{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:20:53Z","timestamp":1776766853676,"version":"3.51.2"},"reference-count":36,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Experimental results on the multiplicative orders of Gauss periods in finite fields are presented. These results indicate that Gauss periods have high order and are often primitive (self-dual) normal elements in finite fields. It is shown that Gauss periods can be exponentiated in quadratic time. An application is an efficient pseudorandom bit generator.<\/p>","DOI":"10.1090\/s0025-5718-98-00935-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"343-352","source":"Crossref","is-referenced-by-count":30,"title":["Gauss periods: orders and cryptographical applications"],"prefix":"10.1090","volume":"67","author":[{"given":"Shuhong","family":"Gao","sequence":"first","affiliation":[]},{"given":"Joachim","family":"von zur Gathen","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Panario","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"L.M. Adleman and H.W. Lenstra, Jr., Finding irreducible polynomials over finite fields,  Proc. 18th Annual ACM Symp. on Theory of Computing (1986), 350\u2013355.","DOI":"10.1145\/12130.12166"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/BF00196789","article-title":"An implementation for a fast public-key cryptosystem","volume":"3","author":"Agnew, G. B.","year":"1991","journal-title":"J. Cryptology","ISSN":"https:\/\/id.crossref.org\/issn\/0933-2790","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"G.B. Agnew, R.C. Mullin and S.A. Vanstone, An implementation of elliptic curve cryptosystems over \ud835\udc39_{2\u00b9\u2075\u2075},  IEEE J. on Selected Areas in Communications, 11 (1993), 804\u2013813.","DOI":"10.1109\/49.223883"},{"key":"4","doi-asserted-by":"crossref","unstructured":"T. Beth, Efficient zero-knowledge identification scheme for smart cards,  Advances in Cryptology (Proc. Eurocrypt \u201888), Springer LNCS 330 (1988), 77\u201384.","DOI":"10.1007\/3-540-45961-8_7"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/3-540-57936-2_27","article-title":"Constructive problems for irreducible polynomials over finite fields","author":"Blake, Ian F.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540579362"},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"850","DOI":"10.1137\/0213053","article-title":"How to generate cryptographically strong sequences of pseudorandom bits","volume":"13","author":"Blum, Manuel","year":"1984","journal-title":"SIAM J. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-5397","issn-type":"print"},{"key":"7","series-title":"Contemporary Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/022","volume-title":"Factorizations of $b^n \\pm1$","volume":"22","author":"Brillhart, John","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0821850784","edition":"2"},{"issue":"6","key":"8","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1109\/tit.1976.1055638","article-title":"New directions in cryptography","volume":"IT-22","author":"Diffie, Whitfield","year":"1976","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1109\/TIT.1985.1057074","article-title":"A public key cryptosystem and a signature scheme based on discrete logarithms","volume":"31","author":"ElGamal, Taher","year":"1985","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"key":"10","doi-asserted-by":"crossref","unstructured":"S. Gao, J. von zur Gathen and D. Panario, Gauss periods, primitive normal bases, and fast exponentiation in finite fields, preliminary version in  Proc. Latin\u201995, Valpara\u00edso, Chile, Springer LNCS 911 (1995), 311\u2013322; full version in Technical Report 296\/95, Department of Computer Science, University of Toronto, 1995.","DOI":"10.1007\/3-540-59175-3_98"},{"key":"11","doi-asserted-by":"crossref","unstructured":"S. Gao and D. Panario, Tests and constructions of irreducible polynomials over finite fields, in Foundations of Computational Mathematics, F. Cucker and M. Shub (Eds.), Springer Verlag, 1997, 346\u2013361.","DOI":"10.1007\/978-3-642-60539-0_27"},{"issue":"211","key":"12","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.2307\/2153492","article-title":"On orders of optimal normal basis generators","volume":"64","author":"Gao, Shuhong","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"13","unstructured":"J. von zur Gathen and F. Pappalardi, Density estimates related to Gauss periods, 1996, preprint."},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1007\/BFb0015425","article-title":"Orders of Gauss periods in finite fields","author":"von zur Gathen, Joachim","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/3540605738"},{"key":"15","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4939-7560-0","volume-title":"Disquisitiones arithmeticae","author":"Gauss, Carl Friedrich","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0387962549"},{"issue":"200","key":"16","doi-asserted-by":"publisher","first-page":"639","DOI":"10.2307\/2153081","article-title":"Primitive polynomials over finite fields","volume":"59","author":"Hansen, Tom","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"17","isbn-type":"print","volume-title":"Finite fields","author":"Jungnickel, Dieter","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3411161116"},{"key":"18","doi-asserted-by":"crossref","unstructured":"L. Lamport, Password authentication with insecure communication,  Comm. ACM, 24 (1981), 770\u2013772.","DOI":"10.1145\/358790.358797"},{"key":"19","doi-asserted-by":"crossref","unstructured":"C.H. Lim and P.J. Lee, Another method for attaining security against adaptively chosen ciphertext attacks,  Advances in Cryptology (Proc. Crypto \u201993), Springer LNCS 773 (1994), 420\u2013434.","DOI":"10.1007\/3-540-48329-2_36"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1137\/0217021","article-title":"The discrete logarithm hides \ud835\udc42(\ud835\udc59\ud835\udc5c\ud835\udc54\ud835\udc5b) bits","volume":"17","author":"Long, Douglas L.","year":"1988","journal-title":"SIAM J. