first collects r = (P(w) u : u ∈ Q) ∈ R Q . Then, he runs the extraction procedure given in Figure 3, on input σ and r, and he outputs his result. Remark 3.6. By considering σ(C) instead of C, we loose the F q -linearity, but one can check that verification structures still make sense and provide the result claimed in Lemma 3.4.

Input: σ ∈ S(F q ) n and r ∈ R Q . Output: m ∈ F k q or a failure symbol ⊥. 1. Define r = σ -1 (r). 2. On challenges u ∈ Q such that V (u, r u ) = 0, assign r u ←⊥.
The next result states that the map C → σ(C) does not modify the distance between codewords. Lemma 3.7. Let C ⊆ F n q be a linear code, (Q, V ) a verification structure for C, and σ ∈ S(F q ) n . Then it holds that:

• the distribution of distances in C and σ(C) are the same,

• the distribution of distances in R(C) and R(σ(C)) are the same. Proof. Since every σ i is one-to-one, for any c, c ∈ C we get

d(c, c ) = |{i ∈ [1, n], c i = c i }| = |{i ∈ [1, n], σ i (c i ) = σ i (c i )}| = d(σ(c), σ(c )) .
The proof for response codes relies on the same argument.

Remark these results imply that, if C is linear, then the minimum distance of R(σ(C)) is the minimum weight of R(C).

Definition 3.8. Let ε ∈ [0, 1] and (Q, V ) be a verification structure for a code C ⊆ F n q . We say

r ∈ R Q is ε-close to (Q, V ) if: wt(V (r)) := |{u ∈ Q, V (u, r u ) = 0}| ≤ εN . Let now c ∈ C and β ∈ [0, 1]. We say that r ∈ R Q is a β-liar for (Q, V, c) if: |{u ∈ Q, V (u, r u ) = 0 and r u = R(u, c)}| ≤ βN .
Bounded-distance error-and-erasure decoder. Let A ⊆ F n q be any code of minimum distance d, and let a ∈ A be corrupted with b errors and e erasures, resulting in a word r ∈ (F q ∪ {⊥}) n . Then, it is well-known that, as long as 2b + e < d, it is possible to retrieve a from r thanks to a so-called bounded-distance error-and-erasure decoding algorithm. This is precisely the decoding algorithm that we employ in Figure 3 on the code A = R(C).

Our framework allows us to reformulate the extraction success in terms of a probability to decode corrupted codewords. More precisely: Proposition 3.9. Let σ ∈ S(F q ) n , m ∈ F k q and denote by d the minimum distance of R(C), of length N . Let also r ∈ R Q be the response word, output of a proving algorithm P taking as input w = σ(C(m)). Finally, assume that r is ε-close to (Q, V σ ) and a β-liar for (Q, V σ , w), with (ε + 2β)N < d. Then, Extract(r, σ) = m, where Extract(r, σ) is defined in Figure 3.

Proof. Recall that r ∈ (R ∪ {⊥}) Q represents the word we get from r after the second step of the algorithm given in Figure 3. Let us now translate our assumptions on r in coding-theoretic terminology:

• r is ε-close to (Q, V σ ) means that there are at most εN challenges u ∈ Q for which we know that the coordinate r u is not authentic. This justifies that we assign erasure symbols to these coordinates.

• r is a β-liar for (Q, V, c) means that there are at most βN other corrupted values r u , but we cannot identify them. Therefore we can assimilate these coordinates to errors.

To sum up, we see r as a corruption of R(C(m)) with at most εN erasures and at most βN errors, where N = |Q|. Since we assume that (ε + 2β)N < d, we know from the previous discussion that the decoding succeeds to retrieve m.

Bounding the extraction failure

According to Definition 2.3, our PoR scheme is weakly (ε, τ )-sound if for every polynomial-time algorithm P outputting a response word r (w) from a file w, we have

Pr σ,m   decoding r (w) into m fails and wt(V σ (r (w) )) ≤ εN m ← R F k q σ ← R S(F q ) n w = σ(C(m))   ≤ τ .
Using Proposition 3.9, the security analysis of our PoR scheme reduces to measuring the ability of the Prover to produce a response word r which is ε-close to (Q, V σ ) and a β-liar for (Q, V σ , w), with (ε + 2β)N ≥ d.

For fixed r ∈ R Q , σ ∈ S(F q ) n and w = σ(C(m)) the authentic file given to the prover, we define three subsets of Q:

• D(r, w) := {u ∈ Q, r u = R(w) u } and D(r, w) := |D(r, w)| = wt(r -R(w))
. This represents challenges u on which the response word r differs from the authentic one R(w).

• E(r, σ) := {u ∈ Q, V σ (u, r u ) = 0} and E(r, σ) := |E(r, σ)| = wt(V σ (r))
. These are challenges u on which the associated coordinate r u is not accepted by the verification map (it corresponds to erasures in the decoding process).

• B(r, σ, w) := {u ∈ Q, r u = R(w) u and V σ (u, r u ) = 0} and B(r, σ, m) := |B(r, σ, m)|. These are the challenges u on which the associated coordinate r u is accepted by the verification map, but differs from the authentic response s u (it corresponds to errors in the decoding process).

One can easily check that, for every σ, the sets E(r, σ) and B(r, σ, w) define a partition of D(r, w).

The probability of extraction failure can thus be written as:

Pr   2D(r, w) -E(r, σ) ≥ d min (R(C)) and E(r, σ) ≤ εN m ← R F k q σ ← R S(F q ) n w = σ(C(m))   .
(3)

For w ∈ F n q , let us define the set of admissible permutations and messages:

Φ w := {(σ, m) ∈ S(F q ) n × F k q , w = σ(C(m))} ,
so that Equation (3) rewrites:

Pr 2D(r, w) -E(r, σ) ≥ d min (R(C)) E(r, σ) ≤ εN (σ, m) ← R Φ w .
Later on, we will use the notation Pr Φw to refer to the fact that (σ, m) is uniformly drawn from Φ w . Similarly we will use notation E Φw for the expectancy and Var Φw for the variance.

Given r ∈ R Q , we also define α(r, w) := max u∈D(r,w)

Pr Φw (V σ (u, r u ) = 0)

and α := max (r,w) α(r, w) where (r, w) are such that D(r, w) = 0. The parameter α ∈ (0, 1) is called the bias of the verification structure (Q, V ) for C. It corresponds to the maximum probability that a response is accepted but not authentic.

Lemma 3.10. For all r ∈ R Q and w ∈ F n q , we have:

E Φw (E(r, σ)) ≥ (1 -α)D(r, w) .
Proof. A simple computation shows:

E Φw (E(r, σ)) = E Φw u∈D(r,w) 1 V σ (u,ru) =0 = u∈D(r,w) Pr Φw (V σ (u, r u ) = 0) ≥ u∈D(r,w) (1 -α) ≥ (1 -α)D(r, w) .
Lemma 3.10 essentially means that, if an adversary to our PoR scheme wants its response word to be (in average) ε-close to the verification structure, then he should modify at most D(r, w) ≤ εN 1-α responses. Below we take advantage of this result and we measure the probability of an extraction failure.

First, for δ, ε ∈ (0, 1), let p(r, w; ε, δ) := Pr Φw (2D(r, w) -E(r, σ) ≥ δN and E(r, σ) ≤ εN ) = Pr Φw (E(r, σ) ≤ min{εN, 2D(r, w) -δN }) .

The probability p(r, w; ε, δ) represents the probability that the extraction fails for a response code of relative distance δ and an adversarial response word r associated to w, which is ε-close to the verification structure. Let us bound p(r, w; ε, δ).

Proposition 3.11. Let δ, ε ∈ (0, 1) such that δ 1-α 1+α > ε. Let also r ∈ R Q and w ∈ F n q . Then we have:

p(r, w; ε, δ) ≤ Var Φw (E(r, σ)) 1+α 2 δ 1-α 1+α -ε 2 N 2 .
Proof. We distinguish three cases.

