{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T05:40:13Z","timestamp":1777009213300,"version":"3.51.4"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T00:00:00Z","timestamp":1764115200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T00:00:00Z","timestamp":1764115200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"University of Bergen"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    APN functions offer optimal resistance to differential attacks and are instrumental in the design of block ciphers in cryptography. While finding APN functions is very difficult in general, a promising way to construct APN functions is through symmetric matrices called Quadratic APN matrices (QAM). It is known that the search space for the QAM method can be reduced by means of orbit partitions induced by linear equivalences. This paper builds upon and improves these approaches in the case of homogeneous quadratic functions over\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathbb {F}_{2^n}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    with coefficients in the subfield\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathbb {F}_{2^m}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . We propose an innovative approach for computing orbit partitions for cases where it is infeasible due to the large search space, resulting in the applications for the dimensions\n                    <jats:inline-formula>\n                      <jats:tex-math>$$(n,m)=(8,4)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$(n,m)=(9,3)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . We find and classify, up to CCZ-equivalence, all quadratic APN functions for the cases of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$(n,m)=(8,2)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$(n,m)=(10,1)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Also, we show that exhaustive searches for\n                    <jats:inline-formula>\n                      <jats:tex-math>$$(n,m) = (10,2)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$(n,m)=(8,4)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    are infeasible for the QAM method using currently available means, following partial searches for these cases.\n                  <\/jats:p>","DOI":"10.1007\/s12095-025-00851-1","type":"journal-article","created":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T08:19:30Z","timestamp":1764145170000},"page":"727-762","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Pushing the QAM method for finding APN functions further"],"prefix":"10.1007","volume":"18","author":[{"given":"Nadiia","family":"Ichanska","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simon","family":"Berg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikolay","family":"Kaleyski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuyin","family":"Yu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,11,26]]},"reference":[{"key":"851_CR1","unstructured":"Beierle, C., Langevin, P., Leander, G., Polujan, A. and Rasoolzadeh, S.: Millions of inequivalent quadratic apn functions in eight variables, (2025)"},{"key":"851_CR2","unstructured":"Berg, S.: The list of all found $$27$$ inequivalent classes of quadratic functions over $$\\mathbb{F}_{2^8}$$ with coefficients in $$\\mathbb{F}_{4}$$. https:\/\/github.com\/Simon-Berg\/thesis, (2023)"},{"key":"851_CR3","doi-asserted-by":"crossref","unstructured":"Bosma, W., Cannon, J. and Playoust, C.: The Magma algebra system. The user language, (1997)","DOI":"10.1006\/jsco.1996.0125"},{"issue":"2","key":"851_CR4","doi-asserted-by":"publisher","first-page":"150","DOI":"10.1016\/j.ffa.2008.10.001","volume":"15","author":"L Budaghyan","year":"2009","unstructured":"Budaghyan, L., Carlet, C., Leander, G.: Constructing new APN functions from known ones. Finite Fields and Their Appl. 15(2), 150\u2013159 (2009)","journal-title":"Finite Fields and Their Appl."},{"issue":"11","key":"851_CR5","doi-asserted-by":"publisher","first-page":"7081","DOI":"10.1109\/TIT.2020.3007513","volume":"66","author":"L Budaghyan","year":"2020","unstructured":"Budaghyan, L., Helleseth, T., Kaleyski, N.: A new family of APN quadrinomials. IEEE Trans. Inf. Theory 66(11), 7081\u20137087 (2020)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"851_CR6","unstructured":"Canteaut, A., Couvreur, A., Perrin, L: Recovering or testing extended-affine equivalence. Cryptology ePrint Archive, Paper 2021\/225, (2021)"},{"key":"851_CR7","doi-asserted-by":"crossref","unstructured":"Carlet, C.: Boolean Functions for Cryptography and Coding Theory. Cambridge University Press, (2021)","DOI":"10.1017\/9781108606806"},{"key":"851_CR8","unstructured":"Davidova, D., Kaleyski, N.: Classification of all DO planar polynomials with prime field coefficients over $$GF(3^n)$$ for $$n$$ up to $$7$$. Cryptology ePrint Archive, Paper 2022\/1059, (2022). https:\/\/eprint.iacr.org\/2022\/1059"},{"key":"851_CR9","unstructured":"Edel, Y., Pott, A.: On the equivalence of nonlinear functions. In: Enhancing cryptographic primitives with techniques from error correcting codes, pp. 87\u2013103. IOS Press, (2009)"},{"key":"851_CR10","unstructured":"Weng, G., Tan, Y., Gong, G.: On quadratic almost perfect nonlinear functions and their related algebraic object. Technical Report CACR 2013-18, University of Waterloo, Centre for Applied Cryptographic Research, (2013). Available at: https:\/\/cacr.uwaterloo.ca\/techreports\/2013\/cacr2013-18.pdf"},{"issue":"3","key":"851_CR11","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1007\/s10801-011-0309-1","volume":"35","author":"S Yoshiara","year":"2012","unstructured":"Yoshiara, S.: Equivalences of quadratic APN functions. J. Algebraic Comb. 35(3), 461\u2013475 (2012)","journal-title":"J. Algebraic Comb."},{"key":"851_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2020.101733","volume":"68","author":"Y Yu","year":"2020","unstructured":"Yu, Y., Kaleyski, N., Budaghyan, L., Li, Y.: Classification of quadratic APN functions with coefficients in $$\\mathbb{F}_2$$ for dimensions up to 9. Finite Fields and Their Appl. 68, 101733 (2020)","journal-title":"Finite Fields and Their Appl."},{"key":"851_CR13","doi-asserted-by":"crossref","unstructured":"Yu, Y., Li, J., Ichanska, N., Kaleyski, N.: Construction of quadratic APN functions with coefficients in $$\\mathbb{F}_2$$ in dimensions $$10$$ and $$11$$. Cryptology ePrint Archive, Paper 2024\/1778, (2024)","DOI":"10.1007\/s00200-025-00711-8"},{"key":"851_CR14","unstructured":"Yu, Y., Perrin, L.: Constructing more quadratic APN functions with the QAM method. Cryptology ePrint Archive, Paper 2021\/574, (2021)"},{"key":"851_CR15","unstructured":"Yu, Y., Wang, M., Li, Y.: A matrix approach for constructing quadratic APN functions. Cryptology ePrint Archive, Paper 2013\/007, (2013)"},{"issue":"2","key":"851_CR16","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1007\/s10623-014-9955-3","volume":"73","author":"Y Yu","year":"2014","unstructured":"Yu, Y., Wang, M., Li, Y.: A matrix approach for constructing quadratic APN functions. Des. Codes Crypt. 73(2), 587\u2013600 (2014)","journal-title":"Des. Codes Crypt."}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-025-00851-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-025-00851-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-025-00851-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T04:50:11Z","timestamp":1777006211000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-025-00851-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,26]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["851"],"URL":"https:\/\/doi.org\/10.1007\/s12095-025-00851-1","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,26]]},"assertion":[{"value":"8 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 November 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}