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Selecting Cryptographic Key Sizes
 TO APPEAR IN THE JOURNAL OF CRYPTOLOGY, SPRINGERVERLAG
, 2001
"... In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter ..."
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Cited by 257 (6 self)
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In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter settings, combined with existing data points about the cryptosystems.
Discrete Logarithms in Finite Fields and Their Cryptographic Significance
, 1984
"... Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q  1, for which u = g k . The wellknown problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its appl ..."
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Cited by 87 (6 self)
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Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q  1, for which u = g k . The wellknown problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...
On MemoryBound Functions for Fighting Spam
 In Crypto
, 2002
"... In 1992, Dwork and Naor proposed that email messages be accompanied by easytocheck proofs of computational effort in order to discourage junk email, now known as spam. They proposed specific CPUbound functions for this purpose. Burrows suggested that, since memory access speeds vary across ma ..."
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Cited by 81 (2 self)
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In 1992, Dwork and Naor proposed that email messages be accompanied by easytocheck proofs of computational effort in order to discourage junk email, now known as spam. They proposed specific CPUbound functions for this purpose. Burrows suggested that, since memory access speeds vary across machines much less than do CPU speeds, memorybound functions may behave more equitably than CPUbound functions; this approach was first explored by Abadi, Burrows, Manasse, and Wobber [8].
Efficient DES key search
 School of Computer Science, Carleton University
, 1994
"... Abstract. Despite recent improvements in analytic techniques for attacking the Data Encryption Standard (DES), exhaustive key search remains the most practical and efficient attack. Key search is becoming alarmingly practical. We show how to build an exhaustive DES key search machine for $1 million ..."
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Cited by 66 (0 self)
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Abstract. Despite recent improvements in analytic techniques for attacking the Data Encryption Standard (DES), exhaustive key search remains the most practical and efficient attack. Key search is becoming alarmingly practical. We show how to build an exhaustive DES key search machine for $1 million that can find a key in 3.5 hours on average. The design for such a machine is described in detail for the purpose of assessing the resistance of DES to an exhaustive attack. This design is based on mature technology to avoid making guesses about future capabilities. With this approach, DES keys can be found one to two orders of magnitude faster than other recently proposed designs. The basic machine design can be adapted to attack the standard DES modes of operation for a small penalty in running time. The issues of development cost and machine reliability are examined as well. In light of this work, it would be prudent in many applications to use DES in a tripleencryption mode. 1.
Twofish: A 128Bit Block Cipher
 in First Advanced Encryption Standard (AES) Conference
, 1998
"... Twofish is a 128bit block cipher that accepts a variablelength key up to 256 bits. The cipher is a 16round Feistel network with a bijective F function made up of four keydependent 8by8bit Sboxes, a fixed 4by4 maximum distance separable matrix over GF(2 8 ), a pseudoHadamard transform, bit ..."
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Cited by 56 (8 self)
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Twofish is a 128bit block cipher that accepts a variablelength key up to 256 bits. The cipher is a 16round Feistel network with a bijective F function made up of four keydependent 8by8bit Sboxes, a fixed 4by4 maximum distance separable matrix over GF(2 8 ), a pseudoHadamard transform, bitwise rotations, and a carefully designed key schedule. A fully optimized implementation of Twofish encrypts on a Pentium Pro at 17.8 clock cycles per byte, and an 8bit smart card implementation encrypts at 1660 clock cycles per byte. Twofish can be implemented in hardware in 14000 gates. The design of both the round function and the key schedule permits a wide variety of tradeoffs between speed, software size, key setup time, gate count, and memory. We have extensively cryptanalyzed Twofish; our best attack breaks 5 rounds with 2 22.5 chosen plaintexts and 2 51 effort.
Breaking Ciphers with COPACOBANA  A CostOptimized Parallel Code Breaker
 IN WORKSHOP ON CRYPTOGRAPHIC HARDWARE AND EMBEDDED SYSTEMS — CHES 2006,YOKOHAMA
, 2006
"... Cryptanalysis of symmetric and asymmetric ciphers is computationally extremely demanding. Since the security parameters (in particular the key length) of almost all practical crypto algorithms are chosen such that attacks with conventional computers are computationally infeasible, the only promising ..."
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Cited by 35 (14 self)
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Cryptanalysis of symmetric and asymmetric ciphers is computationally extremely demanding. Since the security parameters (in particular the key length) of almost all practical crypto algorithms are chosen such that attacks with conventional computers are computationally infeasible, the only promising way to tackle existing ciphers (assuming no mathematical breakthrough) is to build specialpurpose hardware. Dedicating those machines to the task of cryptanalysis holds the promise of a dramatically improved costperformance ratio so that breaking of commercial ciphers comes within reach. This contribution presents the design and realization of the COPACOBANA (CostOptimized Parallel Code Breaker) machine, which is optimized for running cryptanalytical algorithms and can be realized for less than US $ 10,000. It will be shown that, depending on the actual algorithm, the architecture can outperform conventional computers by several orders in magnitude. COPACOBANA hosts 120 lowcost FPGAs and is able to, e.g., perform an exhaustive key search of the Data Encryption Standard (DES) in less than nine days on average. As a realworld application, our architecture can be used to attack machine readable travel documents (ePass). COPACOBANA is intended, but not necessarily restricted to solving problems related to cryptanalysis. The hardware architecture is suitable for computational problems which are parallelizable and have low communication requirements. The hardware can be used, e.g., to attack elliptic curve cryptosystems and to factor numbers. Even though breaking fullsize RSA (1024 bit or more) or elliptic curves (ECC with 160 bit or more) is out of reach with COPACOBANA, it can be used to analyze cryptosystems with a (deliberately chosen) small bitlength to provide reliable security estimates of RSA and ECC by extrapolation.
