{"abstracts":[{"sha1":"fac41d375aa675bb704a3e542a5df6ac3091ea24","content":"At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool\ncalled Boomerang Connectivity Table (BCT) for measuring the resistance of a\nblock cipher against the boomerang attack (which is an important cryptanalysis\ntechnique introduced by Wagner in 1999 against block ciphers). Next, Boura and\nCanteaut introduced an important parameter (related to the BCT) for\ncryptographic Sboxes called boomerang uniformity. In this context, we present a\nbrief state-of-the-art on the notion of boomerang uniformity of vectorial\nfunctions (or Sboxes) and provide new results. More specifically, we present a\nslightly different (and more convenient) formulation of the boomerang\nuniformity and show that the row sum and the column sum of the boomerang\nconnectivity table can be expressed in terms of the zeros of the second-order\nderivative of the permutation or its inverse. Most importantly, we specialize\nour study of boomerang uniformity to quadratic permutations in even dimension\nand generalize the previous results on quadratic permutation with optimal BCT\n(optimal means that the maximal value in the Boomerang Connectivity Table\nequals the lowest known differential uniformity). As a consequence of our\ngeneral result, we prove that the boomerang uniformity of the binomial\ndifferentially 4-uniform permutations presented by Bracken, Tan, and Tan\nequals 4. This result gives rise to a new family of optimal Sboxes.","mimetype":"text/plain","lang":"en"},{"sha1":"6f070feb4590632918f19b24f89059c4fe906df8","content":"At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool\ncalled Boomerang Connectivity Table (BCT) for measuring the resistance of a\nblock cipher against the boomerang attack (which is an important cryptanalysis\ntechnique introduced by Wagner in 1999 against block ciphers). Next, Boura and\nCanteaut introduced an important parameter (related to the BCT) for\ncryptographic Sboxes called boomerang uniformity. In this context, we present a\nbrief state-of-the-art on the notion of boomerang uniformity of vectorial\nfunctions (or Sboxes) and provide new results. More specifically, we present a\nslightly different (and more convenient) formulation of the boomerang\nuniformity and show that the row sum and the column sum of the boomerang\nconnectivity table can be expressed in terms of the zeros of the second-order\nderivative of the permutation or its inverse. Most importantly, we specialize\nour study of boomerang uniformity to quadratic permutations in even dimension\nand generalize the previous results on quadratic permutation with optimal BCT\n(optimal means that the maximal value in the Boomerang Connectivity Table\nequals the lowest known differential uniformity). As a consequence of our\ngeneral result, we prove that the boomerang uniformity of the binomial\ndifferentially $4$-uniform permutations presented by Bracken, Tan, and Tan\nequals $4$. This result gives rise to a new family of optimal Sboxes.","mimetype":"application/x-latex","lang":"en"}],"refs":[],"contribs":[{"index":0,"raw_name":"Sihem Mesnager","role":"author"},{"index":1,"raw_name":"Chunming Tang","role":"author"},{"index":2,"raw_name":"Maosheng Xiong","role":"author"}],"license_slug":"ARXIV-1.0","language":"en","version":"v1","ext_ids":{"arxiv":"1903.00501v1"},"release_year":2019,"release_date":"2019-03-01","release_stage":"submitted","release_type":"article","work_id":"ceymrngiubfktc54neckk36roq","title":"On the boomerang uniformity of (quadratic) permutations over F_2^n","state":"active","ident":"ubxro22o5jhj5eekmoym7pxyli","revision":"9ab271fb-cd04-4118-ad6a-abdb211b6d66","extra":{"arxiv":{"base_id":"1903.00501","categories":["cs.CR"]}}}