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A memory checker for a data structure provides a method to check that the output of the data structure operations is consistent with the input even if the data is stored on some insecure medium. In [8] we present a general solution for all data structures that are based on insert(i,v) and delete(j) commands. In particular this includes stacks, queues, deques (double-ended queues) and lists. Here, we describe more time and space efficient solutions for stacks, queues and deques. Each algorithm takes only a single function evaluation of a pseudorandomlike function like DES or a collision-free hash function like MD5 or SHA for each push/pop resp. enqueue/dequeue command making our methods applicable to smart cards.
We present efficient non-malleable commitment schemes based on standard assumptions such as RSA and Discrete-Log, and under the condition that the network provides publicly available RSA or Discrete-Log parameters generated by a trusted party. Our protocols require only three rounds and a few modular exponentiations. We also discuss the difference between the notion of non-malleable commitment schemes used by Dolev, Dwork and Naor [DDN00] and the one given by Di Crescenzo, Ishai and Ostrovsky [DIO98].
We address to the problem to factor a large composite number by lattice reduction algorithms. Schnorr has shown that under a reasonable number theoretic assumptions this problem can be reduced to a simultaneous diophantine approximation problem. The latter in turn can be solved by finding sufficiently many l_1--short vectors in a suitably defined lattice. Using lattice basis reduction algorithms Schnorr and Euchner applied Schnorrs reduction technique to 40--bit long integers. Their implementation needed several hours to compute a 5% fraction of the solution, i.e., 6 out of 125 congruences which are necessary to factorize the composite. In this report we describe a more efficient implementation using stronger lattice basis reduction techniques incorporating ideas of Schnorr, Hoerner and Ritter. For 60--bit long integers our algorithm yields a complete factorization in less than 3 hours.
Based on the quadratic residuosity assumption we present a non-interactive crypto-computing protocol for the greater-than function, i.e., a non-interactive procedure between two parties such that only the relation of the parties' inputs is revealed. In comparison to previous solutions our protocol reduces the number of modular multiplications significantly. We also discuss applications to conditional oblivious transfer, private bidding and the millionaires' problem.
We propose a new security measure for commitment protocols, called Universally Composable (UC) Commitment. The measure guarantees that commitment protocols behave like an \ideal commitment service," even when concurrently composed with an arbitrary set of protocols. This is a strong guarantee: it implies that security is maintained even when an unbounded number of copies of the scheme are running concurrently, it implies non-malleability (not only with respect to other copies of the same protocol but even with respect to other protocols), it provides resilience to selective decommitment, and more. Unfortunately two-party uc commitment protocols do not exist in the plain model. However, we construct two-party uc commitment protocols, based on general complexity assumptions, in the common reference string model where all parties have access to a common string taken from a predetermined distribution. The protocols are non-interactive, in the sense that both the commitment and the opening phases consist of a single message from the committer to the receiver.
We review the representation problem based on factoring and show that this problem gives rise to alternative solutions to a lot of cryptographic protocols in the literature. And, while the solutions so far usually either rely on the RSA problem or the intractability of factoring integers of a special form (e.g., Blum integers), the solutions here work with the most general factoring assumption. Protocols we discuss include identification schemes secure against parallel attacks, secure signatures, blind signatures and (non-malleable) commitments.
We show that non-interactive statistically-secret bit commitment cannot be constructed from arbitrary black-box one-to-one trapdoor functions and thus from general public-key cryptosystems. Reducing the problems of non-interactive crypto-computing, rerandomizable encryption, and non-interactive statistically-sender-private oblivious transfer and low-communication private information retrieval to such commitment schemes, it follows that these primitives are neither constructible from one-to-one trapdoor functions and public-key encryption in general. Furthermore, our separation sheds some light on statistical zeroknowledge proofs. There is an oracle relative to which one-to-one trapdoor functions and one-way permutations exist, while the class of promise problems with statistical zero-knowledge proofs collapses in P. This indicates that nontrivial problems with statistical zero-knowledge proofs require more than (trapdoor) one-wayness.
