{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T04:20:05Z","timestamp":1775794805321,"version":"3.50.1"},"publisher-location":"Berlin, Heidelberg","reference-count":28,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783662498958","type":"print"},{"value":"9783662498965","type":"electronic"}],"license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016]]},"DOI":"10.1007\/978-3-662-49896-5_22","type":"book-chapter","created":{"date-parts":[[2016,4,27]],"date-time":"2016-04-27T00:45:32Z","timestamp":1461717932000},"page":"616-648","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":56,"title":["Randomness Complexity of Private Circuits for Multiplication"],"prefix":"10.1007","author":[{"given":"Sonia","family":"Bela\u00efd","sequence":"first","affiliation":[]},{"given":"Fabrice","family":"Benhamouda","sequence":"additional","affiliation":[]},{"given":"Alain","family":"Passel\u00e8gue","sequence":"additional","affiliation":[]},{"given":"Emmanuel","family":"Prouff","sequence":"additional","affiliation":[]},{"given":"Adrian","family":"Thillard","sequence":"additional","affiliation":[]},{"given":"Damien","family":"Vergnaud","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,4,28]]},"reference":[{"key":"22_CR1","unstructured":"https:\/\/github.com\/fabrice102\/private_multiplication"},{"key":"22_CR2","doi-asserted-by":"crossref","unstructured":"Barker, E.B., Kelsey, J.M.: Sp 800\u201390a. recommendation for random number generation using deterministic random bit generators. Technical report, Gaithersburg, MD, USA (2012)","DOI":"10.6028\/NIST.SP.800-90a"},{"key":"22_CR3","unstructured":"Barthe, G., Bela\u00efd, S., Dupressoir, F., Fouque, P.A., Gr\u00e9goire, B.: Compositional verification of higher-order masking: Application to a verifying masking compiler. Cryptology ePrint Archive, Report 2015\/506 (2015). \n                    http:\/\/eprint.iacr.org\/2015\/506"},{"key":"22_CR4","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/978-3-662-46800-5_18","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2015","author":"G Barthe","year":"2015","unstructured":"Barthe, G., Bela\u00efd, S., Dupressoir, F., Fouque, P.-A., Gr\u00e9goire, B., Strub, P.-Y.: Verified proofs of higher-order masking. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 457\u2013485. Springer, Heidelberg (2015)"},{"key":"22_CR5","doi-asserted-by":"crossref","unstructured":"Ben-Or, M., Goldwasser, S., Kilian, J., Wigderson, A.: Multi-prover interactive proofs: How to remove intractability assumptions. In: 20th ACM STOC, pp. 113\u2013131. ACM Press, May 1988","DOI":"10.1145\/62212.62223"},{"key":"22_CR6","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1007\/978-3-662-45608-8_18","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2014","author":"B Bilgin","year":"2014","unstructured":"Bilgin, B., Gierlichs, B., Nikova, S., Nikov, V., Rijmen, V.: Higher-order threshold implementations. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part II. LNCS, vol. 8874, pp. 326\u2013343. Springer, Heidelberg (2014)"},{"key":"22_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/978-3-319-06734-6_17","volume-title":"Progress in Cryptology \u2013 AFRICACRYPT 2014","author":"B Bilgin","year":"2014","unstructured":"Bilgin, B., Gierlichs, B., Nikova, S., Nikov, V., Rijmen, V.: A more efficient AES threshold implementation. In: Pointcheval, D., Vergnaud, D. (eds.) AFRICACRYPT 2014. LNCS, vol. 8469, pp. 267\u2013284. Springer, Heidelberg (2014)"},{"key":"22_CR8","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1007\/3-540-48405-1_26","volume-title":"Advances in Cryptology - CRYPTO 1999","author":"S Chari","year":"1999","unstructured":"Chari, S., Jutla, C.S., Rao, J.R., Rohatgi, P.: Towards sound approaches to counteract power-analysis attacks. