Results 1 
9 of
9
NonParallelizable and NonInteractive Client Puzzles from Modular Square Roots
"... Abstract—Denial of Service (DoS) attacks aiming to exhaust the resources of a server by overwhelming it with bogus requests have become a serious threat. Especially protocols that rely on public key cryptography and perform expensive authentication handshakes may be an easy target. A wellknown coun ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract—Denial of Service (DoS) attacks aiming to exhaust the resources of a server by overwhelming it with bogus requests have become a serious threat. Especially protocols that rely on public key cryptography and perform expensive authentication handshakes may be an easy target. A wellknown countermeasure against DoS attacks are client puzzles. The victimized server demands from the clients to commit computing resources before it processes their requests. To get service, a client must solve a cryptographic puzzle and submit the right solution. Existing client puzzle schemes have some drawbacks. They are either parallelizable, coarsegrained or can be used only interactively. In case of interactive client puzzles where the server poses the challenge an attacker might mount a counterattack on the clients by injecting fake packets containing bogus puzzle parameters. In this paper we introduce a novel scheme for client puzzles which relies on the computation of square roots modulo a prime. Modular square root puzzles are nonparallelizable, i. e., the solution cannot be obtained faster than scheduled by distributing the puzzle to multiple machines or CPU cores, and they can be employed both interactively and noninteractively. Our puzzles provide polynomial granularity and compact solution and verification functions. Benchmark results demonstrate the feasibility of our approach to mitigate DoS attacks on hosts in 1 or even 10 GBit networks. In addition, we show how to raise the efficiency of our puzzle scheme by introducing a bandwidthbased cost factor for the client. Keywords—client puzzles, Denial of Service (DoS), network protocols, authentication, computational puzzles
A HighSpeed Square Root Algorithm in Extension Fields
"... Abstract. A square root (SQRT) algorithm in GF (p m)(m=r0r1 ···rn−12 d, ri: odd prime, d>0: integer) is proposed in this paper. First, the TonelliShanks algorithm is modified to compute the inverse SQRT in GF (p 2d where most of the computations are performed in the corresponding subfields GF (p 2 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. A square root (SQRT) algorithm in GF (p m)(m=r0r1 ···rn−12 d, ri: odd prime, d>0: integer) is proposed in this paper. First, the TonelliShanks algorithm is modified to compute the inverse SQRT in GF (p 2d where most of the computations are performed in the corresponding subfields GF (p 2i) for 0 i d − 1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF (p m) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the TonelliShanks algorithm and the proposed algorithm in GF (p 22), GF (p 44) and GF (p 88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF (p 22), by 45 times in GF (p 44), and by 70 times in GF (p 88), compared to the TonelliShanks algorithm, which is supported by the evaluation of the number of computations. 1
New cube root algorithm based on third order linear recurrence relation in finite field, preprint, available from http://eprint.iacr
"... In this paper, we present a new cube root algorithm in finite field Fq with q a power of prime, which extends the CipollaLehmer type algorithms [4, 5]. Our cube root method is inspired by the work of Müller [8] on quadratic case. For given cubic residue c ∈ Fq with q ≡ 1 (mod 9), we show that there ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In this paper, we present a new cube root algorithm in finite field Fq with q a power of prime, which extends the CipollaLehmer type algorithms [4, 5]. Our cube root method is inspired by the work of Müller [8] on quadratic case. For given cubic residue c ∈ Fq with q ≡ 1 (mod 9), we show that there is an irreducible polynomial f(x) = x 3 − ax 2 + bx − 1 with root α ∈ Fq3 efficient cube root algorithm based on third order linear recurrence sequence arising from f(x). Complexity estimation shows that our algorithm is better than previously proposed CipollaLehmer type algorithms. such that T r(α q2 +q−2 9) is a cube root of c. Consequently we find an
Square Root Algorithm in Fq for q ≡ 2 s + 1 (mod 2 s+1)
"... We present a square root algorithm in Fq which generalizes Atkins’s square root algorithm [6] for q ≡ 5 (mod 8) and Kong et al.’s algorithm [8] for q ≡ 9 (mod 16). Our algorithm precomputes a primitive 2 sth root of unity ξ where s is the largest positive integer satisfying 2 s q − 1, and is appli ..."
