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We call a distribution on n bit strings (", e) locally random, if for every choice of e · n positions the induced distribution on e bit strings is in the L1 norm at most " away from the uniform distribution on e bit strings. We establish local randomness in polynomial random number generators (RNG) that are candidate one way functions. Let N be a squarefree integer and let f1, . . . , f be polynomials with coe±- cients in ZZN = ZZ/NZZ. We study the RNG that stretches a random x 2 ZZN into the sequence of least significant bits of f1(x), . . . , f(x). We show that this RNG provides local randomness if for every prime divisor p of N the polynomials f1, . . . , f are linearly independent modulo the subspace of polynomials of degree · 1 in ZZp[x]. We also establish local randomness in polynomial random function generators. This yields candidates for cryptographic hash functions. The concept of local randomness in families of functions extends the concept of universal families of hash functions by Carter and Wegman (1979). The proofs of our results rely on upper bounds for exponential sums.
We propose two improvements to the Fiat Shamir authentication and signature scheme. We reduce the communication of the Fiat Shamir authentication scheme to a single round while preserving the e±ciency of the scheme. This also reduces the length of Fiat Shamir signatures. Using secret keys consisting of small integers we reduce the time for signature generation by a factor 3 to 4. We propose a variation of our scheme using class groups that may be secure even if factoring large integers becomes easy.
We introduce novel security proofs that use combinatorial counting arguments rather than reductions to the discrete logarithm or to the Diffie-Hellman problem. Our security results are sharp and clean with no polynomial reduction times involved. We consider a combination of the random oracle model and the generic model. This corresponds to assuming an ideal hash function H given by an oracle and an ideal group of prime order q, where the binary encoding of the group elements is useless for cryptographic attacks In this model, we first show that Schnorr signatures are secure against the one-more signature forgery : A generic adversary performing t generic steps including l sequential interactions with the signer cannot produce l+1 signatures with a better probability than (t 2)/q. We also characterize the different power of sequential and of parallel attacks. Secondly, we prove signed ElGamal encryption is secure against the adaptive chosen ciphertext attack, in which an attacker can arbitrarily use a decryption oracle except for the challenge ciphertext. Moreover, signed ElGamal encryption is secure against the one-more decryption attack: A generic adversary performing t generic steps including l interactions with the decryption oracle cannot distinguish the plaintexts of l + 1 ciphertexts from random strings with a probability exceeding (t 2)/q.
Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target ciphertext. We also prove security against the novel one-more-decyption attack. Our security proofs are in a new model, corresponding to a combination of two previously introduced models, the Random Oracle model and the Generic model. The security extends to the distributed threshold version of the scheme. Moreover, we propose a very practical scheme for private information retrieval that is based on blind decryption of ElGamal ciphertexts.
Let b1, . . . , bm 2 IRn be an arbitrary basis of lattice L that is a block Korkin Zolotarev basis with block size ¯ and let ¸i(L) denote the successive minima of lattice L. We prove that for i = 1, . . . ,m 4 i + 3 ° 2 i 1 ¯ 1 ¯ · kbik2/¸i(L)2 · ° 2m i ¯ 1 ¯ i + 3 4 where °¯ is the Hermite constant. For ¯ = 3 we establish the optimal upper bound kb1k2/¸1(L)2 · µ3 2¶m 1 2 1 and we present block Korkin Zolotarev lattice bases for which this bound is tight. We improve the Nearest Plane Algorithm of Babai (1986) using block Korkin Zolotarev bases. Given a block Korkin Zolotarev basis b1, . . . , bm with block size ¯ and x 2 L(b1, . . . , bm) a lattice point v can be found in time ¯O(¯) satisfying kx vk2 · m° 2m ¯ 1 ¯ minu2L kx uk2.
With ubiquitous use of digital camera devices, especially in mobile phones, privacy is no longer threatened by governments and companies only. The new technology creates a new threat by ordinary people, who now have the means to take and distribute pictures of one’s face at no risk and little cost in any situation in public and private spaces. Fast distribution via web based photo albums, online communities and web pages expose an individual’s private life to the public in unpreceeded ways. Social and legal measures are increasingly taken to deal with this problem. In practice however, they lack efficiency, as they are hard to enforce in practice. In this paper, we discuss a supportive infrastructure aiming for the distribution channel; as soon as the picture is publicly available, the exposed individual has a chance to find it and take proper action.
Korrektur zu: C.P. Schnorr: Security of 2t-Root Identification and Signatures, Proceedings CRYPTO'96, Springer LNCS 1109, (1996), pp. 143-156 page 148, section 3, line 5 of the proof of Theorem 3. Die Korrektur wurde präsentiert als: "Factoring N via proper 2 t-Roots of 1 mod N" at Eurocrypt '97 rump session.
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}).
We present a novel practical algorithm that given a lattice basis b1, ..., bn finds in O(n exp 2 *(k/6) exp (k/4)) average time a shorter vector than b1 provided that b1 is (k/6) exp (n/(2k)) times longer than the length of the shortest, nonzero lattice vector. We assume that the given basis b1, ..., bn has an orthogonal basis that is typical for worst case lattice bases. The new reduction method samples short lattice vectors in high dimensional sublattices, it advances in sporadic big jumps. It decreases the approximation factor achievable in a given time by known methods to less than its fourth-th root. We further speed up the new method by the simple and the general birthday method. n2