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 398\u2013412. Springer, Heidelberg (1999)"},{"key":"22_CR9","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1007\/978-3-662-43933-3_21","volume-title":"Fast Software Encryption","author":"J-S Coron","year":"2014","unstructured":"Coron, J.-S., Prouff, E., Rivain, M., Roche, T.: Higher-order side channel security and mask refreshing. In: Moriai, S. (ed.) FSE 2013. LNCS, vol. 8424, pp. 410\u2013424. Springer, Heidelberg (2014)"},{"key":"22_CR10","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1007\/978-3-642-55220-5_24","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2014","author":"A Duc","year":"2014","unstructured":"Duc, A., Dziembowski, S., Faust, S.: Unifying leakage models: from probing attacks to noisy leakage. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 423\u2013440. Springer, Heidelberg (2014)"},{"key":"22_CR11","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/978-3-662-46800-5_16","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2015","author":"A Duc","year":"2015","unstructured":"Duc, A., Faust, S., Standaert, F.-X.: Making masking security proofs concrete. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 401\u2013429. Springer, Heidelberg (2015)"},{"key":"22_CR12","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/978-3-662-46803-6_6","volume-title":"Advances in Cryptology - EUROCRYPT 2015","author":"S Dziembowski","year":"2015","unstructured":"Dziembowski, S., Faust, S., Skorski, M.: Noisy leakage revisited. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part II. LNCS, vol. 9057, pp. 159\u2013188. Springer, Heidelberg (2015)"},{"key":"22_CR13","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1007\/3-540-48059-5_15","volume-title":"Cryptographic Hardware and Embedded Systems","author":"L Goubin","year":"1999","unstructured":"Goubin, L., Patarin, J.: DES and differential power analysis the \u201cduplication\u201d method. In: Ko\u00e7, \u00c7.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 158\u2013172. Springer, Heidelberg (1999)"},{"key":"22_CR14","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1007\/978-3-642-39206-1_49","volume-title":"Automata, Languages, and Programming","author":"Y Ishai","year":"2013","unstructured":"Ishai, Y., Kushilevitz, E., Li, X., Ostrovsky, R., Prabhakaran, M., Sahai, A., Zuckerman, D.: Robust pseudorandom generators. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 576\u2013588. Springer, Heidelberg (2013)"},{"key":"22_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1007\/978-3-540-45146-4_27","volume-title":"Advances in Cryptology - CRYPTO 2003","author":"Y Ishai","year":"2003","unstructured":"Ishai, Y., Sahai, A., Wagner, D.: Private circuits: securing hardware against probing attacks. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 463\u2013481. Springer, Heidelberg (2003)"},{"key":"22_CR16","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1007\/3-540-68697-5_9","volume-title":"Advances in Cryptology - CRYPTO \u201996","author":"PC Kocher","year":"1996","unstructured":"Kocher, P.C.: Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104\u2013113. Springer, Heidelberg (1996)"},{"issue":"3","key":"22_CR17","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1145\/357172.357176","volume":"4","author":"L Lamport","year":"1982","unstructured":"Lamport, L., Shostak, R.E., Pease, M.C.: The byzantine generals problem. ACM Trans. Program. Lang. Syst. 4(3), 382\u2013401 (1982)","journal-title":"ACM Trans. Program. Lang. Syst."},{"key":"22_CR18","series-title":"Lecture Notes in Computer Science","first-page":"275","volume-title":"Advances in Cryptology - EUROCRYPT 1988","author":"PJ Lee","year":"1988","unstructured":"Lee, P.J., Brickell, E.F.: An observation on the security of McEliece\u2019s public-key cryptosystem. In: G\u00fcnther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 275\u2013280. Springer, Heidelberg (1988)"},{"issue":"5","key":"22_CR19","doi-asserted-by":"publisher","first-page":"1354","DOI":"10.1109\/18.21270","volume":"34","author":"JS Leon","year":"1988","unstructured":"Leon, J.S.: A probabilistic algorithm for computing minimum weights of large error-correcting codes. IEEE Trans. Inf. Theor. 