Galois geometriesGeometry and cryptography
Applications of Galois Geometries to CodingTheory and Cryptography
Leo Storme
Ghent UniversityDept. of Mathematics
Krijgslaan 281 - Building S229000 Ghent
Belgium
Albena, July 1, 2013
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
OUTLINE
1 GALOIS GEOMETRIES1. Affine spaces2. Projective spaces
2 GEOMETRY AND CRYPTOGRAPHY1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
FINITE FIELDS
q = prime number.
Prime fields Fq = {0,1, . . . ,q − 1} (mod q).Binary field F2 = {0,1}.Ternary field F3 = {0,1,2} = {−1,0,1}.
Finite fields Fq: q prime power.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
AFFINE SPACE AG(n,q)
V (n,q) = n-dimensional vector space over Fq.AG(n,q) = V (n,q) plus parallelism.k -dimensional affine subspace = (translate) ofk -dimensional vector space.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
PARALLELISM IN AFFINE SPACE AG(n,q)
Let Πk be k -dimensional vector space of V (n,q).Πk + b, for b ∈ V (n,q), are the affine k -subspaces parallelto Πk .Two parallel affine k -subspaces are disjoint or equal.Parallelism leads to partitions of AG(n,q) into (parallel)affine k -subspaces.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
AFFINE PLANE AG(2,3) OF ORDER 3
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
FROM V (3,q) TO PG(2,q)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
FROM V (3,q) TO PG(2,q)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
THE FANO PLANE PG(2,2)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
THE FANO PLANE PG(2,2)
Gino Fano (1871-1952)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
THE PLANE PG(2,3)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
FROM V (4,q) TO PG(3,q)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
FROM V (4,q) TO PG(3,q)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
PG(3,2)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
FROM V (n + 1,q) TO PG(n,q)
1 From V (1,q) to PG(0,q) (projective point),2 From V (2,q) to PG(1,q) (projective line),3 · · ·4 From V (i + 1,q) to PG(i ,q) (i-dimensional projective
subspace),5 · · ·6 From V (n,q) to PG(n − 1,q) ((n − 1)-dimensional
subspace = hyperplane),7 From V (n + 1,q) to PG(n,q) (n-dimensional space).
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
LINK BETWEEN AFFINE AND PROJECTIVE SPACES
AG(n,q) = PG(n,q) minus one hyperplane (the hyperplaneat infinity).
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Affine spaces2. Projective spaces
LINK BETWEEN AG(2,3) AND PG(2,3)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
OUTLINE
1 GALOIS GEOMETRIES1. Affine spaces2. Projective spaces
2 GEOMETRY AND CRYPTOGRAPHY1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
SECRET SHARING SCHEME
1 Secret sharing scheme: cryptographic equivalent of vaultthat needs several keys to be opened.
2 Secret S divided into shares.3 Authorised sets: have access to secret S by putting their
shares together.4 Unauthorised sets: have no access to secret S by putting
their shares together.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
(n, k)-THRESHOLD SCHEME
1 n participants.2 Each group of k participants can reconstruct secret S, but
less than k participants have no way to learn anythingabout secret S.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
SHAMIR’S k -OUT-OF-n SECRET SHARING SCHEME
1 Fq = finite field of order q.2 Dealer chooses polynomial
f (X ) = f0 + f1X + · · ·+ fk−1X k−1 ∈ Fq[X ], and,3 gives participant number i , point (xi , f (xi)) on graph of f
(xi 6= 0).4 Value f (0) = f0 is secret S.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
SHAMIR’S k -OUT-OF-n SECRET SHARING SCHEME
1 Set of k participants can reconstructf (X ) = f0 + f1X + · · ·+ fk−1X k−1 by interpolating theirshares (xi , f (xi)). Then they can compute secret f (0).
2 If k ′ < k persons try to reconstruct secret, for every y ∈ Fq,there are exactly |Fq|k−k ′−1 polynomials of degree at mostk − 1 which pass through their shares and the point (0, y).Thus they gain no information about f (0).