1. 2D(r, w) -δN < 0. The event E(r, σ) ≤ min{εN, 2D(r, w) -δN } never occurs since E(r, σ) ≥ 0. Hence p(r, w; ε, δ) = 0. 2. εN ≤ 2D(r, w) -δN. The inequality E(r, σ) ≤ εN implies

E(r, σ) -E Φw (E) ≤ εN -(1 -α)D(r, w) ≤ εN -(1 -α) ε + δ 2 N ≤ - 1 + α 2 δ 1 -α 1 + α -ε N .
Hence, using Chebychev's inequality,

p(r, w; ε, δ) = Pr Φw (E(r, σ) ≤ εN ) ≤ Pr Φw |E(r, σ) -E Φw (E)| ≥ 1 + α 2 δ 1 -α 1 + α -ε N ≤ Var Φw (E(r, σ)) 1+α 2 δ 1-α 1+α -ε 2 N 2 . 3. 0 ≤ 2D(r, w) -δN < εN . In this case, E(r, σ) ≤ 2D(r, w) -δN implies E(r, σ) -E Φw (E) ≤ (1 + α)D(r, w) -δN ≤ (1 + α) ε + δ 2 N -δN ≤ - 1 + α 2 δ 1 -α 1 + α -ε N.
Therefore, similarly to the previous case, we obtain the claimed result.

For any u ∈ D(r, w), denote by X u the {0, 1}-random variable "1 V σ (u,ru)=0 " when σ is uniformly drawn from Φ w . It holds that E(r, σ) = u∈D(r,w) (1 -X u ).

Recall that two real random variables Y, Z are uncorrelated if E(Y Z) = E(Y )E(Z). For instance, two independent random variables are uncorrelated.

Lemma 3.12. Let r ∈ R Q and w ∈ F n q . If the random variables {X u } u∈D(r,w) are pairwise uncorrelated, then:

Var Φw (E(r, σ)) ≤ D(r, w) .

Proof. By assumption, {X u } u∈D(r,w) are pairwise uncorrelated, hence Var Φw (E(r, σ)) = u∈D(r,w) Var Φw (1 -X u ). The trivial bound Var Φw (1 -X u ) ≤ 1 gives the result.

As a corollary of Proposition 3.11 and Lemma 3.12, under the same hypothesis and assuming

δ 1-α 1+α > ε, we get p(r, w; ε, δ) ≤ 4 N ((1 -α)δ -(1 + α)ε) 2
since D(r, w) ≤ N . Moreover, if lim N →∞ δ > 0 and lim N →∞ α = 0, then p(r, w; ε, δ) = O(1/N ).

Therefore, we end up with the following theorem. Theorem 3.13. Let (Q, V ) be a verification structure for C with bias α. Denote by N = |Q| and δ = d min (R(C))/N the relative distance of the associated response code. Finally, assume that, for any r ∈ R Q and any w ∈ F n q the variables {X u } u∈D(r,w) are pairwise uncorrelated. Then, for any ε < δ 1-α 1+α , the PoR scheme associated to C and (Q, V ) is (ε, τ )-sound, where

τ = 4 N ((1 -α)δ -(1 + α)ε) 2 .
For asymptotically small α, a code C equipped with a verification structure satisfying the conditions of Theorem 3.13 thus gives an (ε, τ )-sound PoR scheme for every ε < (1 + o(1))δ and τ = O(1/N ).

According to Theorem 3.13, we thus need to look for (sequences of) codes C and associated verification structures (Q, V ) such that:

1. the response code R(C) admits a good relative distance δ = d min (R(C))/N 2. the bias α is small, 3. random variables {X u } u∈D(r,w) are pairwise uncorrelated. Subsections 3.4 and 3.5 characterize conditions under which the last two points are fulfilled. Then, in Section 5 we discuss which response codes can achieve good relative distance.

Then,

E(X u X v ) = 1 q 2t a,b∈(F t q ) 2 Pr(σ -1 (r u ) |u ∈ K u | σ -1 (r u ) |u∩v = a) × Pr(σ -1 (r v ) |v ∈ K v | σ -1 (r v ) |u∩v = b) .
and we conclude since

E(X u ) = q -t a∈F t q Pr(σ -1 (r u ) |u ∈ K u | σ -1 (r u ) |u∩v = a) .

Performance

Efficient scrambling of the encoded file

In the PoR scheme we propose, the storage cost of an n-tuple of permutations in S(F q ) n is excessive, since it is superlinear in the original file size. In this subsection, we propose a storageefficient way to scramble the codeword c ∈ C produced by the Verifier.

Precisely we want to define a family of maps (σ (κ) ) κ , where σ (κ) : C → F n q , c → w ∈ F n q , with the following requirements:

• for every κ, the map σ (κ) is efficiently computable and requires a low storage,

• for every κ and every c ∈ C, if w = σ (κ) (c), then for every i ∈ [1, n] the local inverse map w i → c i is efficiently computable,

• if κ is randomly generated but unknown, then given the knowledge of w = σ (κ) (c) and C, it is hard to produce a response word r ∈ R Q such that, for many u ∈ Q, both V σ (κ) (u, r u ) = 0 and r u = w |u hold. To be more specific, and in light of the security analysis of Section 3.3, we require that it is hard to distinguish σ (κ) (c) from a random (z 1 , . . . , z n ) ∈ F n q , where symbols z i are picked independently and uniformly at random. We here propose to derive σ (κ) from a suitable block cipher, yielding the explicit construction given below. Of course, other proposals can be envisioned.

The construction. Let IV denote a random initialisation vector for AES in CTR mode (IV could be a nonce concatenated with a random value). Vector IV is kept secret by the Verifier, as well as a randomly chosen key κ for the cipher. Let also f be a permutation polynomial over F q of degree d > 1. For instance one could choose f (x) = x d with gcd(d, q -1) = 1. Notice that polynomial f can be made public.

Let s = 256 log 2 q be the number of F q -symbols one can store in a 256-bit word2 . Up to appending a few random bits to c, we assume that s | n, and we define t = n/s. Let us fix a partition of [1, n] into s-tuples i = (i 1 , . . . , i s ); it can be for instance (1, . . . , s), (s + 1, . . . , 2t), . . . , ((t -1)s + 1, . . . , n). Notice that this partition does not need to be chosen at random. Given c = (c 1 , . . . , c n ) ∈ C and i an element of the above partition, we now define

b i = f (c i 1 ) | • • • | f (c is ) ⊕ AES κ (IV ⊕ i) ∈ {0, 1} 256 .
If log 2 q 256, trailing zeroes can be added to evaluations of f . Finally, the pseudo-random permutation σ is defined by: σ(c) := (b 1 , . . . , b t ) .

Design rationale. AES is a natural choice when one needs a (secret)-keyed pseudo-random permutation. Also notice that with this construction, one only needs to store the key κ and the vector IV , since the other objects (the polynomial f , the partition) are made public. Hence our objectives in terms of storage are met.

We now point out the necessity to use i as a part of the input of the AES cipher. Assume that we do not. Then, the local permutation σ j , 1 ≤ j ≤ n, would not depend on j. As a consequence, for certain class of codes the local verification map r u → V σ (u, r u ) would not depend on u, and a malicious Prover would then be able to produce accepted answers while storing only a small piece of the file w (e.g. w |u for only one u ∈ Q).

Another mandatory feature is the non-linearity of the permutation polynomial f . Indeed, assume for instance that f = id. Then, given the knowledge of w = σ(c), it would be very easy for a malicious Prover to produce a word w = w, such that r = R(w ) is always accepted by the Verifier. Simply, the Prover defines w = w + c , where c is any non-zero codeword of C. Hence, one sees that the polynomial f must be non-linear in order to prevent such kind of attacks.

