7,773 research outputs found
Structure computation and discrete logarithms in finite abelian p-groups
We present a generic algorithm for computing discrete logarithms in a finite
abelian p-group H, improving the Pohlig-Hellman algorithm and its
generalization to noncyclic groups by Teske. We then give a direct method to
compute a basis for H without using a relation matrix. The problem of computing
a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian
group G is addressed, yielding a Monte Carlo algorithm to compute the structure
of G using O(|G|^0.5) group operations. These results also improve generic
algorithms for extracting pth roots in G.Comment: 23 pages, minor edit
Root Extraction in Finite Abelian Groups
We formulate the Root Extraction problem in finite Abelian -groups and
then extend it to generic finite Abelian groups. We provide algorithms to solve
them. We also give the bounds on the number of group operations required for
these algorithms. We observe that once a basis is computed and the discrete
logarithm relative to the basis is solved, root extraction takes relatively
fewer "bookkeeping" steps. Thus, we conclude that root extraction in finite
Abelian groups is no harder than solving discrete logarithms and computing
basis
Speeding Up Elliptic Curve Discrete Logarithm Computations with Point Halving
Pollard rho method and its parallelized variants are at present known as the best generic algorithms for computing elliptic curve discrete logarithms. We propose new iteration function for the rho method by exploiting the fact that point halving is more efficient than point addition for elliptic curves over binary fields. We present a careful analysis of the alternative rho method with new iteration function. Compared to the previous -adding walk, generally the new method can achieve a significant speedup for computing elliptic curve discrete logarithms over binary fields. For instance, for certain NIST-recommended curves over binary fields, the new method is about 27\% faster than the previous best methods in single-instance Pollard rho method. When running several instances of Pollard rho method concurrently, and computing the inversions using the simultaneous inversion algorithm by Peter Montgomery, the new method is about 12-17\% faster than the previous best methods
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
Quantum resource estimates for computing elliptic curve discrete logarithms
We give precise quantum resource estimates for Shor's algorithm to compute
discrete logarithms on elliptic curves over prime fields. The estimates are
derived from a simulation of a Toffoli gate network for controlled elliptic
curve point addition, implemented within the framework of the quantum computing
software tool suite LIQ. We determine circuit implementations for
reversible modular arithmetic, including modular addition, multiplication and
inversion, as well as reversible elliptic curve point addition. We conclude
that elliptic curve discrete logarithms on an elliptic curve defined over an
-bit prime field can be computed on a quantum computer with at most qubits using a quantum circuit of at most Toffoli gates. We are able to classically simulate the
Toffoli networks corresponding to the controlled elliptic curve point addition
as the core piece of Shor's algorithm for the NIST standard curves P-192,
P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to
recent resource estimates for Shor's factoring algorithm. The results also
support estimates given earlier by Proos and Zalka and indicate that, for
current parameters at comparable classical security levels, the number of
qubits required to tackle elliptic curves is less than for attacking RSA,
suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added.
ASIACRYPT 201
On Black-Box Ring Extraction and Integer Factorization
The black-box extraction problem over rings has (at least) two important interpretations in cryptography: An efficient algorithm for this problem implies (i) the equivalence of computing discrete logarithms and solving the Diffie-Hellman problem and (ii) the in-existence of secure ring-homomorphic encryption schemes.
In the special case of a finite field, Boneh/Lipton and Maurer/Raub showed that there exist algorithms solving the black-box extraction problem in subexponential time. It is unknown whether there exist more efficient algorithms.
In this work we consider the black-box extraction problem over finite rings of characteristic , where has at least two different prime factors. We provide a polynomial-time reduction from factoring to the black-box extraction problem for a large class of finite commutative unitary rings. Under the factoring assumption, this implies the in-existence of certain efficient generic reductions from computing discrete logarithms to the Diffie-Hellman problem on the one side, and might be an indicator that secure ring-homomorphic encryption schemes exist on the other side
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 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})
Quantum Period Finding is Compression Robust
We study quantum period finding algorithms such as Simon and Shor (and its
variants Eker{\aa}-H{\aa}stad and Mosca-Ekert). For a periodic function
these algorithms produce -- via some quantum embedding of -- a quantum
superposition , which requires a certain amount
of output qubits that represent . We show that one can lower this
amount to a single output qubit by hashing down to a single bit in an
oracle setting.
Namely, we replace the embedding of in quantum period finding circuits by
oracle access to several embeddings of hashed versions of . We show that on
expectation this modification only doubles the required amount of quantum
measurements, while significantly reducing the total number of qubits. For
example, for Simon's algorithm that finds periods in our hashing technique reduces the required output
qubits from down to , and therefore the total amount of qubits from
to . We also show that Simon's algorithm admits real world applications
with only qubits by giving a concrete realization of a hashed version of
the cryptographic Even-Mansour construction. Moreover, for a variant of Simon's
algorithm on Even-Mansour that requires only classical queries to Even-Mansour
we save a factor of (roughly) in the qubits.
Our oracle-based hashed version of the Eker{\aa}-H{\aa}stad algorithm for
factoring -bit RSA reduces the required qubits from
down to . We also show a real-world (non-oracle)
application in the discrete logarithm setting by giving a concrete realization
of a hashed version of Mosca-Ekert for the Decisional Diffie Hellman problem in
, thereby reducing the number of qubits by even a linear
factor from downto
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