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Reduce the Size Complexity of the Private KeySpecial Constructions for Hadamard matricesCAIAF2011D. E. SimosPrivate-KeyCryptosystemsMotivationCryptographicAlgorithmsEncryptionSchemesCryptanalysisConclusionHadamard matrices with one Circulant Core1 A Hadamard matrix of order n = p + 1 which can be written as1 1 · · · 11. C1or1. C11 −1 · · · − 1where C = (c ij ) is a circulant matrix of order p, is said to haveone circulant core2 Existence: Infinite families i.e. Paley, (1933) Stanton, Sprottand Whiteman, (1958, 1962) Marshall Hall Jr., (1956)12 / 26Hadamard Core CipherKey k: The binary vector A c = [a 1 , a 2 , . . . , a p ] which denotesthe first row of the circulant matrix C and consist of of p bitsSize of the key: O(n), since it consists of p = n − 1 bitsEncryption-Decryption: as before using the Hadamard matrixn = p + 1 as an encryption matrix
Add Randomness in the Encryption ProcessSchemes from Plotkin arraysCAIAF2011D. E. SimosPrivate-KeyCryptosystemsMotivationCryptographicAlgorithmsEncryptionSchemesCryptanalysisConclusionPlotkin Cipher (Koukouvinos and Simos, AMIS, 2011)Encryption: Divide a message m of arbitrary length into blocksm 1 , . . . , m q of length 4 (padding the last block with zeros ifnecessary)Randomness: Random vectors g 1 , . . . , g q of length 4 areconstructed using pseudorandom generatorsEncryption matrix: The Plotkin array of order 8 and type(1, 1, 1, 1, 1, 1, 1, 1), denoted by P , whereP P T = (A 2 + B 2 + . . . + H 2 )I 8Encryption process: The matrix P is applied successively tom i ⊕ g iCiphertext: c = P (m 1 ⊕ g 1 ) ⊕ . . . ⊕ P (m q ⊕ g q )Decryption: Divide c into blocks c 1 , . . . , c q of size 8 andcompute P T c i /fKey k: The chosen entries A, B, . . . , H of P ; (integer numbers)13 / 26
Page 1 and 2: CAIAF2011D. E. SimosPrivate-KeyCryp
Page 3 and 4: Outline of the TalkCAIAF2011D. E. S
Page 5 and 6: Why Interested in Combinatorial Des
Page 7 and 8: Private-Key Ciphersbased on Combina
Page 9 and 10: Hadamard MatricesCAIAF2011D. E. Sim
Page 11 and 12: Plotkin arraysCAIAF2011D. E. SimosP
Page 13 and 14: Decryption AlgorithmCAIAF2011D. E.
Page 15 and 16: Encryption SchemeCAIAF2011D. E. Sim
Page 17: Ciphers from Hadamard matricesCAIAF
Page 21 and 22: Electronic Codebook Mode (ECB)CAIAF
Page 24 and 25: Cryptanalysis of Brute-Force Attack
Page 26: Cryptanalysis of Chosen-Plaintext A
Page 29 and 30: Cryptanalysis of Ciphertext-only At
Page 31 and 32: Security Comparison for Block Ciphe
Page 33 and 34: SummaryCAIAF2011D. E. SimosPrivate-
Page 35: Questions - CommentsCAIAF2011D. E.

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