How to Forge DESEncrypted Messages in 2^28 Steps
, 1996
"... In this paper we suggest keycollision attacks, and show that the theoretic strength of a cipher cannot exceed the square root of the size of the key space. As a result, in some circumstances, some DES keys can be recovered while they are still in use, and these keys can then be used to forge messag ..."
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Cited by 17 (0 self)
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In this paper we suggest keycollision attacks, and show that the theoretic strength of a cipher cannot exceed the square root of the size of the key space. As a result, in some circumstances, some DES keys can be recovered while they are still in use, and these keys can then be used to forge messages: in particular, one key of DES can be recovered with complexity 2 28 , and one key of (threekey) tripleDES can be recovered with complexity 2 84 .
Cryptanalysis with COPACOBANA
 IEEE TRANSACTIONS ON COMPUTERS
, 2008
"... Cryptanalysis of ciphers usually involves massive computations. The security parameters of cryptographic algorithms are commonly chosen so that attacks are infeasible with available computing resources. Thus, in the absence of mathematical breakthroughs to a cryptanalytical problem, a promising way ..."
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Cited by 17 (5 self)
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Cryptanalysis of ciphers usually involves massive computations. The security parameters of cryptographic algorithms are commonly chosen so that attacks are infeasible with available computing resources. Thus, in the absence of mathematical breakthroughs to a cryptanalytical problem, a promising way for tackling the computations involved is to build specialpurpose hardware exhibiting a (much) better performancecost ratio than offtheshelf computers. This contribution presents a variety of cryptanalytical applications utilizing the CostOptimized Parallel Code Breaker (COPACOBANA) machine, which is a highperformance lowcost cluster consisting of 120 fieldprogrammable gate arrays (FPGAs). COPACOBANA appears to be the only such reconfigurable parallel FPGA machine optimized for code breaking tasks reported in the open literature. Depending on the actual algorithm, the parallel hardware architecture can outperform conventional computers by several orders of magnitude. In this work, we will focus on novel implementations of cryptanalytical algorithms, utilizing the impressive computational power of COPACOBANA. We describe various exhaustive key search attacks on symmetric ciphers and demonstrate an attack on a security mechanism employed in the electronic passport (epassport). Furthermore, we describe timememory tradeoff techniques that can, e.g., be used for attacking the popular A5/1 algorithm used in GSM voice encryption. In addition, we introduce efficient implementations of more complex cryptanalysis on asymmetric cryptosystems, e.g., Elliptic Curve Cryptosystems (ECCs) and number cofactorization for RSA. Even though breaking RSA or elliptic curves with parameter lengths used in most practical applications is out of reach with COPACOBANA, our attacks on algorithms with artificially short bit lengths allow us to extrapolate more reliable security estimates for realworld bit lengths. This is particularly useful for deriving estimates about the longevity of asymmetric key lengths.
The full cost of cryptanalytic attacks
 Journal of Cryptology
"... Abstract. An open question about the asymptotic cost of connecting many processors to a large memory using three dimensions for wiring is answered, and this result is used to find the full cost of several cryptanalytic attacks. In many cases this full cost is higher than the accepted complexity of a ..."
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Cited by 15 (0 self)
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Abstract. An open question about the asymptotic cost of connecting many processors to a large memory using three dimensions for wiring is answered, and this result is used to find the full cost of several cryptanalytic attacks. In many cases this full cost is higher than the accepted complexity of a given algorithm based on the number of processor steps. The full costs of several cryptanalytic attacks are determined, including Shanks ’ method for computing discrete logarithms in cyclic groups of prime order n, which requires n 1/2+o(1) processor steps, but when all factors are taken into account, has full cost n 2/3+o(1). Other attacks analyzed are factoring with the number field sieve, generic attacks on block ciphers, attacks on double and triple encryption, and finding hash collisions. In many cases parallel collision search gives a significant asymptotic advantage over wellknown generic attacks.
Exploiting Parallelism in Hardware Implementation of the DES
 Proc. CRYPTO '91
, 1992
"... The Data Encryption Standard algorithm has features which may be used to advantage in parallelizing an implementation. The kernel of the algorithm, a single round, may be decomposed into several parallel computations resulting in a structure with minimal delay. These rounds may also be computed in a ..."
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Cited by 13 (2 self)
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The Data Encryption Standard algorithm has features which may be used to advantage in parallelizing an implementation. The kernel of the algorithm, a single round, may be decomposed into several parallel computations resulting in a structure with minimal delay. These rounds may also be computed in a pipelined parallel structure for operations modes which do not require cryptext feedback. Finally, system I/O may be performed in parallel with the encryption computation for further gain. Although several of these ideas have been discussed before separately, the composite presentation is novel. 1 Introduction 1 The Data Encryption Standard (DES) is probably the most widely used publicly available secretkey algorithm. Since its introduction by the National Bureau of Standards (NBS) in 1977[FIPS46], DES implementations have improved greatly in encryption rate. Yet, typical computer communication rates have also increased significantly during the same period. Today's highperformance comput...