# (Short Paper) How to Solve DLOG Problem with Auxiliary Input

• Akinaga Ueda
• Kaoru Kurosawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11049)

## Abstract

Let $$\mathbb {G}$$ be a group of prime order p with a generator g. We first show that if $$p-1=d_1 \ldots d_t$$ and $$g,g^x$$, $$g^{x^{(p-1)/d_1}}, \ldots , \ g^{x^{(p-1)/(d_1 \ldots d_{t-1})}}$$ are given, then x can be computed in time $$O(\sqrt{d_1}+ \ldots + \sqrt{d_t} )$$ exponentiations. Further suppose that $$p-1=d_1^{e_1} \ldots d_t^{e_t}$$, where $$d_1, \ldots , d_t$$ are relatively prime. We then show that x can be computed in time $$O(e_1\sqrt{d_1}+\ldots + e_t\sqrt{d_t})$$ exponentiations if g and $$g^{x^{(p-1)/d_i}}, \ldots , g^{x^{(p-1)/d_i^{e_i-1}}}$$ are given for $$i=1, \ldots , t$$.

## Keywords

Discrete logarithm Auxiliary inputs Cheon algorithm

## References

1. 1.
den Boer, B.: Diffie-Hellman is as strong as discrete log for certain primes. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 530–539. Springer, New York (1990).
2. 2.
Cheon, J.H.: Discrete logarithm problems with auxiliary inputs. J. Cryptol. 23(3), 457–476 (2010)
3. 3.
Galbraith, S.D.: Mathematics of Public Key Cryptography. Cambridge University Press, Cambridge (2012)
4. 4.
Pollard, J.M.: Monte Carlo methods for index computation (mod p). Math. Comput. 32(143), 918–924 (1978)
5. 5.
Mitsunari, S., Sakai, R., Kasahara, M.: A new traitor tracing. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 85(2), 481–484 (2002)Google Scholar
6. 6.
Boneh, D., Boyen, X.: Short signatures without random oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 56–73. Springer, Heidelberg (2004).
7. 7.
Boneh, D., Boyen, X.: Efficient selective-ID secure identity-based encryption without random oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 223–238. Springer, Heidelberg (2004).
8. 8.
Boneh, D., Gentry, C., Waters, B.: Collusion resistant broadcast encryption with short ciphertexts and private keys. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 258–275. Springer, Heidelberg (2005).