We show lower bounds for the signature size of incremental schemes which are secure against substitution attacks and support single block replacement. We prove that for documents of n blocks such schemes produce signatures of \Omega(n^(1/(2+c))) bits for any constant c>0. For schemes accessing only a single block resp. a constant number of blocks for each replacement this bound can be raised to \Omega(n) resp. \Omega(sqrt(n)). Additionally, we show that our technique yields a new lower bound for memory checkers.
Given a real vector alpha =(alpha1 ; : : : ; alpha d ) and a real number E > 0 a good Diophantine approximation to alpha is a number Q such that IIQ alpha mod Zk1 ", where k \Delta k1 denotes the 1-norm kxk1 := max 1id jx i j for x = (x1 ; : : : ; xd ). Lagarias [12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector ff 2 Q d , a rational number " ? 0 and a number N 2 N+ , decide whether there exists a number Q with 1 Q N and kQff mod Zk1 ". We prove that, unless ...
Given x small epsilon, Greek Rn an integer relation for x is a non-trivial vector m small epsilon, Greek Zn with inner product <m,x> = 0. In this paper we prove the following: Unless every NP language is recognizable in deterministic quasi-polynomial time, i.e., in time O(npoly(log n)), the ℓinfinity-shortest integer relation for a given vector x small epsilon, Greek Qn cannot be approximated in polynomial time within a factor of 2log0.5 − small gamma, Greekn, where small gamma, Greek is an arbitrarily small positive constant. This result is quasi-complementary to positive results derived from lattice basis reduction. A variant of the well-known L3-algorithm approximates for a vector x small epsilon, Greek Qn the ℓ2-shortest integer relation within a factor of 2n/2 in polynomial time. Our proof relies on recent advances in the theory of probabilistically checkable proofs, in particular on a reduction from 2-prover 1-round interactive proof-systems. The same inapproximability result is valid for finding the ℓinfinity-shortest integer solution for a homogeneous linear system of equations over Q.
We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2^((n+2)/4) best possible. Given a real vector x=(x_1,...,x_{n-1},1) in R^n this CFA generates a sequence of vectors (p_1^(k),...,p_{n-1}^(k),q^(k)) in Z^n, k=1,2,... with increasing integers |q^{(k)}| satisfying for i=1,...,n-1 | x_i - p_i^(k)/q^(k) | <= 2^((n+2)/4) sqrt(1+x_i^2) |q^(k)|^(1+1/(n-1)) By a theorem of Dirichlet this bound is best possible in that the exponent 1+1/(n-1) can in general not be increased.
We generalize the concept of block reduction for lattice bases from l2-norm to arbitrary norms. This extends the results of Schnorr. We give algorithms for block reduction and apply the resulting enumeration concept to solve subset sum problems. The deterministic algorithm solves all subset sum problems. For up to 66 weights it needs in average less then two hours on a HP 715/50 under HP-UX 9.05.
We propose a fast variant of the Gaussian algorithm for the reduction of two dimensional lattices for the l1-, l2- and l-infinite- norm. The algorithm runs in at most O(nM(B) logB) bit operations for the l-infinite- norm and in O(n log n M(B) logB) bit operations for the l1 and l2 norm on input vectors a, b 2 ZZn with norm at most 2B where M(B) is a time bound for B-bit integer multiplication. This generalizes Schönhages monotone Algorithm [Sch91] to the centered case and to various norms.
We present an efficient variant of LLL-reduction of lattice bases in the sense of Lenstra, Lenstra, Lov´asz [LLL82]. We organize LLL-reduction in segments of size k. Local LLL-reduction of segments is done using local coordinates of dimension 2k. Strong segment LLL-reduction yields bases of the same quality as LLL-reduction but the reduction is n-times faster for lattices of dimension n. We extend segment LLL-reduction to iterated subsegments. The resulting reduction algorithm runs in O(n3 log n) arithmetic steps for integer lattices of dimension n with basis vectors of length 2O(n), compared to O(n5) steps for LLL-reduction.