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
REALISATION OF SHAMIR’S k -OUT-OF-n SECRET
SHARING SCHEME
ut ut
ut
ut
ut
S1
S2
S3
S4
S5 rs
secret point
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
GEOMETRICAL REALISATION OF SHAMIR’S k -OUT-OF-nSECRET SHARING SCHEME (BLAKLEY)
1 Secret S = point of PG(3,q).2 Shares = planes of PG(3,q) such that exactly three of
them only intersect in S.3 Classical example: Normal rational curve of planes
X0 + tX1 + t2X2 + t3X3 = 0, t ∈ Fq,
andX3 = 0.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
GEOMETRICAL REALISATION OF SHAMIR’S k -OUT-OF-nSECRET SHARING SCHEME (BLAKLEY)
1 Secret S = point of PG(k ,q).2 Shares = hyperplanes of PG(k ,q) such that exactly k of
them only intersect in S.3 Classical example: Normal rational curve of hyperplanes
X0 + tX1 + t2X2 + · · ·+ tkXk = 0, t ∈ Fq,
andXk = 0.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
GEOMETRICAL REALISATION OF SHAMIR’S k -OUT-OF-nSECRET SHARING SCHEME (BLAKLEY)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
GEOMETRICAL REALISATION OF SHAMIR’S k -OUT-OF-nSECRET SHARING SCHEME
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
GEOMETRICAL REALISATION OF SHAMIR’S k -OUT-OF-nSECRET SHARING SCHEME
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
CODING-THEORETICAL REALISATION OF SHAMIR’S
k -OUT-OF-n SECRET SHARING SCHEME
(McEliece and Sarwate)1 C : [n + 1, k ,n − k + 2]q MDS code.2 For secret c0 ∈ Fq, dealer creates codeword
c = (c0, c1, . . . , cn) ∈ C. Share of participant number i issymbol ci .
3 Since C is MDS code with minimum distance n − k + 2,codeword c can be uniquely reconstructed if only ksymbols are known.
4 So any set of k persons can compute secret c0.5 On the other hand, less than k persons do not learn
anything about secret, since for any possible secret c′, thesame number of codewords that fit to secret c′ and theirshares exist.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
MORE GENERAL SECRET SHARING SCHEME
DEFINITION
Support of c = (c1, . . . , cn) ∈ Fnq :
sup(c) = {i | ci 6= 0}.
Let C be linear code. Nonzero codeword c ∈ C is calledminimal if
∀c′ ∈ C \ {0} : sup(c′) ⊆ sup(c) =⇒ c′ = ρc,
ρ ∈ Fq \ {0}.
(In binary case, c minimal if no non-zero codeword c′ withsup(c′) ⊂ sup(c), sup(c′) 6= sup(c))
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
MORE GENERAL SECRET SHARING SCHEME
LEMMA (MASSEY)
Let C be an [n + 1, k ]q-code. Secret sharing scheme isconstructed from C by choosing codeword c = (c0, . . . , cn).Secret is c0 and shares of participants are coordinates ci(1 ≤ i ≤ n).Minimal authorized sets of secret sharing scheme correspondto minimal codewords of C⊥ with 0 in their supports.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
BINARY REED-MULLER CODES
DEFINITION
Binary r -th order Reed-Muller code RM(r ,m) (0 ≤ r ≤ m) = setof all binary vectors f of length n = 2m associated with Booleanpolynomials f (x1, x2, ..., xm) of degree at most r :
c = (f (0, . . . ,0), . . . , f (1, . . . ,1)).
Minimum weight d = 2m−r .Minimum weight codewords of RM(r ,m) = incidencevectors of AG(m − r ,2) in AG(m,2).
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
BINARY REED-MULLER CODES
THEOREM (KASAMI, TOKURA, AND AZUMI)
Let f (x1, ..., xm) be Boolean function of degree at most r , wherer ≥ 2, such that |sup(f )| < 2m−r+1. Then f can be transformedby an affine transformation into
f = x1 · · · xr−2(xr−1xr +· · ·+xr+2µ−3xr+2µ−2), 2 ≤ 2µ ≤ m−r +2,
or
f = x1 · · · xr−µ(xr−µ+1 · · · xr +xr+1 · · · xr+µ), 3 ≤ µ ≤ r , µ ≤ m−r .