Parameters

We here consider a PoR built upon a code C ⊆ F n q , with verification structure (Q, V ) satisfying R = F q and V (R) = F s q . We also assume that we use an n-tuple of pseudo-random permutations as described in the previous subsection.

Communication complexity. At each verification step, the client sends an -tuple of coordinates (u 1 , . . . , u ), u i ∈ [1, n]. The server then answers with corresponding symbols w u i ∈ F q . Therefore the upload communication cost is log 2 n bits, while the download communication cost is log 2 q, thus a total of (log 2 n + log 2 q) bits.

Computation complexity. In the initialisation phase, following the encryption described in section 4.1, the client essentially has:

• to compute the codeword c ∈ C associated to its message,

• to make n evaluations of the permutation polynomial f over F q , and • to compute t = n log 2 q 256 AES ciphertexts to produce the word w to be sent to the server.

Given a generator matrix of C, the codeword c can be computed in O(kn) operations over F q with a matrix-vector product. Notice that quasi-linear-time encoding algorithms exist for some classes of codes. Besides, if a monomial or a sparse permutation polynomial is used, then the cost of each evaluation is O((log 2 q) 3 ). If we denote by c the bitcost of an AES encryption, we get a total bitcost of O(nk(log 2 q) 2 + n(log 2 q) 3 + cn log 2 q) for the initialisation phase. Recall this is a worst-case scenario in which the encoding process is inefficient.

At each verification step, an honest server only needs to read symbols from the file it stores. Hence its computation complexity is O( ). The client has to compute a matrix-vector product over F q , where the matrix has size s × and the vector has size , thus a computation cost of O( s) operations over F q .

Storage needs. The client stores 2 × 256 bits for secret material κ and IV to use in AES. The server storage overhead exactly corresponds to the redundancy of the linear code C, that is (n -dim C) log 2 q bits. Other features. Our PoR scheme is unbounded-use, since every challenge reveals nothing about the secret data held by the client. It does not feature dynamic updates of files. Though, we must emphasize that the file w the client produces can be split among several servers, and the verification step remains possible even if the servers do not communicate with each other. Indeed, computing a response to a challenge does not require to mix distinct symbols w i of the uploaded file. Therefore, our scheme is well-suited for the storage of large static distributed databases.

Client storage 512 bits Server total storage n log 2 q bits Communication complexity (verif.) log 2 (nq) bits Client computation complexity (verif.) decryptions, s operations over F q Server computation complexity (verif.) reads, no computation Figure 4: Summary of parameters of our PoR construction, for an original file of size k log 2 q bits, and a code C of dimension k over F q equipped with a verification structure (Q, V ) such that |u| = , and rank V (u, •) ≤ s for all u ∈ Q.

Instantiations

In this section we present several instantiations of our PoR construction. We first recall basics and notation from coding theory.

The code Rep( ) ⊆ F q denotes the repetition code (1, . . . , 1) . We recall that Rep( ) ⊥ is the parity code Par( ) := {c ∈ F q , i=1 c i = 0}. Let C, C be two linear codes over F q of respective parameters [n, k, d] and [n, k , d ]. Their tensor product C ⊗ C is the F q -linear code generated by words (c i c j :

1 ≤ i ≤ n, 1 ≤ j ≤ n ) ∈ F nn q .
It has dimension kk and minimum distance dd . We also denote by

C ⊗s := C ⊗ • • • ⊗ C s times ⊆ F n s q
the s-fold tensor product of C with itself.

Tensor-product codes

The upcoming subsection illustrates our construction with a non practical but simple instance. The next ones lead to practical PoR instances.

A simple but non-practical instance

Let n = N , and

Q = {u i = {i + 1, i + 2, . . . , (i + 1) }, i ∈ [0, N -1]}. The set Q defines a partition of [1, n].
We define the code

C = {c ∈ F n q , j∈u c j = 0, ∀u ∈ Q} ⊆ F n q
In other words, C = Par( ) ⊗ F N q , and a parity-check matrix H for C is given by:

H =       1 • • • 1 0 • • • • • • • • • • • • • • • 0 0 • • • 0 1 • • • 1 . . . . . . . . . . . . . . . . . . . . . 0 0 • • • • • • • • • • • • • • • 0 1 • • • 1       .
The verification map V : Q×F q → F q is defined by V (u, b) := j=1 b u j , for all (u, b) ∈ Q×F q . By construction (see the fundamental Example 3.2), the pair (Q, V ) defines a verification structure for C.

Lemma 5.1. Let C = Par( )⊗F N q as above. Then the response code R(C) has minimum distance 1.

Proof. We see that the restriction map R sends the codeword (1, -1, 0, 0, . . . , 0) ∈ C to a word of weight 1. Besides, R is injective so d min (R(C)) > 0.

Since δ = d min (R(C))/N = 1/N → 0 when N goes to infinity, an attempt to build a PoR scheme from C cannot be practical.

Higher order tensor-product codes

Let A ⊆ F q be a non-degenerate [ , k A , d A ] q -linear code, and define C = A ⊗s ⊆ F n q where n = s . Notice it will be more convenient to see coordinates of words w ∈ F n q as elements of [1, ] s . For a ∈ [1, ] s and 1 ≤ i ≤ s, we define L i,a ⊂ [1, ] s , the "i-th axis-parallel line with basis a", as

L i,a := {x ∈ [1, ] s such that x j = a j , ∀j = i} .
By definition of C, a word c lies in C if and only if, for every L = L i,a , the restriction c |L ∈ A. This means that we can define:

• a set of queries Q = {L i,a , i ∈ [1, s], a ∈ [1, ] s }; • a verification map V : Q × R → F -k A q (L, r) → Hr
where H is a parity-check matrix for A whose columns are ordered according to the line L.

By the previous discussion, it is clear that c ∈ C implies that V (L, c |L ) = 0 for every L ∈ Q (in fact these two assertions are equivalent). Hence, (Q, V ) defines a verification structure for C, and we have N = |Q| = s s-1 .

Lemma 5.2. Let C = A ⊗s as above. Then, R(C) has minimum distance s • d s-1 A .

Proof. Let us first prove that the minimum distance of R(C) is larger than s • d s-1 A . Let r = R(c) ∈ R(C), and assume r = 0. Then, there exists

L ∈ Q such that 0 = r L = c |L ∈ A. Therefore c x = 0 for some x ∈ L ⊂ [1, ] s . Consider the set S i,x = {y ∈ [1, ] s , y i = x i } .
Very informally, the set S i,x corresponds to the hyperplane passing through x and "orthogonal" to the i-th axis. By definition of C = A ⊗s , we know that c |S i,x ∈ A ⊗(s-1) \{0} for every 1 ≤ i ≤ s. Denote by U i = supp(c |S i,x ) = {u (i,1) , . . . , u (i,t i ) }, with t i ≥ d min (A ⊗(s-1) ) = (d A ) s-1 . Every u (i,j) ∈ U i defines a line L i,u (i,j) on which c |L i,u (i,j) is a non-zero codeword of A. Equivalently, r is non-zero on index L i,u (i,j) ∈ Q. Therefore,

wt(r) = |{L ∈ Q, r L = 0}| ≥ s i=1 L i,u (i,j) , 1 ≤ j ≤ t i ≥ s i=1 t i ≥ s(d A ) s-1 .
Let us now build a word r ∈ R(C) of weight s(d A ) s-1 . Let w ∈ A \ {0} be a minimum-weight codeword of A, and define W := supp(w) ⊆ A. Define c = w ⊗s ∈ C; then supp(c) = W s . Let finally r = R(c). We see that r L i,x = 0 if and only if x ∈ W s . Hence we get

wt(r) = |{L ∈ Q, r L = 0}| = s i=1 {L i,x , x ∈ W s } = s • d s-1 A .
since each line L i,x is counted d A times when x runs over W s .

