Finite Fields, Applications and Open Problems Daniel Panario School - PowerPoint PPT Presentation

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Finite Fields, Applications and Open Problems Daniel Panario School of Mathematics and Statistics Carleton University daniel@math.carleton.ca LAWCI School, Campinas, July 2018 Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Summary Lecture 1: Applications in Combinatorics Brief review of finite fields. Introduction to combinatorics objects (designs, latin squares, several types of arrays). Classical results (latin squares and sudokus; Costas arrays). Orthogonal arrays and their constructions based on finite fields. Some applications in cryptography/coding theory (brief): secret sharing and combinatorial designs; orthogonal arrays and codes. Orthogonal array variants (covering arrays, ordered orthogonal arrays) and their constructions based on finite fields. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Summary (cont.) Lecture 2: Applications in cryptography Applications of finite fields (brief). Differential map, differential uniformity, and differential cryptanalysis. Example of S -box function and its characteristics. Perfect nonlinear (PN) and almost perfect nonlinear (APN) functions. Permutation polynomials and their cycle decomposition. Iterations of functions. Generating pseudorandom sequences: how random is a sequence, requirements for sequences in cryptography. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Applications in cryptography Cryptosystems: Diffie-Hellman method to share a key; ElGamal digital signature method; RSA (permutation polynomials over finite fields); Elliptic and hyperelliptic curve cryptosystem; Chor-Rivest cryptosystem; Powerline cryptosystem; Goppa-code cryptosystem; Shamir’s secret sharing; etc. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Applications in cryptography (cont.) Security: discrete logarithm problem; index calculus method and its variants (Waterloo, Coppersmith); linear and differential cryptanalysis (PN and APN functions). Stream ciphers: WG (Welch-Gong); RC4; etc. Block ciphers: AES (advanced encryption standard): Rijndael; SAFER (Secure And Fast Encryption Routine); RC6 (permutation polynomials over integer rings). Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Applications in coding theory Classical applications: BCH codes; Reed-Solomon codes; burst error-correcting codes; convolution codes; codes based on algebraic curves; etc. Recent applications: LDPC (low density parity check) codes; turbo codes. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Applications in engineering LFSR (feedback shift register sequences); pseudorandom number generators (LFSR, polynomials); radar and sonar (sequences over finite fields, Costas arrays); digital signal processing: transforms (discrete Fourier, Hadamard, trigonometric); ad-hoc (like concert hall acoustics); etc. For more information on LFSR and sequences, see Golomb and Gong (2005) book. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Applications in mathematics Finite geometries: affine and projective geometries; constructions of projective planes with a finite number of points and lines. Combinatorial designs: BIBD (balance incomplete block designs), latin squares and MOLS (mutually orthogonal latin squares), orthogonal and covering arrays, etc. There are also recent applications to bioinformatics (dynamical systems over finite fields). For more information see (shameless advertisement coming): Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Applications in mathematics Finite geometries: affine and projective geometries; constructions of projective planes with a finite number of points and lines. Combinatorial designs: BIBD (balance incomplete block designs), latin squares and MOLS (mutually orthogonal latin squares), orthogonal and covering arrays, etc. There are also recent applications to bioinformatics (dynamical systems over finite fields). For more information see (shameless advertisement coming): Handbook of Finite Fields by Gary Mullen and Daniel Panario published by CRC in 2013. Finite Fields, Applications and Open Problems Daniel Panario

Mullen • Panario Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences DISCRETE MATHEMATICS AND ITS APPLICATIONS DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN Series Editor KENNETH H. ROSEN HANDBOOK OF FINITE FIELDS HANDBOOK OF copy to come FINITE FIELDS Gary L. Mullen Daniel Panario K13417 K13417_Draft.indd 1 9/20/12 9:20 AM Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Differential Map Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Differential map and uniformity Brief recall of substitution-permutation networks and differential cryptanalysis Cipher AES (Advanced Encryption Standard) APN (Almost Perfect Nonlinear) functions Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences SPN (Substitution Permutation Networks) A substitution-permutation network consists of R rounds and the secret key is broken into R + 1 subkeys. At each round, the data stream is mixed with a subkey and fed into a series of substitution boxes (S-boxes), then the resulting output bits are mixed by a permutation box (P-box). S-boxes are functions which act on a subset of the input bits into a round; their primary purpose is to increase the confusion of the cipher. P-boxes act as a shuffling of the bits between rounds; their purpose is to diffuse characteristics of the data stream. The output of the final round’s S-boxes is mixed with a final round key to create the ciphertext. A diagram of a basic 16 -bit, 4 -round SPN is given next. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences An S-box is a look-up table which substitutes small blocks of bits for another block of bits. In most cases (but not in all cases, e.g. DES), we consider S-boxes as maps from F n 2 → F n 2 . Since permutations and adding round keys are all linear relations between bits, S-boxes are the only possibly non-linear component of the network. This non-linearity is crucial to the security of the cipher. Key-mixing is done by the XOR operation of the key bits with the input bits of the round. The XOR operation is self-inverse. Each S-box is a one-to-one function, and so can be inverted, and each P-box is a permutation, so decryption involves applying the inverse permutation. Since each component of the network is invertible, decryption is performed by running the ciphertext backwards through the cipher. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Differential cryptanalisis Differential cryptanalysis was introduced by Biham and Shamir in 1991, as an attack against DES. It has been used to reduce the number of DES keys to be tested from 2 55 (brute-force) to 2 47 . Though less successful than linear cryptanalysis for DES, differential cryptanalysis scales very well to other ciphers. Differential cryptanalysis is a chosen plaintext attack, where an attacker has access to the keyed cipher and is able to encrypt any plaintext. The main goal of differential cryptanalysis is to exploit highly probabilistic relationships between differences of plaintexts with the difference of inputs into the last round’s cipher. As in linear cryptanalysis, differential cryptanalysis can be used to recover bits of the final round’s key. Finite Fields, Applications and Open Problems Daniel Panario

Differential map PN and APN functions Permutation polynomials Iteration of Functions Random sequences Ciphers Finite Fields, Applications and Open Problems Daniel Panario