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However, the high computational cost of bootstrapping remains a major bottleneck, especially in the multi-key scenario where blind rotation is the dominant overhead. To address this, we propose a novel and parallel-friendly blind rotation scheme based on the NTRU assumption for efficient MKFHE bootstrapping. Our core technical contribution is a grouped inner product algorithm optimized for automorphism-based blind rotation, which reorganizes hybrid product storage and extends the external product to be compatible with both NTRU and MK-RLWE ciphertexts. Our parallelized algorithm reduces the time complexity from\n                    <jats:italic>O<\/jats:italic>\n                    (\n                    <jats:italic>n<\/jats:italic>\n                    ) to\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$O(\\sqrt{n})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msqrt>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Our scheme demonstrates significant improvements over prior MKFHE works in both computational efficiency and storage requirements. At a 100-bit security level with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$k=8$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>8<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    participants, our scheme achieves a ciphertext bootstrapping time of 0.048 seconds, representing a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$6.8 \\times$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>6.8<\/mml:mn>\n                            <mml:mo>\u00d7<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    speedup compared to Kwak et al.\u2019s state-of-the-art work. Furthermore, our scheme substantially reduces storage overhead, requiring only 81.5MB for evaluation keys (\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$1.7 \\times$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>1.7<\/mml:mn>\n                            <mml:mo>\u00d7<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    smaller) and 64KB for re-linearization keys (\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$6.0 \\times$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>6.0<\/mml:mn>\n                            <mml:mo>\u00d7<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    smaller) relative to Kwak et al.\u2019s implementation.\n                  <\/jats:p>","DOI":"10.1186\/s42400-026-00563-w","type":"journal-article","created":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T07:19:20Z","timestamp":1773386360000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Accelerating MKFHE bootstrapping via parallel-friendly NTRU-based blind rotation"],"prefix":"10.1186","volume":"9","author":[{"given":"Yiran","family":"Dai","sequence":"first","affiliation":[]},{"given":"Binwu","family":"Xiang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5948-0780","authenticated-orcid":false,"given":"Yi","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Jiang","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,13]]},"reference":[{"key":"563_CR1","doi-asserted-by":"publisher","unstructured":"Gentry C (2009) Fully homomorphic encryption using ideal lattices. 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