Proposition 5.3. Let δ > 0 and A be an [ , (1 -δ) + 1, δ] q MDS code. Define C = A ⊗s and (Q, V ) as above. If every Φ w is sufficiently uniform, then the PoR scheme associated to C and (Q, V ) is (ε, τ )-sound for τ = O( 1 (δ ) s s ) and every ε < ε 0 where ε 0 = (1 + O(q -δ +1 ))δ s , when → ∞.

Proof. First, the relative distance of R(C) is δ s according to Lemma 5.2. Then, the random variables {X u } u∈D are pairwise uncorrelated because the inequality

max u =v∈Q 2 |u ∩ v| = 1 < (1 -δ) + 2 = min u∈Q d min ((C |u ) ⊥ )
allows us to apply Proposition 3.18. Besides, if every Φ w is sufficiently uniform, then the bias α satisfies α = O(q -δ +1 ); hence 1-α 1+α = 1 + O(q -δ +1 ). Therefore we can use Theorem 3.13 and we get the desired result.

Parameters. We mainly focus on the download communication complexity in the verification step and on the server storage overhead, since these are the most crucial parameters which depend on the family of codes C we use. Besides, we consider that it is more relevant to analyse the ratio between these quantities and the file size than their absolute values.

Here, for an initial file of size |F | = ((1 -δ)q + 1) s log 2 q bits, we get

• a redundancy rate n log 2 q |F | = q (1-δ)q+1 s ≤ 1 (1-δ) s ; • a communication complexity rate log 2 q |F | = q ((1-δ)q+1) s ≤ 1 (1-δ) s q 1-s .
Example 5.4. We present various parameters of PoR instances admitting 0.10 ≤ ε 0 ≤ 0.16, for files of size approaching 10 4 , 10 6 and 10 9 bits. Here A is a [q, (1 -δ)q + 1, δq] q MDS code (e.g. a Reed-Solomon code), and C = A ⊗s . The previous example shows that, while the communication rate is reasonable for these PoR instances over large files, the storage needs remain large.

Besides, the relative distance of R(C) is q 2 +2 q(q+1) → 1 according to Lemma 5.5. If every Φ w is sufficiently uniform, the bias α satisfies α ∈ O(1/q); hence 1-α 1+α = 1 + O(1/q). Therefore we can use Theorem 3.13 and we get the desired result.

Parameters. For an initial file of size |F | = 1 2 (q -1)(q -2) log 2 q bits, we get

• a redundancy rate q 2 log 2 q |F | = 2 (1-1/q)(1-2/q) → 2;

• a communication complexity rate q log 2 q |F | = 2 q 1

(1-1/q)(1-2/q) = O(1/q).

Storage improvements via lifted codes

The redundancy rate of Reed-Muller codes presented above stays stuck above 2. Affine lifted codes, introduced by Guo, Kopparty and Sudan [START_REF] Guo | New Affine-Invariant Codes from Lifting[END_REF], allow to break this barrier while keeping the same verification structure. Generically, they are defined as follows:

Lift(m, d) := {(f (P)) P∈F m q , f ∈ F q [X 1 , . . . , X m ], ∀ affine line L ⊂ F m q , (f (Q)) Q∈L ∈ RS q (d + 1)} .
We refer to [START_REF] Guo | New Affine-Invariant Codes from Lifting[END_REF] for more details about the construction. Here we focus on Lift(2, q -2), since it can be compared to RM(2, q -2). Indeed, one sees that

RM(2, q -2) ⊆ Lift(2, q -2) (4)
and Equation (4) turns into a proper inclusion as long as q is not a prime. Besides, by definition of lifted codes, Lift(2, q-2) admits the same verification structure as the one presented previously for RM(2, q -2).

Lemma 5.7. The response code of Lift(2, q -2) has minimum distance at least q 2 -q + 2.

Proof. The rationale is similar to the proof of Lemma 5.5. Let 0 = c ∈ C, c = (f (P)) P∈F 2 q , f ∈ F q [X, Y ], and denote by L 1 , . . . , L a ⊂ F 2 q the lines on which f vanishes. 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 (R(Lift(2, q -2))) ≥ q 2 + q -2q + 2 = q 2 -q + 2.

We believe the bound given in Lemma 5.7 is not tight, but it is sufficient to have d min (R(Lift(2, q-2)))/N → 1. Similarly to Proposition 5.6, we can then prove that practical PoRs can be constructed with the family of lifted codes Lift(2, q -2).

Proposition 5.8. Let C = Lift(2, q -2), and (Q, V ) its associated verification structure. If every Φ w is sufficiently uniform, then the PoR scheme associated to C and (Q, V ) is (ε, τ )-sound for every ε < 1 and τ

= O( 1 (1-ε)q 2 ), when q → ∞
The crucial improvement is that lifted codes potentially have much higher dimension than Reed-Muller codes. For q = 2 e , the dimension of Lift(2, q -2) can be proved to equal 4 e -3 e [GKS13].

Example 5.9. We present parameters of PoRs based on Reed-Muller codes and lifted codes, using files of size approaching 10 4 , 10 6 and 10 9 bits. Note that this family of codes has been used in the PoR proposal of [START_REF] Lavauzelle | New Proofs of Retrievability using Locally Decodable Codes[END_REF].

On more generic families of codes

We have presented two rather small families of codes producing practical instances of PoR. Let us give a short summary of approximate lower bounds on crucial PoR parameters that have been shown in previous sections.

Family of codes over F q redundancy rate communication complexity rate s-fold tensor product (Sec. 5.1.2)

(1 -δ) -s q -(s-1) (1 -δ) -s plane RM (Sec. 5.2.1) 2 2q -1 plane lifted code (Sec. 5.2.2)

1 + q log 2 (3)-2 q -1 + q log 2 (3)-3
Now we quickly mention other families of codes that could be interesting to consider.

Multi-variate generalisation. We have only presented Reed-Muller and lifted codes embedded into the affine plane F 2 q . One could of course consider a broader ambient space F m q , m > 2. Lines would have smaller relative weight compared to the ambient space, and thus we would decrease the communication complexity of our PoR schemes. We must however care about the storage overhead which can drastically increase if m gets large: for instance, any Reed-Muller code RM q (m, q -2) has rate ≤ 1/m!.

Lower degree generalisation. In order to increase the soundness of our PoR schemes, one could consider Reed-Muller codes RM q (2, d) (as well as related lifted codes) with a lower degree d < q -2. The communication complexity remains unchanged; however we could observe overwhelming storage overhead if d is too small. Combinatorial generalisation. Codes Lift(2, q -2) can be viewed as codes from designs (see [START_REF] Assmus | Designs and Their Codes[END_REF] for more details), where the underlying block design is the classical affine plane. Considering designs with smaller block size would lead to PoRs with smaller communication complexity. But once again, this could be expensive in terms of storage, since only a few designs produce high dimensional codes.

Conclusion

We have proposed a security model for PoRs in line of previous work, together with a generic codebased framework. We have then sharply quantified the extraction failure of our PoR construction as a function of code parameters. Specialising this construction for particular families of codes, we provided instances with practical parameters. We hope our work will be an incentive for further proposals of code instances, aiming at better PoR parameters.

Influence of N on the estimator. Table 1 shows experimentally that, for M large enough and fixed q, the number N has few influence on the estimator (N being respectively the number 10 3 10 4 10 5 10 6 M 10 -2 10 -1 ξ M (p Φ )

t = 2
expected 1/(q-1), q=8 expected 1/(q-1), q=64 q=8, Test 1 q=8, Test 3 q=8, Test 2 q=64, Test 1 q=64, Test 3 q=64, Test 2 expected 1/(q-1), q=8 expected 1/(q-1), q=64 q=8, Test 1 q=8, Test 3 q=8, Test 2 q=64, Test 1 q=64, Test 3 q=64, Test 2 of responses r u sampled in Test 2, and the number of challenges u sampled in Test 1). The minor increase of the values can be thought as a standard deviation due to the fact that the number of samples M = 100, 000 is finite.