We introduce algorithms for lattice basis reduction that are improvements of the famous L3-algorithm. If a random L3-reduced lattice basis b1,b2,...,bn is given such that the vector of reduced Gram-Schmidt coefficients ({µi,j} 1<= j< i<= n) is uniformly distributed in [0,1)n(n-1)/2, then the pruned enumeration finds with positive probability a shortest lattice vector. We demonstrate the power of these algorithms by solving random subset sum problems of arbitrary density with 74 and 82 many weights, by breaking the Chor-Rivest cryptoscheme in dimensions 103 and 151 and by breaking Damgard's hash function.
We call a vector x/spl isin/R/sup n/ highly regular if it satisfies =0 for some short, non-zero integer vector m where <...> is the inner product. We present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//spl isin/N finds a highly regular nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no short relation m~ of length less than /spl alpha//2 exists for points x~ within half the x'-distance from x. The integer relation m for x' is for random x up to an average factor 2/sup /spl alpha//2/ a shortest integer relation for x'. Our algorithm uses, for arbitrary real input x, at most O(n/sup 4/(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n/sup 5/+n/sup 3/(log /spl alpha/)/sup 2/+log(/spl par/qx/spl par//sup 2/)) many bits where q is the common denominator for x.
We study the following problem: given x element Rn either find a short integer relation m element Zn, so that =0 holds for the inner product <.,.>, or prove that no short integer relation exists for x. Hastad, Just Lagarias and Schnorr (1989) give a polynomial time algorithm for the problem. We present a stable variation of the HJLS--algorithm that preserves lower bounds on lambda(x) for infinitesimal changes of x. Given x \in {\RR}^n and \alpha \in \NN this algorithm finds a nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no very short relation exists for points \bar{x} within half the x'--distance from x. On the other hand if x'=x then m is, up to a factor 2^{n/2}, a shortest integer relation for \mbox{x.} Our algorithm uses, for arbitrary real input x, at most \mbox{O(n^4(n+\log \alpha))} many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most \mbox{O(n^5+n^3 (\log \alpha)^2 + \log (\|q x\|^2))} many bits where q is the common denominator for x.
Black box cryptanalysis applies to hash algorithms consisting of many small boxes, connected by a known graph structure, so that the boxes can be evaluated forward and backwards by given oracles. We study attacks that work for any choice of the black boxes, i.e. we scrutinize the given graph structure. For example we analyze the graph of the fast Fourier transform (FFT). We present optimal black box inversions of FFT-compression functions and black box constructions of collisions. This determines the minimal depth of FFT-compression networks for collision-resistant hashing. We propose the concept of multipermutation, which is a pair of orthogonal latin squares, as a new cryptographic primitive that generalizes the boxes of the FFT. Our examples of multipermutations are based on the operations circular rotation, bitwise xor, addition and multiplication.
Parallel FFT-hashing
(1994)
We propose two families of scalable hash functions for collision resistant hashing that are highly parallel and based on the generalized fast Fourier transform (FFT). FFT hashing is based on multipermutations. This is a basic cryptographic primitive for perfect generation of diffusion and confusion which generalizes the boxes of the classic FFT. The slower FFT hash functions iterate a compression function. For the faster FFT hash functions all rounds are alike with the same number of message words entering each round.
We report on improved practical algorithms for lattice basis reduction. We propose a practical floating point version of theL3-algorithm of Lenstra, Lenstra, Lovász (1982). We present a variant of theL3-algorithm with "deep insertions" and a practical algorithm for block Korkin—Zolotarev reduction, a concept introduced by Schnorr (1987). Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC1 + computer.
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.
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}).
Let G be a group of prime order q with generator g. We study hardcore subsets H is include in G of the discrete logarithm (DL) log g in the model of generic algorithms. In this model we count group operations such as multiplication, division while computations with non-group data are for free. It is known from Nechaev (1994) and Shoup (1997) that generic DL-algorithms for the entire group G must perform p2q generic steps. We show that DL-algorithms for small subsets H is include in G require m/ 2 + o(m) generic steps for almost all H of size #H = m with m <= sqrt(q). Conversely, m/2 + 1 generic steps are su±cient for all H is include in G of even size m. Our main result justifies to generate secret DL-keys from seeds that are only 1/2 * log2 q bits long.
