New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs
Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements...
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| Autor*in: |
Weng, Jiang [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag Berlin Heidelberg 2016 |
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| Übergeordnetes Werk: |
Enthalten in: Applicable algebra in engineering, communication and computing - Springer Berlin Heidelberg, 1990, 28(2016), 2 vom: 19. Aug., Seite 99-108 |
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| Übergeordnetes Werk: |
volume:28 ; year:2016 ; number:2 ; day:19 ; month:08 ; pages:99-108 |
| Links: |
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| DOI / URN: |
10.1007/s00200-016-0301-z |
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| Katalog-ID: |
OLC2075500150 |
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| 520 | |a Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. | ||
| 650 | 4 | |a Elliptic curve discrete logarithm problem | |
| 650 | 4 | |a Auxiliary inputs | |
| 650 | 4 | |a Baby-step giant-step | |
| 650 | 4 | |a Cheon’s algorithm | |
| 700 | 1 | |a Dou, Yunqi |4 aut | |
| 700 | 1 | |a Ma, Chuangui |4 aut | |
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10.1007/s00200-016-0301-z doi (DE-627)OLC2075500150 (DE-He213)s00200-016-0301-z-p DE-627 ger DE-627 rakwb eng 510 620 004 VZ 510 004 600 VZ 11 ssgn Weng, Jiang verfasserin aut New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. Elliptic curve discrete logarithm problem Auxiliary inputs Baby-step giant-step Cheon’s algorithm Dou, Yunqi aut Ma, Chuangui aut Enthalten in Applicable algebra in engineering, communication and computing Springer Berlin Heidelberg, 1990 28(2016), 2 vom: 19. Aug., Seite 99-108 (DE-627)130915807 (DE-600)1051032-1 (DE-576)025004964 0938-1279 nnns volume:28 year:2016 number:2 day:19 month:08 pages:99-108 https://doi.org/10.1007/s00200-016-0301-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4126 GBV_ILN_4247 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4318 AR 28 2016 2 19 08 99-108 |
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10.1007/s00200-016-0301-z doi (DE-627)OLC2075500150 (DE-He213)s00200-016-0301-z-p DE-627 ger DE-627 rakwb eng 510 620 004 VZ 510 004 600 VZ 11 ssgn Weng, Jiang verfasserin aut New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. Elliptic curve discrete logarithm problem Auxiliary inputs Baby-step giant-step Cheon’s algorithm Dou, Yunqi aut Ma, Chuangui aut Enthalten in Applicable algebra in engineering, communication and computing Springer Berlin Heidelberg, 1990 28(2016), 2 vom: 19. Aug., Seite 99-108 (DE-627)130915807 (DE-600)1051032-1 (DE-576)025004964 0938-1279 nnns volume:28 year:2016 number:2 day:19 month:08 pages:99-108 https://doi.org/10.1007/s00200-016-0301-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4126 GBV_ILN_4247 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4318 AR 28 2016 2 19 08 99-108 |
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10.1007/s00200-016-0301-z doi (DE-627)OLC2075500150 (DE-He213)s00200-016-0301-z-p DE-627 ger DE-627 rakwb eng 510 620 004 VZ 510 004 600 VZ 11 ssgn Weng, Jiang verfasserin aut New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. Elliptic curve discrete logarithm problem Auxiliary inputs Baby-step giant-step Cheon’s algorithm Dou, Yunqi aut Ma, Chuangui aut Enthalten in Applicable algebra in engineering, communication and computing Springer Berlin Heidelberg, 1990 28(2016), 2 vom: 19. Aug., Seite 99-108 (DE-627)130915807 (DE-600)1051032-1 (DE-576)025004964 0938-1279 nnns volume:28 year:2016 number:2 day:19 month:08 pages:99-108 https://doi.org/10.1007/s00200-016-0301-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4126 GBV_ILN_4247 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4318 AR 28 2016 2 19 08 99-108 |
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10.1007/s00200-016-0301-z doi (DE-627)OLC2075500150 (DE-He213)s00200-016-0301-z-p DE-627 ger DE-627 rakwb eng 510 620 004 VZ 510 004 600 VZ 11 ssgn Weng, Jiang verfasserin aut New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. Elliptic curve discrete logarithm problem Auxiliary inputs Baby-step giant-step Cheon’s algorithm Dou, Yunqi aut Ma, Chuangui aut Enthalten in Applicable algebra in engineering, communication and computing Springer Berlin Heidelberg, 1990 28(2016), 2 vom: 19. Aug., Seite 99-108 (DE-627)130915807 (DE-600)1051032-1 (DE-576)025004964 0938-1279 nnns volume:28 year:2016 number:2 day:19 month:08 pages:99-108 https://doi.org/10.1007/s00200-016-0301-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4126 GBV_ILN_4247 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4318 AR 28 2016 2 19 08 99-108 |
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10.1007/s00200-016-0301-z doi (DE-627)OLC2075500150 (DE-He213)s00200-016-0301-z-p DE-627 ger DE-627 rakwb eng 510 620 004 VZ 510 004 600 VZ 11 ssgn Weng, Jiang verfasserin aut New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. Elliptic curve discrete logarithm problem Auxiliary inputs Baby-step giant-step Cheon’s algorithm Dou, Yunqi aut Ma, Chuangui aut Enthalten in Applicable algebra in engineering, communication and computing Springer Berlin Heidelberg, 1990 28(2016), 2 vom: 19. Aug., Seite 99-108 (DE-627)130915807 (DE-600)1051032-1 (DE-576)025004964 0938-1279 nnns volume:28 year:2016 number:2 day:19 month:08 pages:99-108 https://doi.org/10.1007/s00200-016-0301-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4126 GBV_ILN_4247 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4318 AR 28 2016 2 19 08 99-108 |
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Enthalten in Applicable algebra in engineering, communication and computing 28(2016), 2 vom: 19. Aug., Seite 99-108 volume:28 year:2016 number:2 day:19 month:08 pages:99-108 |
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Enthalten in Applicable algebra in engineering, communication and computing 28(2016), 2 vom: 19. Aug., Seite 99-108 volume:28 year:2016 number:2 day:19 month:08 pages:99-108 |
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new algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs |
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New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs |
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Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. © Springer-Verlag Berlin Heidelberg 2016 |
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Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. © Springer-Verlag Berlin Heidelberg 2016 |
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Abstract The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$ of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$ in the case of $$d|(p\pm 1)$$ with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$ group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$ storage ($$i=\frac{1}{2}$$ or 1 for $$d|(p-1)$$ case or $$d|(p+1)$$ case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$ and multiplicative cyclic group $$K=\langle k \rangle $$ are given, where d is a prime, $$\varphi (d)$$ is the order of K and $$\varphi $$ is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$ can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$ group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$ storage. © Springer-Verlag Berlin Heidelberg 2016 |
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