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
BINARY REED-MULLER CODES
First type of codewords(1)
f = x1 · · · xr−2(xr−1xr +· · ·+xr+2µ−3xr+2µ−2),2 ≤ 2µ ≤ m−r+2,
In PG(m − r + 2,2) defined by X1 = X0, . . . ,Xr−2 = X0,cone Ψ with vertex PG(m − r + 1− 2µ,2) at infinity, andbase non-singular parabolic quadric Q(2µ,2) in 2µdimensions having non-singular hyperbolic quadric atinfinity.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
QUADRATIC CONE
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1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
BINARY REED-MULLER CODES
Second type of codewords(2)
f = x1 · · · xr−µ(xr−µ+1 · · · xr +xr+1 · · · xr+µ),3 ≤ µ ≤ r , µ ≤ m−r .
(Symmetric difference): Union of two (m − r)-dimensionalaffine spaces α and β, but not (m − r − µ)-dimensionalaffine intersection space α ∩ β.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
SYMMETRIC DIFFERENCE
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Galois geometriesGeometry and cryptography
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COUNTING NON-MINIMAL CODEWORDS IN RM(r ,m)
Non-minimal codeword c = c1 + c2, with c1, c2 non-zerocodewords having disjoint supports.For w(c) < 3 · 2m−r , c1 codeword of smallest weight 2m−r ,and c2 codeword of weight 2m−r or quadric or symmetricdifference.Number of non-minimal codewords c of weight 2 · 2m−r
calculated by Borissov, Manev, and Nikova.Number of non-minimal codewords c of weight2 · 2m−r < w(c) < 3 · 2m−r calculated by Schillewaert,Storme, and Thas.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
COUNTING NON-MINIMAL CODEWORDS IN RM(r ,m)
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
PROBLEM OF AUTHENTICATION
1 Problem: Alice wants to send Bob a message m.2 Attacker intercepts m and sends alternated message m′ to
Bob.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
PROBLEM OF AUTHENTICATION
How can Bob be sure that message he gets is correct?Introduce authentication!
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
EXAMPLE OF MESSAGE AUTHENTICATION CODE
1 ` = line of PG(2,q).2 Message m = point of `.3 Authentication key K = point in PG(2,q)\`.4 Authentication tag = line through message m and key K .
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
EXAMPLE OF AUTHENTICATION CODE
1 If attacker wants to create message (m,K ) withoutknowing key K , he must guess an affine line through m.There are q possibilities, i.e. the chance for correct attackis 1
q .2 If attacker already knows authenticated message (m,K ),
he knows that key K must lie on the line mK .But for every of q affine points on line mK , there exists linethrough m. So he cannot do better than guess the keywhich gives probability of 1
q for successful attack.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
SECURITY OF AUTHENTICATION CODE
1 pi = probability of attacker to construct pair (m,K ) withoutknowledge of key K , if he only knows i different pairs(mj ,K ).
2 Smallest value r for which pr+1 = 1 is called order ofauthentication code.
3 For r = 1, p0 = probability of impersonation attack andprobability p1 = probability of substitution attack.
THEOREM
If MAC has attack probabilities pi = 1/ni (0 ≤ i ≤ r ), then|K| ≥ n0 · · · nr .
MAC that satisfies this theorem with equality is called perfect.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
GEOMETRICAL CONSTRUCTION OF PERFECT MAC
DEFINITION
Generalised dual arc D of order l with dimensionsd1 > d2 > · · · > dl+1 of PG(n,q) is set of subspaces ofdimension d1 such that:
1 each j subspaces intersect in subspace of dimension dj ,1 ≤ j ≤ l + 1,
2 each l + 2 subspaces have no common intersection.(n,d1, . . . ,dl+1) = parameters of dual arc.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
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GENERALISED DUAL ARCS
THEOREM
There exists generalised dual arc in PG((n+d+1
d+1
)− 1,q), with
dimensions di =(n+d+1−i
d+1−i
)− 1, i = 0, . . . ,d + 1.