N q = 8 q = 64 1 0.1418 0.0152 5 0.1433 0.0163 10 0.1443 0.0165 50 0.1455 0.0169 100 0.1452 0.0167 500 0.1464 0.0169 1/(q -1) = 0.1429 0.01587 N q = 8 q = 64 1 0.1414 0.0158 5 0.1431 0.0162 10 0.1452 0.0166 50 0.1450 0.0168 100 0.1458 0.0168 500 0.1470 0.0168 1/(q -1) = 0.1429 0.01587 Table 1: Estimators using Test 1 (on the left) and Test 2 (on the right) with M = 100, 000 and t = 2, for q ∈ {8, 64} and various values of N . The quantity 1/(q -1) represents an estimated upper bound on α that ξ M (p Φ ) should approximate.

Influence of q on the estimator. In Table 2, we show that estimator ξ M (p Φ ) converges to an expected value 1/(q -1), for any value of q. 10 3 10 4 10 5 10 6 M 10 -2 10 -1 ξ M (p Φ )

t =
expected 1/(q-1), q=8 expected 1/(q-1), q=64 2: Estimators using Test 3 with M = 1, 000, 000 and t = 2, for various values of prime powers q. The quantity 1/(q -1) represents an estimated upper bound on α that ξ M (p Φ ) should approximate.

  [PSU13] Paterson et al. noticed that in the Shacham-Waters scheme [SW13], given the knowledge of m and w, an unbounded Prover may be able to 1. compute (or at least randomly guess) a key κ such that Init(m, κ) = w, 2. build m = m such that Init(m , κ) = w , and 3. set P (w ) = r which (a) successfully passes every audit and (b) leads to the extraction of m = m. Hence, we choose to use a weaker but still realistic security model, where informally, the Prover only knows what he stores (that is, w) and has no information on the initial message m. The following security model thus remains conform with the one given by Paterson et al. [PSU13]. Definition 2.3 (security model, weak version). Let ε, τ ∈ [0, 1].

Figure 2 :

 2 Figure 2: Definition of our PoR scheme

  3. Run a bounded-distance error-and-erasure decoding algorithm for R(C) with input r ∈ (R ∪ {⊥}) Q . It outputs either a word m ∈ F k q , or the failure symbol ⊥. 4. Return this output.

Figure 3 :

 3 Figure 3: Our extraction procedure Extract(r, σ).

Figure 5 :

 5 Figure5: Estimators for various values of M ∈ [10 3 , 10 6 ], of q ∈ {8, 64}, and of Test i, i ∈ {1, 2, 3}. Support size t = 2 is fixed. For Tests 1 and 2, the parameter N is set to 10. Black horizontal lines represent the expected value of α.

Figure 6 :

 6 Figure6: Estimators for various values of M ∈ [10 3 , 10 6 ], of q ∈ {8, 64}, and of Test i, i ∈ {1, 2, 3}. Support size t = 3 is fixed. For Tests 1 and 2, the parameter N is set to 10. Black horizontal lines represent the expected value of α.

  al. [ABC + 11]. Besides, protocols of Lillibridge et al. [LEB + 03] and Naor and Rothblum

  Figure7: Estimators for various values of M ∈ [10 3 , 10 6 ], of q ∈ {8, 64}, and of Test i, i ∈ {1, 2, 3}. Support size t = is fixed. For Tests 1 and 2, the parameter N is set to 10. Black horizontal lines represent the expected value of α.

	q	ξ M (p Φ ) 1/(q -1)
	4 8 16 0.0665 0.333 0.143 32 0.032 64 0.0161 128 0.00791 0.007874 0.3333 0.1429 0.06667 0.03226 0.01587	q 7 17 0.0627 ξ M (p Φ ) 1/(q -1) 0.166 0.1667 0.0625 31 0.0335 0.03333 257 0.00398 0.004000
	256 0.00382 0.003922
	Table
		q=8, Test 2 q=8, Test 3 q=8, Test 1 q=64, Test 2 q=64, Test 3 q=64, Test 1

in the sense that its behaviour only depends on the values of challenges u, and not on previous calls to the verification procedure

in the scheme we propose, we will always have log(q) < 256

Acknowledgements

This work is partially funded by French ANR-15-CE39-0013-01 "Manta". The authors would like to thank Daniel Augot who shared fruitful discussions on the definition of proofs-of-retrievability, as well as Alain Couvreur for his suggestion leading to the proof of Lemma 5.7.

Estimating α

In this section we prove that, assuming Φ w approximates the uniform distribution over S(F q ) n in a sense that we make precise later, the bias α can be bounded according to parameters of the verification structure.

Let us fix r ∈ R Q , w ∈ F n q and u ∈ Q. We recall that α is defined by: α = max r,w max u∈D(r,w)

Pr Φw (V σ (u, r u ) = 0)

where randomness comes from σ ← R Φ w = {(σ, m) ∈ S(F q ) n × F k q , w = σ(C(m))}. We notice that this is equivalent to write σ ← R {σ ∈ S(F q ) n , σ -1 (w) ∈ C}.

For convenience, we will view r u ∈ R = F q as a vector indexed by u = (u 1 , . . . , u ), so that we can easily denote by r u [u j ] ∈ F q its j-th coordinate, 1 ≤ j ≤ . We define the code K u := ker V (u, •) ⊆ F q , and up to re-indexing coordinates, C |u ⊆ K u . This allows us to write that for every σ, we have V σ (u, r u ) = 0 if and only if σ -1 u (r u ) ∈ K u . Finally, we denote by Z u := {i ∈ u, r u [i] = R(w) u [i]} the set of coordinates of r u that are not authentic.

Informally, the reason why we consider an event Y u (σ) conditioned by supp(σ -1 u (r u )) = Z u is that the Prover is free to choose any support Z u on which he can modify the original file. More formally, this constraint will help us to bound the probability Pr Φw (V σ (u, r u ) = 0) in Lemma 3.14.

We say that Φ w is sufficiently uniform if, for every u ∈ Q, we have:

when the file size n log q → ∞. In other words, Φ w is sufficiently uniform if it is a good approximation of the whole set of n-tuples of permutations, when considering the probability that Y u (σ) happens.

Lemma 3.14. Let r, w, u and Z u be defined as above. Let also

Proof. For every σ such that (σ, m) ∈ Φ w , we know that σ -1 u (R(w) u ) ∈ K u , and we recall that V σ (u, r u ) = 0 if and only if σ -1 u (r u ) ∈ K u . Since K u is linear , and up to considering σ -1 u (R(w) u -r u ) instead, we can assume without loss of generality that σ -1 u (r u )[i] = 0 for every i ∈ u \ Z u . In other words we assume that supp(σ

since A u counts the number of codewords in K u whose support is Z u .

Therefore we get

Lemma 3.15. Let S u be the F q -vector space {x ∈ K u , supp(x) = Z u } and assume that S u = {0}. We have:

Proof. We prove that, if A u > q e for some integer e ≥ 0, then d min (S u ) ≤ |Z u | -e, which clearly induces our result. If A u > q e , then dim S u > e since |S u | ≥ A u . The Singleton bound then provides:

Finally, we get the following upper bound on α.

where γ u = max γ u .

Proof. Remark that S u , defined previous lemma, is a subcode of K u shortened on u \ Z u . Hence d min (K u ) ≤ d min (S u ), and we can apply previous results and obtain the desired bound:

where γ = max u γ u .

