Applications of finite geometry in coding theory and cryptography

Construction 1Let PG(V ) be an n-dimensional(space with basis e i (0 ≤ i ≤ n).(n+d+1 ) )Let PG(W) be and+1 − 1 -dimensional space with basis e i0,...,i d(0 ≤ i 0 ≤i 1 ≤ · · · ≤ i d ≤ n).To simplify notations, we will write e i0,...,i dwith 0 ≤ i 0 , . . .,i d ≤ n when we meanthe vector e iσ(0) ,...,i σ(d)where σ is a permutation with 0 ≤ i σ(0) ≤ i σ(1) ≤ · · · ≤ i σ(d) ≤n.Let θ : V d+1 → W be the multilinear mapping.θ : (n∑i 0=0x (0)i 0e i0 , . . .,n∑i d =0x (d)i de id ) ↦→∑0≤i 0,...,i d ≤nx (0)i 0· . . . · x (d)i de i0,...,i d. (4)For each point P = [x] of PG(V ), we define a subspace D(P) of PG(W) byD(P) = 〈θ(x, v 1 , . . .,v d ) | v 1 , . . . , v d ∈ V 〉 . (5)Theorem 15The set D = {D(P) | P ∈ PG(V )} is a generalised dual arc with dimensions d i =)− 1, i = 0, . . . , d + 1.( n+d+1−id+1−iProof. Since θ is a multilinear form, we getD(P 0 ) ∩ · · · ∩ D(P k−1 ) = 〈θ(x 0 , . . . , x k−1 , v k , . . .,v d ) | v k , . . . , v d ∈ V 〉and hence dim(D(P 0 ) ∩ · · · ∩ D(P k−1 )) = ( )n+d+1−kd+1−k − 1. (The −1 is because theprojective dimension is one less than the vector space dimension).□The link between dual arcs and MACs is:Theorem 16Let π be a hyperplane of PG(n + 1, q) and let D be a generalised dual arc of order l inπ with parameters (n, d 1 , . . . , d l+1 ).The elements of D are the messages and the points of PG(n + 1, q) not in π arethe keys. The authentication tag that belongs to a message and a key is the generated(d 1 + 1)-dimensional subspace.This defines a perfect MAC of order r = l + 1 with attack probabilitiesp i = q di+1−di .Proof. After i message tag pairs (m 1 , t 1 ), . . . , (m i , t i ) are sent, the attacker knows thatthe key must lie in the (d i + 1)-dimensional space π = t 1 ∩ · · · ∩ t i . This space containsq di+1 different keys. A message m i+1 intersects m 1 ∩ · · · ∩ m i in a d i+1 -dimensionalspace π ′ . Two keys K and ¯K generate the same authentication tag if and only if K and¯K generate together with π ′ the same (d i+1 +1)-dimensional space. Thus the keys formgroups of size q di+1+1 and keys from the same group give the same authentication tag.The attacker has to guess a group. The probability to guess the correct group isp i = q di+1+1 /q di+1 .□