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-5397","issn-type":"print"},{"key":"21","unstructured":"A.J. Menezes, I.F. Blake, X.H. Gao, R.C. Mullin, S.A. Vanstone, and T. Yaghoobian, Applications of Finite Fields,  Kluwer Academic Publishers, 1993."},{"issue":"208","key":"22","doi-asserted-by":"publisher","first-page":"759","DOI":"10.2307\/2153296","article-title":"Primitive normal polynomials over finite fields","volume":"63","author":"Morgan, Ilene H.","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"23","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/0166-218X(88)90090-X","article-title":"Optimal normal bases in \ud835\udc3a\ud835\udc39(\ud835\udc5d\u207f)","volume":"22","author":"Mullin, R. C.","year":"1988","journal-title":"Discrete Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0166-218X","issn-type":"print"},{"key":"24","unstructured":"National Institute for Standard and Technology,  Federal Information Processing Standard for Digital Signature Standard (DSS), FIPS 186, May 1994."},{"key":"25","doi-asserted-by":"crossref","unstructured":"R. Peralta, Simultaneous security of bits in the discrete log,  Advances in Cryptology (Proc. Eurocrypt \u201885), Springer LNCS 219 (1986), 62\u201372.","DOI":"10.1007\/3-540-39805-8_8"},{"key":"26","isbn-type":"print","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/0-387-34805-0_22","article-title":"Efficient identification and signatures for smart cards","author":"Schnorr, C.-P.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0387973176"},{"key":"27","doi-asserted-by":"crossref","unstructured":"C.P. Schnorr, Efficient signature generation by smart cards,  J. of Cryptology, 4 (1991), 161\u2013174.","DOI":"10.1007\/BF00196725"},{"key":"28","unstructured":"V. Shoup, Exponentiation in \ud835\udc3a\ud835\udc39(2\u207f) using fewer polynomial multiplications, preprint, 1994."},{"issue":"31","key":"29","first-page":"155","article-title":"Konstruktion von Normalbasen","author":"Wassermann, Alfred","year":"1990","journal-title":"Bayreuth. Math. Schr.","ISSN":"https:\/\/id.crossref.org\/issn\/0172-1062","issn-type":"print"},{"issue":"44","key":"30","first-page":"147","article-title":"Zur Arithmetik in endlichen K\u00f6rpern","author":"Wassermann, Alfred","year":"1993","journal-title":"Bayreuth. Math. Schr.","ISSN":"https:\/\/id.crossref.org\/issn\/0172-1062","issn-type":"print"},{"key":"31","doi-asserted-by":"crossref","unstructured":"Y. Zheng and J. Seberry, Practical approaches for attaining security against adaptively chosen ciphertext attacks,  Advances in Cryptology (Proc. Crypto \u201992), Springer LNCS 740 (1993), 292\u2013304.","DOI":"10.1007\/3-540-48071-4_20"},{"key":"32","doi-asserted-by":"crossref","unstructured":"Y. Zheng and J. Seberry, Immunizing public key cryptosystems against chosen ciphertext attacks,  IEEE J. on Selected Areas in Communications, 11 (1993), 715\u2013724.","DOI":"10.1109\/49.223871"},{"key":"33","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/S0019-9958(68)90973-X","article-title":"On primitive trinomials (\ud835\udc5a\ud835\udc5c\ud835\udc512)","volume":"13","author":"Zierler, Neal","year":"1968","journal-title":"Information and Control","ISSN":"https:\/\/id.crossref.org\/issn\/0019-9958","issn-type":"print"},{"key":"34","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1016\/S0019-9958(69)90356-8","article-title":"On primitive trinomials (\ud835\udc5a\ud835\udc5c\ud835\udc512). II","volume":"14","author":"Zierler, Neal","year":"1969","journal-title":"Information and Control","ISSN":"https:\/\/id.crossref.org\/issn\/0019-9958","issn-type":"print"},{"issue":"205","key":"35","doi-asserted-by":"publisher","first-page":"385","DOI":"10.2307\/2153416","article-title":"A table of primitive binary polynomials","volume":"62","author":"\u017divkovi\u0107, Miodrag","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"207","key":"36","doi-asserted-by":"publisher","first-page":"301","DOI":"10.2307\/2153576","article-title":"Table of primitive binary polynomials. II","volume":"63","author":"\u017divkovi\u0107, Miodrag","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00935-1\/S0025-5718-98-00935-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00935-1\/S0025-5718-98-00935-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:41:17Z","timestamp":1776721277000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00935-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":36,"journal-issue":{"issue":"221","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["S0025-5718-98-00935-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00935-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1998]]}}}