If every Φ w is sufficiently uniform, then by definition we have γ = o(1) when the file size n log q → ∞. This assumption is significant since we desire to have a small bias α, which is deeply linked to the soundness of PoRs (see Theorem 3.13). In Appendix A we present experimental estimates of α, validating that the assumption that Φ w is sufficiently uniform.

Pairwise uncorrelation of {X u } u∈D

This section is devoted to proving that variables {X u } u∈D(r,w) are pairwise uncorrelated if the supports of challenges u ∈ D(r, w) have small pairwise intersection. For this purpose, let us recall that for fixed r ∈ R Q , w and u ∈ D(r, w), the random variable X u represents 1 V σ (u,ru)=0 , when σ is uniformly picked in Φ w .

We first state a technical lemma that will be useful to prove Proposition 3.18 below. For clarity we denote by d ⊥ (C) the minimum distance of the dual code C ⊥ of a linear code C.

Lemma 3.17. Let C ⊆ F n q be a linear code and

2. for every non-zero a ∈ F T q , there exists a non-zero c (a) ∈ C such that

3. for every a ∈ F T q , N a = q k-t where k = dim C.

Proof.

1. The fact that V 0 = {v ∈ F X q , v |T = 0} is actually the well-known definition of the shortening of a code. It is easy to prove that it defines a linear code. 2. Let a ∈ F T q be non-zero, and let us first prove that there exists c (a) ∈ C such that c (a)

|T = a. If it were not the case, then by definition we would have C |T = F t q . But this is impossible since C ⊥ contains no non-zero codeword of weight less that t. It is then easy to check that

we get the expected result.

Proof. Recall that K u = ker V (u, •), and that by definition of a verification structure, we have

Set t = |u ∩ v| and let (a, b) ∈ (F t q ) 2 . We denote by Z(σ, a, b) the event

We first notice that {σ -1 |u∩v , σ ∈ Φ w } = S(F q ) t . Indeed, we can here use an argument similar to the proof of Lemma 3.17: the constraint σ -1 (w) ∈ C is ineffective on σ -1 |u∩v , since |u ∩ v| ≤ t < d ⊥ (C |z ) for every z ∈ Q. Therefore, for every (a, b) ∈ (F t q ) 2 , we have

and it follows that:

Recall now that t < min{d ⊥ (C |u ), u ∈ Q} ≤ min{d ⊥ (K u ), u ∈ Q}. Hence, for fixed a and b, the variables

) are independent (once again it is a consequence of the structure results of Lemma 3.17). Therefore:

Reed-Muller and related codes

Low-degree Reed-Muller codes are known to admit many distinct low-weight parity-check equations, whose supports correspond to affine subspaces of the ambient space. Therefore they seem naturally adapted to our construction. Let us first consider the plane (or bivariate) Reed-Muller code case.

5.2.1

The plane Reed-Muller code RM q (2, q -2)

Let C be the Reed-Muller code

It is well known that C has length q 2 and dimension (q -1)(q -2)/2. Besides, for every line

The restriction of f on an affine line L can be interpolated as a univariate polynomial f |L of degree at most a. Our claim follows since z∈Fq z i = 0 for every i ≤ q -2.

Therefore, we can define Q as the set of affine lines L of F 2 q , and V (L, r) = j=1 r j ∈ F q . From the previous discussion we see that (Q, V ) is a verification structure for C. Also notice there are q(q + 1) distinct affine lines in F 2 q , hence N = q(q + 1).

Lemma 5.5. Let C = RM q (2, q -2) equipped with its verification structure defined as above.

Then, the response code R(C) has minimum distance q 2 + 2.

Proof. Any non-zero codeword c ∈ C consists in the evaluation of a non-zero polynomial f (X, Y ) ∈ F q [X, Y ] of degree at most q -2. Denote by L 1 , . . . , L a ⊂ F 2 q , the affine lines on which f vanishes; i.e. f (P ) = 0 for every P ∈ L i , 1 ≤ i ≤ a. We claim that a ≤ q -2. Indeed, since f has total degree < q -1, it also vanishes on closed lines L 1 , . . . , L a , considered as affine lines in F q 2 , where F q denotes the algebraic closure of F q . Denote by g i ∈ F q [X, Y ] the monic polynomial of degree 1 which defines L i . From Hilbert's Nullstellensatz, there exists r > 0 such that ( a i=1 g i )|f r . Since the g i 's have degree 1 and are distinct, we get a ≤ deg f ≤ q -2. Hence, the affine lines different from L 1 , . . . , L a correspond to non-zero coordinates of R(c). There are q(q + 1) -a ≥ q 2 + 2 such lines so d min (R(C)) ≥ q 2 + 2. Now we claim there exists a word r ∈ R(C) of weight N -q + 2 = q 2 + 2. Let L (0) and L (1) be two distinct parallel affine lines respectively defined by X = 0 and X = 1. We build the word c which is -1 on coordinates corresponding to points in L (0) , 1 on those corresponding to points in L (1) and 0 elsewhere. One can check that c ∈ C; indeed c corresponds to the evaluation of z∈Fq\{0,1} (z -X). Now, if we want to compute wt(R(c)), we only need to count the number of lines which do not intersect L (0) nor L (1) . Clearly there are only q -2 such lines. Hence wt(R(c)) = q(q + 1) -(q -2) and this concludes the proof.

Proposition 5.6. Let C = RM(2, q -2), and let (Q, V ) be its associated verification structure. If every Φ w is sufficiently uniform, then the PoR scheme associated to C and (Q, V ) is (ε, τ )-sound for ε = 1 -o(1) and τ = O( 1 (1-ε)q 2 ), when q → ∞.

Proof. One can check that the random variables {X u } u∈D are pairwise uncorrelated since

A Experimental estimate of the bias α

We here confirm our heuristic on the fact that Φ w is sufficiently uniform, by providing experimental estimates of α.

Setup. We consider PoR schemes using Reed-Muller codes C = RM q (2, q -2), 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 Φ w are equivalently distributed. Indeed, if ψ ∈ S(F q ) n satisfies ψ(w) = w , then the distribution of permutations picked from Φ w can be obtained by applying ψ to permutations picked from Φ w . Hence, without loss of generality we assume that w = 0. Proposition 3.16 claims that in this context, α should be O(1/q) since ∆ = 2 and ≤ q. For convenience, we denote by p Φ := P Φw (V σ (u, r u ) = 0), and we recall that α is an upper bound on p Φ (for varying u and r).

We proceed to three kinds of tests in order to estimate α:

• 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 Φ by running M trials and computing the average number of times V σ (u, r u ) = 0 occurs. We denote by ξ M (p Φ ) this estimator. We then collect the maximum value of ξ M (p Φ ) among the N samples of u.

• 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 Φ with M samples. We collect the maximum value of ξ M (p Φ ) among the N values of r u that have been picked.

• Test 3. 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 ξ M (p Φ ).

Influence of M and the chosen test on the estimator. 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, i ∈ {1, 2, 3}, our estimator ξ M (p Φ ) converges to a value close to 1/(q -1).


Daran Zheng

Bo Wang

André Nel

Edmund A Jarzembowski

Haichun Zhang

Su-Chin Chang

Andr E Nel

an archaic group of platystictid damselflies (Odonata: Zygoptera) from mid-Cretaceous Burmese amber

Keywords: 6669DBCD-F2EF-4C84-9623-BD33142AD36B Platystictidae, Zygoptera, Odonata, Cenomanian, Cretaceous, Burmese amber

Mesostictinae subfam. nov.

