### Abstract

Let double-struck G be a group of prime order q, and let g _{1},...,g_{n} be random elements of double-struck G. We say that a vector x = (x_{1},...,x_{2}) ∈ ℤ _{q}
^{n} is a discrete log representation of some some element y ∈ double-struck G (with respect to g_{1},...,g_{n}) if g_{1}
^{x1}⋯g_{n}
^{xn} = y. Any element y has many discrete log representations, forming an affine subspace of ℤ_{q}
^{n}. We show that these representations have a nice continuous leakage-resilience property as follows. Assume some attacker A(g _{1},...,g_{n}, y) can repeatedly learn L bits of information on arbitrarily many random representations of y. That is, A adaptively chooses polynomially many leakage functions f_{i} : ℤ_{q} ^{n} → {0,1}^{L}, and learns the value f_{i}(x _{i}), where x_{i} is a fresh and random discrete log representation of y. A wins the game if it eventually outputs a valid discrete log representation x* of y. We show that if the discrete log assumption holds in double-struck G, then no polynomially bounded A can win this game with non-negligible probability, as long as the leakage on each representation is bounded by L ≈ (n - 2) log q = (1 - 2/n)·|x|. As direct extensions of this property, we design very simple continuous leakage-resilient (CLR) one-way function (OWF) and public-key encryption (PKE) schemes in the so called "invisible key update" model introduced by Alwen et al. at CRYPTO'09. Our CLR-OWF is based on the standard Discrete Log assumption and our CLR-PKE is based on the standard Decisional Diffie-Hellman assumption. Prior to our work, such schemes could only be constructed in groups with a bilinear pairing. As another surprising application, we show how to design the first leakage-resilient traitor tracing scheme, where no attacker, getting the secret keys of a small subset of decoders (called "traitors") and bounded leakage on the secret keys of all other decoders, can create a valid decryption key which will not be traced back to at least one of the traitors.

Original language | English (US) |
---|---|

Title of host publication | Advances in Cryptology, ASIACRYPT 2013 - 19th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings |

Pages | 401-420 |

Number of pages | 20 |

Volume | 8270 LNCS |

Edition | PART 2 |

DOIs | |

State | Published - 2013 |

Event | 19th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2013 - Bengaluru, India Duration: Dec 1 2013 → Dec 5 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Number | PART 2 |

Volume | 8270 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 19th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2013 |
---|---|

Country | India |

City | Bengaluru |

Period | 12/1/13 → 12/5/13 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Advances in Cryptology, ASIACRYPT 2013 - 19th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings*(PART 2 ed., Vol. 8270 LNCS, pp. 401-420). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8270 LNCS, No. PART 2). https://doi.org/10.1007/978-3-642-42045-0_21

**On continual leakage of discrete log representations.** / Agrawal, Shweta; Dodis, Yevgeniy; Vaikuntanathan, Vinod; Wichs, Daniel.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Advances in Cryptology, ASIACRYPT 2013 - 19th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings.*PART 2 edn, vol. 8270 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 2, vol. 8270 LNCS, pp. 401-420, 19th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2013, Bengaluru, India, 12/1/13. https://doi.org/10.1007/978-3-642-42045-0_21

}

TY - GEN

T1 - On continual leakage of discrete log representations

AU - Agrawal, Shweta

AU - Dodis, Yevgeniy

AU - Vaikuntanathan, Vinod

AU - Wichs, Daniel

PY - 2013

Y1 - 2013

N2 - Let double-struck G be a group of prime order q, and let g 1,...,gn be random elements of double-struck G. We say that a vector x = (x1,...,x2) ∈ ℤ q n is a discrete log representation of some some element y ∈ double-struck G (with respect to g1,...,gn) if g1 x1⋯gn xn = y. Any element y has many discrete log representations, forming an affine subspace of ℤq n. We show that these representations have a nice continuous leakage-resilience property as follows. Assume some attacker A(g 1,...,gn, y) can repeatedly learn L bits of information on arbitrarily many random representations of y. That is, A adaptively chooses polynomially many leakage functions fi : ℤq n → {0,1}L, and learns the value fi(x i), where xi is a fresh and random discrete log representation of y. A wins the game if it eventually outputs a valid discrete log representation x* of y. We show that if the discrete log assumption holds in double-struck G, then no polynomially bounded A can win this game with non-negligible probability, as long as the leakage on each representation is bounded by L ≈ (n - 2) log q = (1 - 2/n)·|x|. As direct extensions of this property, we design very simple continuous leakage-resilient (CLR) one-way function (OWF) and public-key encryption (PKE) schemes in the so called "invisible key update" model introduced by Alwen et al. at CRYPTO'09. Our CLR-OWF is based on the standard Discrete Log assumption and our CLR-PKE is based on the standard Decisional Diffie-Hellman assumption. Prior to our work, such schemes could only be constructed in groups with a bilinear pairing. As another surprising application, we show how to design the first leakage-resilient traitor tracing scheme, where no attacker, getting the secret keys of a small subset of decoders (called "traitors") and bounded leakage on the secret keys of all other decoders, can create a valid decryption key which will not be traced back to at least one of the traitors.

AB - Let double-struck G be a group of prime order q, and let g 1,...,gn be random elements of double-struck G. We say that a vector x = (x1,...,x2) ∈ ℤ q n is a discrete log representation of some some element y ∈ double-struck G (with respect to g1,...,gn) if g1 x1⋯gn xn = y. Any element y has many discrete log representations, forming an affine subspace of ℤq n. We show that these representations have a nice continuous leakage-resilience property as follows. Assume some attacker A(g 1,...,gn, y) can repeatedly learn L bits of information on arbitrarily many random representations of y. That is, A adaptively chooses polynomially many leakage functions fi : ℤq n → {0,1}L, and learns the value fi(x i), where xi is a fresh and random discrete log representation of y. A wins the game if it eventually outputs a valid discrete log representation x* of y. We show that if the discrete log assumption holds in double-struck G, then no polynomially bounded A can win this game with non-negligible probability, as long as the leakage on each representation is bounded by L ≈ (n - 2) log q = (1 - 2/n)·|x|. As direct extensions of this property, we design very simple continuous leakage-resilient (CLR) one-way function (OWF) and public-key encryption (PKE) schemes in the so called "invisible key update" model introduced by Alwen et al. at CRYPTO'09. Our CLR-OWF is based on the standard Discrete Log assumption and our CLR-PKE is based on the standard Decisional Diffie-Hellman assumption. Prior to our work, such schemes could only be constructed in groups with a bilinear pairing. As another surprising application, we show how to design the first leakage-resilient traitor tracing scheme, where no attacker, getting the secret keys of a small subset of decoders (called "traitors") and bounded leakage on the secret keys of all other decoders, can create a valid decryption key which will not be traced back to at least one of the traitors.

UR - http://www.scopus.com/inward/record.url?scp=84892419482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892419482&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-42045-0_21

DO - 10.1007/978-3-642-42045-0_21

M3 - Conference contribution

AN - SCOPUS:84892419482

SN - 9783642420443

VL - 8270 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 401

EP - 420

BT - Advances in Cryptology, ASIACRYPT 2013 - 19th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings

ER -