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To expand the applicability of these solutions to computationally weaker devices, it has been advocated that a computationally weaker client (i.e., capable of performing a relatively small number of modular multiplications) delegates such primitive operations to a computationally stronger server. Important requirements for such delegation protocols include privacy of the client\u2019s input exponent and security of the client\u2019s output, in the sense of detecting, except for very small probability, any malicious server\u2019s attempt to convince the client of an incorrect exponentiation result. Only recently, efficient protocols for the delegation of a fixed-based exponentiation, over cyclic and RSA-type groups with certain properties, have been presented and proved to satisfy both requirements.<\/jats:p>\n                  <jats:p>\n                    In this paper we show that a product of\n                    <jats:italic>many<\/jats:italic>\n                    fixed-base exponentiations, over a cyclic groups with certain properties, can be privately and securely delegated by keeping the client\u2019s online number of modular multiplications only slightly larger than in the delegation of a\n                    <jats:italic>single<\/jats:italic>\n                    exponentiation. We use this result to show the\n                    <jats:italic>first<\/jats:italic>\n                    delegations of entire cryptographic schemes: the well-known digital signature schemes by El-Gamal, Schnorr and Okamoto, over the\n                    <jats:italic>q<\/jats:italic>\n                    -order subgroup in \u2124\n                    <jats:sub>\n                      <jats:italic>p<\/jats:italic>\n                    <\/jats:sub>\n                    , for\n                    <jats:italic>p<\/jats:italic>\n                    ,\n                    <jats:italic>q<\/jats:italic>\n                    primes, as well as their variants based on elliptic curves. Previous efficient delegation results were limited to the delegation of single algorithms within cryptographic schemes.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2019-0036","type":"journal-article","created":{"date-parts":[[2020,11,3]],"date-time":"2020-11-03T09:52:35Z","timestamp":1604397155000},"page":"438-459","source":"Crossref","is-referenced-by-count":2,"title":["Delegating a Product of Group Exponentiations with Application to Signature Schemes (Submission to Special NutMiC 2019 Issue of JMC)"],"prefix":"10.1515","volume":"14","author":[{"given":"Giovanni Di","family":"Crescenzo","sequence":"first","affiliation":[{"name":"Perspecta Labs Inc . Basking Ridge , NJ , United States of America"}]},{"given":"Matluba","family":"Khodjaeva","sequence":"additional","affiliation":[{"name":"CUNY John Jay College of Criminal Justice . New York , NY , United States of America"}]},{"given":"Delaram","family":"Kahrobaei","sequence":"additional","affiliation":[{"name":"University of York . Heslington , York , United Kingdom"}]},{"given":"Vladimir","family":"Shpilrain","sequence":"additional","affiliation":[{"name":"City University of New York . New , York, NY , United States of America"}]}],"member":"374","published-online":{"date-parts":[[2020,10,30]]},"reference":[{"key":"2025120600172411961_j_jmc-2019-0036_ref_001_w2aab3b7d254b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"A. Arbit, Y. Livne, Y. Oren, and A. Wool, Implementing Public-Key Cryptography on Passive RFID Tags is Practical. In: International Journal of Information Security, 14(1): pp. 85-99, 2015.","DOI":"10.1007\/s10207-014-0236-y"},{"key":"2025120600172411961_j_jmc-2019-0036_ref_002_w2aab3b7d254b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"M. Atallah, K. Pantazopoulos, J. Rice, and E. 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