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
We enhance the security of Schnorr blind signatures against the novel one-more-forgery of Schnorr [Sc01] andWagner [W02] which is possible even if the discrete logarithm is hard to compute. We show two limitations of this attack. Firstly, replacing the group G by the s-fold direct product G exp(×s) increases the work of the attack, for a given number of signer interactions, to the s-power while increasing the work of the blind signature protocol merely by a factor s. Secondly, we bound the number of additional signatures per signer interaction that can be forged effectively. That fraction of the additional forged signatures can be made arbitrarily small.
We modify the concept of LLL-reduction of lattice bases in the sense of Lenstra, Lenstra, Lovasz [LLL82] towards a faster reduction algorithm. We organize LLL-reduction in segments of the basis. Our SLLL-bases approximate the successive minima of the lattice in nearly the same way as LLL-bases. For integer lattices of dimension n given by a basis of length 2exp(O(n)), SLLL-reduction runs in O(n.exp(5+epsilon)) bit operations for every epsilon > 0, compared to O(exp(n7+epsilon)) for the original LLL and to O(exp(n6+epsilon)) for the LLL-algorithms of Schnorr (1988) and Storjohann (1996). We present an even faster algorithm for SLLL-reduction via iterated subsegments running in O(n*exp(3)*log n) arithmetic steps.
We show that P(n)*(P(n)) for p = 2 with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation Epsilon nor the coproduct Delta are multiplicative. As a consequence the algebra structure of P(n)*(P(n)) is slightly different from what was supposed to be the case. We give formulas for Epsilon(xy) and Delta(xy) and show that the inversion of the formal group of P(n) is induced by an antimultiplicative involution Xi : P(n) -> P(n). Some consequences for multiplicative and antimultiplicative automorphisms of K(n) for p = 2 are also discussed.
The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short nonzero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0.6463 . . . in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408 . . . if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.
Public key signature schemes are necessary for the access control to communication networks and for proving the authenticity of sensitive messages such as electronic fund transfers. Since the invention of the RSA scheme by Rivest, Shamir and Adleman (1978) research has focused on improving the e±ciency of these schemes. In this paper we present an efficient algorithm for generating public key signatures which is particularly suited for interactions between smart cards and terminals.
We present a novel parallel one-more signature forgery against blind Okamoto-Schnorr and blind Schnorr signatures in which an attacker interacts some times with a legitimate signer and produces from these interactions signatures. Security against the new attack requires that the following ROS-problem is intractable: find an overdetermined, solvable system of linear equations modulo with random inhomogenities (right sides). There is an inherent weakness in the security result of POINTCHEVAL AND STERN. Theorem 26 [PS00] does not cover attacks with 4 parallel interactions for elliptic curves of order 2200. That would require the intractability of the ROS-problem, a plausible but novel complexity assumption. Conversely, assuming the intractability of the ROS-problem, we show that Schnorr signatures are secure in the random oracle and generic group model against the one-more signature forgery.
We present a practical algorithm that given an LLL-reduced lattice basis of dimension n, runs in time O(n3(k=6)k=4+n4) and approximates the length of the shortest, non-zero lattice vector to within a factor (k=6)n=(2k). This result is based on reasonable heuristics. Compared to previous practical algorithms the new method reduces the proven approximation factor achievable in a given time to less than its fourthth root. We also present a sieve algorithm inspired by Ajtai, Kumar, Sivakumar [AKS01].
Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and alpha-Kalpha exp(-1) are conjugate in PSl(2,R), but in general the conjugating element alpha cannot be taken in G or a finite index Fuchsian extension of G. We will show that in the case of a normal inclusion in a triangle group G these alpha can be chosen in some triangle group extending G. It turns out that the method leading to this result allows also to answer the question how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface.
We consider Schwarz maps for triangles whose angles are rather general rational multiples of pi. Under which conditions can they have algebraic values at algebraic arguments? The answer is based mainly on considerations of complex multiplication of certain Prym varieties in Jacobians of hypergeometric curves. The paper can serve as an introduction to transcendence techniques for hypergeometric functions, but contains also new results and examples.