Introduction

Platystictidae (shadowdamsels) currently consist of about 250 extant species in eight genera and four tropical subfamilies [START_REF] Wilson | The Platystictidae of Hong Kong and Guangdong, with descriptions of a new genus and two new species (Zygoptera)[END_REF][START_REF] Van Tol | Revision of the Platystictidae of the Philippines (Odonata), excluding the Drepanosticta halterata group, with descriptions of twenty-one new species[END_REF][START_REF] Van Tol | Phylogeny and biogeography of the Platystictidae (Odonata)[END_REF][START_REF] Dow | Telosticta, a new damselfly genus from Borneo and Palawan (Odonata: Zygoptera: Platystictidae)[END_REF][START_REF] Dijkstra | Redefining the damselfly families: a comprehensive molecular phylogeny of Zygoptera (Odonata)[END_REF][START_REF] Haber | Platystictidae-shadowdamsels, forest damsels. Dragonflies and damselflies of Ecuador (Odonata)[END_REF][START_REF] Schorr | World Odonata List[END_REF][START_REF] Suhling | Order Odonata[END_REF][START_REF] Dow | Telosticta fugispinosa sp. nov. from Sabah (Odonata: Zygoptera: Platystictidae)[END_REF][START_REF] Phan | A study of the genus Protosticta Selys, 1855, with descriptions of four new species from Vietnam (Odonata: Platystictidae)[END_REF][START_REF] Bedjani C | Taxonomy and molecular phylogeny of the Platystictidae of Sri Lanka (Insecta: Odonata)[END_REF], and two fossil species of the genus Mesosticta Huang, Azar, Cai & Nel, 2015 from mid-Cretaceous Burmese amber [START_REF] Huang | New damselfly genera in the Cretaceous Burmese amber attributable to the Platystictidae and Platycnemididae Disparoneurinae (Odonata: Zygoptera)[END_REF]Zheng et al. 2016a). The subfamily Palaemnematinae (Palaemnema Selys, 1860) is confined to central and north-western South America, while Sinostictinae (Sinosticta [START_REF] Wilson | The Platystictidae of Hong Kong and Guangdong, with descriptions of a new genus and two new species (Zygoptera)[END_REF], Platystictinae (Platysticta Selys, 1860, Ceylonosticta Fraser, 1931) and Protostictinae (Protosticta Selys, 1885, Drepanosticta Laidlaw, 1917, Telosticta Dow & Orr, 2012and Sulcosticta van Tol, 2005) are restricted to South-East Asia.

Platystictidae were hypothesized to have originated from eastern Africa during the Late Cretaceous, then later migrated to South America, Asia and New Guinea [START_REF] Van Tol | Forest damselflies of the Philippines, their evolution and present status, with the description of Drepanosticta moorei sp. nov. from Luzon (Zygoptera: Platystictidae)[END_REF][START_REF] Van Tol | Phylogeny and biogeography of the Platystictidae (Odonata)[END_REF][START_REF] Herrera | Biogeography of dragonflies and damselflies: highly mobile predators[END_REF]. The recently discovered Mesosticta, however, puts the appearance of Platystictidae back to at least the mid-Cretaceous (Zheng et al. 2016a).

The platystictid damselflies are characterized by a long and very slender abdomen, and a unique wing with a basally recessed 'CuP' [START_REF] Fraser | A reclassification of the order Odonata, based on some new interpretations of the venation of the dragonfly wing[END_REF][START_REF] Jarzembowski | The Mesozoic non calopterygoid Zygoptera: description of new genera and species from the lower Cretaceous of England and Brazil and their phylogenetic significance (Odonata, Zygoptera, Coenagrionoidea, Hemiphlebioidea, Lestoidea)[END_REF][START_REF] Huang | New damselfly genera in the Cretaceous Burmese amber attributable to the Platystictidae and Platycnemididae Disparoneurinae (Odonata: Zygoptera)[END_REF], a long IR1 and a usually diamond-shaped subdiscoidal cell [START_REF] Haber | Platystictidae-shadowdamsels, forest damsels. Dragonflies and damselflies of Ecuador (Odonata)[END_REF]. The platystictid genus Mesosticta (M. burmatica [START_REF] Huang | New damselfly genera in the Cretaceous Burmese amber attributable to the Platystictidae and Platycnemididae Disparoneurinae (Odonata: Zygoptera)[END_REF][START_REF] Huang | New damselfly genera in the Cretaceous Burmese amber attributable to the Platystictidae and Platycnemididae Disparoneurinae (Odonata: Zygoptera)[END_REF]M. electronica Zheng, Zhang, Chang &Wang, 2016) was tentatively attributed to this family due to a basally recessed CuP-crossing [START_REF] Huang | New damselfly genera in the Cretaceous Burmese amber attributable to the Platystictidae and Platycnemididae Disparoneurinae (Odonata: Zygoptera)[END_REF] and the rhomboidal shape of the subdiscoidal triangle (only present in the hindwing of M. electronica; *Corresponding authors. E-mail: dranzheng@gmail.com; suchin@hku.hk Zheng et al. 2016a). Both species, however, do not have a very long IR1, which is present in all recent platystictid damselflies. In this paper, we describe a new platystictid species of Mesosticta from mid-Cretaceous Burmese amber. The new species has a very long IR1, so it confirms the previous attribution of Mesosticta to the Platystictidae. Based on three species of Mesosticta, a new subfamily is established.

Geological background

The piece of amber containing the new damselfly was collected in the Hukawng Valley (26 29 0 N, 96 35 0 E) of Kachin Province, Myanmar (locality in Kania et al. 2015, fig. 1). Abundant insect inclusions have been described from Burmese amber [START_REF] Cruickshank | Geology of an amber locality in the Hukawng Valley, northern Myanmar[END_REF][START_REF] Shi | Age constraint on Burmese amber based on U-Pb dating of zircons[END_REF][START_REF] Kania | Dicranoptycha Osten Sacken, 1860 (Diptera, Limoniidae) from the earliest Upper Cretaceous Burmese amber[END_REF], with rare odonatans including true dragonflies (Sch€ adel & Bechly 2016), damsel-dragonflies [START_REF] Bechly | Burmaphlebia reifi gen. et sp. nov., the first anisozygopteran damsel-dragonfly (Odonata: Epiophlebioptera: Burmaphlebiidae fam. nov.) from Early Cretaceous Burmese amber[END_REF]) and damselflies [START_REF] Poinar | First record of Odonata and a new subfamily of damselflies from Early Cretaceous Burmese amber[END_REF][START_REF] Huang | New damselfly genera in the Cretaceous Burmese amber attributable to the Platystictidae and Platycnemididae Disparoneurinae (Odonata: Zygoptera)[END_REF]Zheng et al. 2016a, b, c, d). The age of the Burmese amber was considered to be Late Albian to Cenomanian based on palynology and ammonite stratigraphy [START_REF] Cruickshank | Geology of an amber locality in the Hukawng Valley, northern Myanmar[END_REF][START_REF] Ross | Chapter 12. Burmese amber[END_REF], while it was radiometrically dated at 98.79 § 0.62 Ma (earliest Cenomanian) based on U-Pb zircon dating of the volcanoclastic matrix [START_REF] Shi | Age constraint on Burmese amber based on U-Pb dating of zircons[END_REF]. The real depositional age of the amber is probably earlier than this radiometric age since the amber shows evidence of redeposition [START_REF] Wang | Brood care in a 100million-year-old scale insect[END_REF]. Here we follow the age given by [START_REF] Shi | Age constraint on Burmese amber based on U-Pb dating of zircons[END_REF], but we treat it as minimal.

Material and methods

The piece of amber containing the damselfly is yellow and transparent. The damselfly is near the amber surface, and well preserved with complete forewings. The abdomen and hindwings were polished away. Photographs were taken using a Zeiss Stereo Discovery V16 microscope system and Zen software. In most instances, incident and transmitted light were used simultaneously. All images are digitally stacked photomicrographic composites of approximately 40 individual focal planes obtained using the free software Combine ZP for a better illustration of the three-dimensional structures. The line drawings were prepared from photographs using image-editing software (CorelDraw X7 and Adobe Photoshop CS6). The specimen is housed in the Nanjing Institute of Geology and Palaeontology, Chinese Academy of Sciences (NIGPAS).

