Security of almost ALL discrete log bits

Let G be a finite cyclic group with generator \alpha and with an encoding so that multiplication is computable in polynomial time. We study the security of bits of the discrete log x when given \exp_{\alpha}(x), assuming that the exponentiation function \exp_{\alpha}(x) = \alpha^x is one-way. We reduce he general problem to the case that G has odd order q. If G has odd order q the security of the least-significant bits of x and of the most significant bits of the rational number \frac{x}{q} \in [0,1) follows from the work of Peralta [P85] and Long and Wigderson [LW88]. We generalize these bits and study the security of consecutive shift bits lsb(2^{-i}x mod q) for i=k+1,...,k+j. When we restrict \exp_{\alpha} to arguments x such that some sequence of j consecutive shift bits of x is constant (i.e., not depending on x) we call it a 2^{-j}-fraction of \exp_{\alpha}. For groups of odd group order q we show that every two 2^{-j}-fractions of \exp_{\alpha} are equally one-way by a polynomial time transformation: Either they are all one-way or none of them. Our key theorem shows that arbitrary j consecutive shift bits of x are simultaneously secure when given \exp_{\alpha}(x) iff the 2^{-j}-fractions of \exp_{\alpha} are one-way. In particular this applies to the j least-significant bits of x and to the j most-significant bits of \frac{x}{q} \in [0,1). For one-way \exp_{\alpha} the individual bits of x are secure when given \exp_{\alpha}(x) by the method of Hastad, N\"aslund [HN98]. For groups of even order 2^{s}q we show that the j least-significant bits of \lfloor x/2^s\rfloor, as well as the j most-significant bits of \frac{x}{q} \in [0,1), are simultaneously secure iff the 2^{-j}-fractions of \exp_{\alpha'} are one-way for \alpha' := \alpha^{2^s}. We use and extend the models of generic algorithms of Nechaev (1994) and Shoup (1997). We determine the generic complexity of inverting fractions of \exp_{\alpha} for the case that \alpha has prime order q. As a consequence, arbitrary segments of (1-\varepsilon)\lg q consecutive shift bits of random x are for constant \varepsilon >0 simultaneously secure against generic attacks. Every generic algorithm using $t$ generic steps (group operations) for distinguishing bit strings of j consecutive shift bits of x from random bit strings has at most advantage O((\lg q) j\sqrt{t} (2^j/q)^{\frac14})

A kilobit hidden SNFS discrete logarithm computation

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes

The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer

A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian `hidden subgroup problem' can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or `flying qubits', instead of entire registers of control qubits.Comment: 16 pages, 3 figures, LaTeX2e, to appear in Proceedings of the 1st NASA International Conference on Quantum Computing and Quantum Communication (Springer-Verlag