Abstract
 Add to MetaCart
We present a square root algorithm in Fq which generalizes Atkins’s square root algorithm [6] for q ≡ 5 (mod 8) and Kong et al.’s algorithm [8] for q ≡ 9 (mod 16). Our algorithm precomputes a primitive 2 sth root of unity ξ where s is the largest positive integer satisfying 2 s q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favorably compared with the algorithms of Atkin, Müller and Kong et al.
On rth Root Extraction Algorithm in Fq For q ≡ lr s + 1 (mod r s+1) with 0 < l < r and Small s
"... We present an rth root extraction algorithm over a finite field Fq. Our algorithm precomputes a primitive r sth root of unity ξ where s is the largest positive integer satisfying r s q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the ..."
Abstract
 Add to MetaCart
We present an rth root extraction algorithm over a finite field Fq. Our algorithm precomputes a primitive r sth root of unity ξ where s is the largest positive integer satisfying r s q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the rth root computation and is favorably compared to the existing algorithms.
Trace Expression of rth Root over Finite Field Email:
"... Efficient computation of rth root in Fq has many applications in computational number theory and many other related areas. We present a new rth root formula which generalizes Müller’s result on square root, and which provides a possible improvement of the CipollaLehmer type algorithms for general ..."
Abstract
 Add to MetaCart
Efficient computation of rth root in Fq has many applications in computational number theory and many other related areas. We present a new rth root formula which generalizes Müller’s result on square root, and which provides a possible improvement of the CipollaLehmer type algorithms for general case. More precisely, for given rth power c ∈ Fq, we show that there exists α ∈ Fqr ( such that T r α (∑r−1 i=0 qi)−r r2)r = c where T r(α) = α + αq + αq2 + · · · + αqr−1 and α is a root of certain irreducible polynomial of degree r over Fq.
Some Connections Between Primitive Roots and Quadratic NonResidues Modulo a Prime
"... Abstract—In this paper we present some interesting connections between primitive roots and quadratic nonresidues modulo a prime. Using these correlations, we propose some polynomial deterministic algorithms for generating primitive roots for primes with special forms (for example, for safe primes). ..."
Abstract
 Add to MetaCart
Abstract—In this paper we present some interesting connections between primitive roots and quadratic nonresidues modulo a prime. Using these correlations, we propose some polynomial deterministic algorithms for generating primitive roots for primes with special forms (for example, for safe primes). Index Terms—primitive roots, LegendreJacobi symbol, quadratic nonresidues, square roots. I.
Modular Square Root Puzzles: Design of NonParallelizable and NonInteractive Client Puzzles
, 2012
"... Denial of Service (DoS) attacks aiming to exhaust the resources of a server by overwhelming it with bogus requests have become a serious threat. Especially protocols that rely on public key cryptography and perform expensive authentication handshakes may be an easy target. A wellknown countermeasur ..."
Abstract
 Add to MetaCart
Denial of Service (DoS) attacks aiming to exhaust the resources of a server by overwhelming it with bogus requests have become a serious threat. Especially protocols that rely on public key cryptography and perform expensive authentication handshakes may be an easy target. A wellknown countermeasure against resource depletion attacks are client puzzles. The victimized server demands from the clients to commit computing resources before it processes their requests. To get service, a client must solve a cryptographic puzzle and submit the right solution. Existing client puzzle schemes have some drawbacks. They are either parallelizable, coarsegrained or can be used only interactively. In case of interactive client puzzles where the server poses the challenge an attacker might mount a counterattack on the clients by injecting faked packets with bogus puzzle parameters bearing the server’s sender address. In this paper we introduce a novel scheme for client puzzles which relies on the computation of square roots modulo a prime. Modular square root puzzles are nonparallelizable, i. e., the solution cannot be obtained faster than scheduled by distributing the puzzle to multiple machines or CPU cores, and they can be employed both interactively and noninteractively. Our puzzles provide polynomial granularity and compact solution and verification functions. Benchmark results demonstrate the feasibility of our approach to mitigate DoS attacks on hosts in 1 or even 10 Gbit networks. In addition, we show how to raise the efficiency of our puzzle scheme by introducing a bandwidthbased cost factor for the client. Furthermore, we also investigate the construction of client puzzles from modular cube roots.