The nomenclature of the dragonfly wing venation used in this paper is based on the interpretations of [START_REF] Riek | A new collection of insects from the Upper Triassic of South Africa[END_REF] and [START_REF] Riek | A new interpretation of dragonfly wing venation based upon Early Carboniferous fossils from Argentina (Insecta: Odonatoidea) and basic character states in pterygote wings[END_REF], as modified by [START_REF] Nel | Les 'Anisozygoptera' fossiles. Phylog enie et classification (Odonata)[END_REF] and [START_REF] Bechly | Morphologische Untersuchungen am Fl€ ugelge€ ader der rezenten Libellen und deren Stammgruppenvertreter (Insecta; Pterygota; Odonata), unter besonderer Ber€ ucksichtigung der Phylogenetischen Systematik und des Grundplanes der Odonata[END_REF]. The phylogeny of extant Zygoptera followed in the present work is based on [START_REF] Dijkstra | Redefining the damselfly families: a comprehensive molecular phylogeny of Zygoptera (Odonata)[END_REF], while the generic characters of Mesosticta are after Zheng et al. (2016a) character (9) to differentiate the species. Within Mesosticta (Fig. 7), M. davidattenboroughi has a crossvein across the basal part of the subdiscoidal cell, differing from M. burmatica which has a crossvein in the distal part of the subdiscoidal cell, and M. electronica which has a free, rhomboidal subdiscoidal cell. M. davidattenboroughi also has the arculus aligned with Ax2, as in M. burmatica but unlike M. electronica which has Ax2 slightly basal of the arculus. The fossil genus Mesosticta has the diagnostic characters of the modern platystictid genera (Fig. 7), viz. a basally recessed 'CuP' (shared by all species), a very long IR1 (in M. davidattenboroughi), and a specialized subdiscoidal area mostly rhomboidal in shape (rhomboidal shape shared with M. electronica). The presence of these characters confirms the previous attribution of Mesosticta to Platystictidae. However, all these characters are not shared by one species as in most recent Platystictidae. Also, Mesosticta shares with the extant platystictid damselflies the following characters: vein RP1 with a very slight angle below the pterostigmal brace; postnodal and postsubnodal crossveins somewhat aligned; longitudinal veins RA, RP1, IR1, RP2 and IR2 strongly converging on wing apex; pterostigma one cell long, thickened and fused with greatly thickened pterostigmal part of RA to form a U-shaped structure.

The differences between the modern platystictid damselflies and Mesosticta (Fig. 7) are that the former have more postnodal and postsubnodal crossveins (over 15 compared to 10 or fewer in Mesosticta), a long MP (at least five cells long in Protostictinae to over half the wing length in Sinostictinae, compared with three or four cells long in Mesosticta -the shortening of the vein MP being a potential synapomorphy of the Mesostictinae), the base of RP2 being more cells distal of the subnodus (at least four cells compared to only three or four cells in Mesosticta) and the nodus lying very basally (23-30% of the wing length compared to at least 35% in the latter). Due to the above differences, we establish a new subfamily Mesostictinae for the fossil platystictid damselflies from Burmese amber, representing the archaic platystictid damselflies.

Figure 1 .

 1 Figure 1. Mesosticta davidattenboroughi sp. nov., holotype, NIGP164541, photograph of specimen.

Figure 2 .

 2 Figure 2. Mesosticta davidattenboroughi sp. nov., holotype, NIGP164541. A, photograph of left forewing; B, photograph of right forewing.

Figure 3 .

 3 Figure 3. Mesosticta davidattenboroughi sp. nov., holotype, NIGP164541, line drawing showing venation of left forewing.

Figure 4 .

 4 Figure 4. Mesosticta davidattenboroughi sp. nov., holotype, NIGP164541, photograph showing detail of left forewing base.

Figure 5 .

 5 Figure 5. Mesosticta davidattenboroughi sp. nov., holotype, NIGP164541, photograph of right forewing nodus.

Figure 6 .

 6 Figure 6. Mesosticta davidattenboroughi sp. nov., holotype, NIGP164541, photograph of right forewing pterostigma.

Figure 7 .

 7 Figure 7. Hypothetical position of Mesosticta davidattenboroughi sp. nov. in phylogenetic tree of Zygoptera. All line drawings are based on forewings (phylogeny based on Dijkstra et al. 2014; line drawing of Sinosticta ogatai Matsuki & Saito, 1996 after Wilson 1997; line drawing of Palaemnema picicaudata Kennedy, 1938 after Kennedy 1938; line drawings of Platysticta deccanensis Laidlaw, 1915 and Protosticta himalaiaca Laidlaw, 1917 after Fraser 1933).

  . Wing abbreviations are as follows: AA, anal anterior; AP, anal posterior; Arc, arculus; Ax, primary antenodal crossvein; Cr, nodal crossvein; CuA, cubitus anterior; CuP, cubitus posterior; DC, discoidal cell; IR, intercalary radial vein; MA, median anterior; MP, median posterior; N, nodus; Pt, pterostigma; RA, radius anterior; RP, radius posterior; ScP, subcosta posterior; Sn, subnodal crossvein. RP1 with slight angle below pterostigmal brace; few postnodal and postsubnodal crossveins (less than 10), somewhat aligned; nodus not in very basal position, at least at 35% of wing length; longitudinal veins RA, RP1, IR1, RP2 and IR2 strongly converging on wing apex; Pt one cell long, thickened and fused with greatly thickened pterostigmal part of RA as U-shaped structure.

		Mesosticta Huang, Azar, Cai & Nel, 2015
	Type species. Mesosticta burmatica Huang, Azar, Cai &
	Nel, 2015.
	Included species. Mesosticta electronica Zheng, Zhang,
	Chang & Wang, 2016; Mesosticta davidattenboroughi sp.
	nov.
	Revised diagnosis. As for subfamily diagnosis.
	Systematic palaeontology	Mesosticta davidattenboroughi sp. nov. (Figs 1-7)
	Type species. NIGP164541, two complete forewings
	attached to body.

Order Odonata Fabricius, 1793 Suborder Zygoptera Selys-Longchamps, 1854 Superfamily Platystictoidea Kennedy, 1920 Family Platystictidae Kennedy, 1920 Subfamily Mesostictinae subfam. nov. Type genus. Mesosticta Huang, Azar, Cai & Nel, 2015. Diagnosis. Base of RP3/4 just basal of Sn; base of RP2 three or four cells distal of Sn; MA long and ending on posterior wing margin just below Pt brace; MP three or four cells long; CuP in basal position; crossvein present basally closing subdiscoidal cell, seperate from CuP; subdiscoidal cell posteriorly closed by AA and not by posterior wing margin; Diagnosis. Forewing characters: IR1 long, originating one cell distal of base of RP2, nearer to N than to Pt; Arc aligned with Ax2; subdiscoidal cell basally crossed by one vein.

Etymology. In honour of Sir David Attenborough, on his 90th birthday, for his appreciation of dragonflies.

Locality and horizon. Hukawng Valley, Kachin Province, Myanmar; lowermost Cenomanian, lowermost Upper Cretaceous.

Mesostictinae subfam. nov., an archaic group of platystictid damselflies

Acknowledgements

This research was supported by the National Natural Science Foundation of China (grant numbers 41572010, 41622201, 41688103), Youth Innovation Promotion Association of the Chinese Academy of Sciences (grant num-ber2011224), and the HKU Seed Funding Program for Basic Research (grant number 201210159058). We offer our sincere gratitude to the two anonymous reviewers for their very useful comments on an earlier version of the manuscript.