34(5), 1354\u20131359 (1988)","journal-title":"IEEE Trans. Inf. Theor."},{"issue":"44","key":"22_CR20","first-page":"114","volume":"42","author":"RJ McEliece","year":"1978","unstructured":"McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Prog. Rep. 42(44), 114\u2013116 (1978)","journal-title":"DSN Prog. Rep."},{"issue":"2","key":"22_CR21","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1007\/s00145-010-9085-7","volume":"24","author":"S Nikova","year":"2011","unstructured":"Nikova, S., Rijmen, V., Schl\u00e4ffer, M.: Secure hardware implementation of nonlinear functions in the presence of glitches. J. Cryptology 24(2), 292\u2013321 (2011)","journal-title":"J. Cryptology"},{"issue":"5","key":"22_CR22","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1109\/TIT.1962.1057777","volume":"8","author":"E Prange","year":"1962","unstructured":"Prange, E.: The use of information sets in decoding cyclic codes. IRE Trans. Inf. Theor. 8(5), 5\u20139 (1962)","journal-title":"IRE Trans. Inf. Theor."},{"key":"22_CR23","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1007\/978-3-642-38348-9_9","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2013","author":"E Prouff","year":"2013","unstructured":"Prouff, E., Rivain, M.: Masking against side-channel attacks: a formal security proof. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 142\u2013159. Springer, Heidelberg (2013)"},{"key":"22_CR24","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"764","DOI":"10.1007\/978-3-662-47989-6_37","volume-title":"Advances in Cryptology \u2013 CRYPTO 2015","author":"O Reparaz","year":"2015","unstructured":"Reparaz, O., Bilgin, B., Nikova, S., Gierlichs, B., Verbauwhede, I.: Consolidating masking schemes. In: Gennaro, R., Robshaw, M.J.B. (eds.) CRYPTO 2015, Part I. LNCS, vol. 9215, pp. 764\u2013783. Springer, Heidelberg (2015)"},{"key":"22_CR25","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1007\/978-3-642-15031-9_28","volume-title":"Cryptographic Hardware and Embedded Systems, CHES 2010","author":"M Rivain","year":"2010","unstructured":"Rivain, M., Prouff, E.: Provably secure higher-order masking of AES. In: Mangard, S., Standaert, F.-X. (eds.) CHES 2010. LNCS, vol. 6225, pp. 413\u2013427. Springer, Heidelberg (2010)"},{"key":"22_CR26","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1007\/BFb0019850","volume-title":"Coding Theory and Applications","author":"J Stern","year":"1988","unstructured":"Stern, J.: A method for finding codewords of small weight. In: Cohen, G.D., Wolfmann, J. (eds.) Coding Theory and Applications. LNCS, vol. 388, pp. 106\u2013113. Springer, Heidelberg (1988)"},{"key":"22_CR27","unstructured":"The Sage Developers: Sage Mathematics Software (Version 6.8) (2015). \n                    http:\/\/www.sagemath.org"},{"key":"22_CR28","doi-asserted-by":"crossref","unstructured":"Yao, A.C.C.: Protocols for secure computations (extended abstract). In: 23rd FOCS, pp. 160\u2013164. IEEE Computer Society Press, November 1982","DOI":"10.1109\/SFCS.1982.38"}],"container-title":["Lecture Notes in Computer Science","Advances in Cryptology \u2013 EUROCRYPT 2016"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-662-49896-5_22","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,27]],"date-time":"2020-04-27T20:05:23Z","timestamp":1588017923000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-662-49896-5_22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"ISBN":["9783662498958","9783662498965"],"references-count":28,"URL":"https:\/\/doi.org\/10.1007\/978-3-662-49896-5_22","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016]]},"assertion":[{"value":"28 April 2016","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}