1 Spaces have dimension d1 =(n+d
d
)− 1.
2 Two spaces intersect in space of dimensiond2 =
(n+d−1d−1
)− 1.
3 Three spaces intersect in space of dimensiond3 =
(n+d−2d−2
)− 1.
4 · · ·
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Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
LINK BETWEEN MAC AND GENERALISED DUAL ARC
1 π = hyperplane of PG(n + 1,q) and D = generalised dualarc of order l in π with parameters (n,d1, . . . ,dl+1).
2 message m = element of D.3 key K = point of PG(n + 1,q) not in π.4 Authentication tag that belongs to message m and key K is
generated (d1 + 1)-dimensional subspace.5 Perfect MAC of order r = l + 1 with attack probabilities
pi = qdi+1−di .
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
ANONYMOUS DATABASE SEARCH
Anonymous database search: query a databaseanonomously.Peer-to-peer community: let users post queries on behalfof each other.Neighbourhood attack: can be modeled as the intersectionof neighbourhoods that may return a single identifiedperson in case of unique neighbourhoods.k-Anonymous neighbourhoods: neighbourhood of personis also neighbourhood of at least k − 1 other persons.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
TRANSVERSAL DESIGNS
Transversal design TDλ(k ,n) = k -uniform structure (P,L)of points and blocks, with |P| = kn, that admits partition ofP in k groups of cardinality n, and that satisfies:
any group and block contain exactly one common point,every pair of points from distinct groups is contained inexactly λ blocks.
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FROM AG(2,n) TO TD1(k ,n)
From affine plane AG(2,n) to transversal design TD1(k ,n),2 ≤ k ≤ n.
Point set P of TD1(k ,n) = points of AG(2,n) on k lines ofone parallel class of AG(2,n),Groups = lines from this parallel class,Blocks of TD1(k ,n) = lines of the other parallel classes ofAG(2,n), restricted to the points in P.
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FROM AG(2,n) TO TD1(k ,n)
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TRANSVERSAL DESIGN TD1(k ,n) AND n-ANONYMOUS
NEIGHBOURHOODS
THEOREM
Transversal design TD1(k ,n) has n-anonymousneighbourhoods.
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THEOREMS
THEOREM (STOKES AND FARRÀS)
Combinatorial (v ,b, r , k)-configuration with n-anonymousneighbourhoods satisfies:
There exists partition G = {gi}mi=1 of the point set such thatthe points in the same part are not collinear and |gi | ≥ n,for all i ∈ {1, . . . ,m},r ≥ n and m ≥ k.
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THEOREMS
THEOREM (STOKES AND FARRÀS)
In combinatorial (v ,b, r , k)-configuration C with n-anonymousneighbourhoods and anonymity partition G = {gi}mi=1 and|gi | = n for all i ∈ {1, . . . ,m},
v = n iff m = k .
In this case, C is transversal design TD1(k ,n), and v = kn andb = n2.
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APPLICATION IN PAY TELEVISION
(Korjik, Ivkov, Merinovich, Barg, and van Tilborg)
subscribers = points of PG(2,q),codes = lines of PG(2,q),subscriber quits: codes of lines become invalid,new issue of codes: only necessary when codes of all linesthrough subscriber become invalid.
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Galois geometriesGeometry and cryptography
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THE FANO PLANE PG(2,2)
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REFERENCES
W.-A. Jackson, K.M. Martin, and C.M. O’Keefe,Geometrical contributions to secret sharing theory. J.Geom. 79 (2004), 102–133.W.-A. Jackson, K.M. Martin, and M.B. Paterson,Applications of Galois geometry to cryptology. Chapter inCurrent research topics in Galois geometry (J. De Beuleand L. Storme, Eds.), NOVA Academic Publishers (2012),215–244.
Leo Storme Galois geometries and cryptography
Galois geometriesGeometry and cryptography
1. Secret sharing scheme2. Message Authentication code (MAC)3. Anonymous database search4. Application in pay television
Thank you very much for your attention!
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