Applications of Galois Geometries to Coding Theory and Cryptography - PowerPoint PPT Presentation

Galois geometries Geometry and cryptography Applications of Galois Geometries to Coding Theory and Cryptography Leo Storme Ghent University Dept. of Mathematics Krijgslaan 281 - Building S22 9000 Ghent Belgium Albena, July 1, 2013 Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces O UTLINE 1 G ALOIS GEOMETRIES 1. Affine spaces 2. Projective spaces 2 G EOMETRY AND CRYPTOGRAPHY 1. Secret sharing scheme 2. Message Authentication code (MAC) 3. Anonymous database search 4. Application in pay television Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces F INITE FIELDS q = prime number. Prime fields F q = { 0 , 1 , . . . , q − 1 } ( mod q ) . Binary field F 2 = { 0 , 1 } . Ternary field F 3 = { 0 , 1 , 2 } = {− 1 , 0 , 1 } . Finite fields F q : q prime power. Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces A FFINE SPACE AG ( n , q ) V ( n , q ) = n -dimensional vector space over F q . AG ( n , q ) = V ( n , q ) plus parallelism. k -dimensional affine subspace = (translate) of k -dimensional vector space. Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces P ARALLELISM IN AFFINE SPACE AG ( n , q ) Let Π k be k -dimensional vector space of V ( n , q ) . Π k + b , for b ∈ V ( n , q ) , are the affine k -subspaces parallel to Π k . 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 geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces A FFINE PLANE AG ( 2 , 3 ) OF ORDER 3 Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces F ROM V ( 3 , q ) TO PG ( 2 , q ) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces F ROM V ( 3 , q ) TO PG ( 2 , q ) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces T HE F ANO PLANE PG ( 2 , 2 ) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces T HE F ANO PLANE PG ( 2 , 2 ) Gino Fano (1871-1952) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces T HE PLANE PG ( 2 , 3 ) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces F ROM V ( 4 , q ) TO PG ( 3 , q ) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces F ROM V ( 4 , q ) TO PG ( 3 , q ) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces PG ( 3 , 2 ) Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces F ROM V ( n + 1 , q ) TO PG ( n , q ) From V ( 1 , q ) to PG ( 0 , q ) (projective point), 1 From V ( 2 , q ) to PG ( 1 , q ) (projective line), 2 · · · 3 From V ( i + 1 , q ) to PG ( i , q ) ( i -dimensional projective 4 subspace), · · · 5 From V ( n , q ) to PG ( n − 1 , q ) ( ( n − 1 ) -dimensional 6 subspace = hyperplane), From V ( n + 1 , q ) to PG ( n , q ) ( n -dimensional space). 7 Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces L INK BETWEEN AFFINE AND PROJECTIVE SPACES AG ( n , q ) = PG ( n , q ) minus one hyperplane (the hyperplane at infinity). Leo Storme Galois geometries and cryptography

Galois geometries 1. Affine spaces Geometry and cryptography 2. Projective spaces L INK BETWEEN AG ( 2 , 3 ) AND PG ( 2 , 3 ) Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television O UTLINE 1 G ALOIS GEOMETRIES 1. Affine spaces 2. Projective spaces 2 G EOMETRY AND CRYPTOGRAPHY 1. Secret sharing scheme 2. Message Authentication code (MAC) 3. Anonymous database search 4. Application in pay television Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television S ECRET SHARING SCHEME Secret sharing scheme : cryptographic equivalent of vault 1 that needs several keys to be opened. Secret S divided into shares . 2 Authorised sets : have access to secret S by putting their 3 shares together. Unauthorised sets : have no access to secret S by putting 4 their shares together. Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television ( n , k ) - THRESHOLD SCHEME n participants. 1 Each group of k participants can reconstruct secret S , but 2 less than k participants have no way to learn anything about secret S . Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME F q = finite field of order q . 1 Dealer chooses polynomial 2 f ( X ) = f 0 + f 1 X + · · · + f k − 1 X k − 1 ∈ F q [ X ] , and, gives participant number i , point ( x i , f ( x i )) on graph of f 3 ( x i � = 0). Value f ( 0 ) = f 0 is secret S . 4 Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME Set of k participants can reconstruct 1 f ( X ) = f 0 + f 1 X + · · · + f k − 1 X k − 1 by interpolating their shares ( x i , f ( x i )) . Then they can compute secret f ( 0 ) . If k ′ < k persons try to reconstruct secret, for every y ∈ F q , 2 there are exactly | F q | k − k ′ − 1 polynomials of degree at most k − 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

ut ut rs ut ut ut 1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television R EALISATION OF S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME secret point S 1 S 3 S 5 S 2 S 4 Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television G EOMETRICAL REALISATION OF S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME (B LAKLEY ) Secret S = point of PG ( 3 , q ) . 1 Shares = planes of PG ( 3 , q ) such that exactly three of 2 them only intersect in S . Classical example: Normal rational curve of planes 3 X 0 + tX 1 + t 2 X 2 + t 3 X 3 = 0 , t ∈ F q , and X 3 = 0 . Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television G EOMETRICAL REALISATION OF S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME (B LAKLEY ) Secret S = point of PG ( k , q ) . 1 Shares = hyperplanes of PG ( k , q ) such that exactly k of 2 them only intersect in S . Classical example: Normal rational curve of hyperplanes 3 X 0 + tX 1 + t 2 X 2 + · · · + t k X k = 0 , t ∈ F q , and X k = 0 . Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television G EOMETRICAL REALISATION OF S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME (B LAKLEY ) Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television G EOMETRICAL REALISATION OF S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME Leo Storme Galois geometries and cryptography

1. Secret sharing scheme Galois geometries 2. Message Authentication code (MAC) Geometry and cryptography 3. Anonymous database search 4. Application in pay television G EOMETRICAL REALISATION OF S HAMIR ’ S k - OUT - OF - n SECRET SHARING SCHEME Leo Storme Galois geometries and cryptography