1,029 research outputs found
Improved Factoring Attacks on Multi-Prime RSA with Small Prime Difference
In this paper, we study the security of multi-prime RSA with small prime difference and propose two improved factoring attacks. The modulus involved in this variant is the product of r distinct prime factors of the same bit-size. Zhang and Takagi (ACISP 2013) showed a Fermat-like factoring attack on multi-prime RSA. In order to improve the previous result, we gather more information about the prime factors to derive r simultaneous modular equations. The first attack is to combine all the equations and solve one multivariate equation by generic lattice approaches. Since the equation form is similar to multi-prime Phi-hiding problem, we propose the second attack by applying the optimal linearization technique. We also show that our attacks can achieve better bounds in the experiments
Public key exponent attacks on multi-prime power modulus using continued fraction expansion method
This paper proposes three public key exponent attacks of breaking the security of the prime power modulus =22 where and are distinct prime numbers of the same bit size. The first approach shows that the RSA prime power modulus =22 for q<<2q using key equation −()=1 where ()= 22(−1)(−1) can be broken by recovering the secret keys / from the convergents of the continued fraction expansion of e/−23/4 +1/2 . The paper also reports the second and third approaches of factoring multi-prime power moduli =2 2 simultaneously through exploiting generalized system of equations −()=1 and −()=1 respectively. This can be achieved in polynomial time through utilizing Lenstra Lenstra Lovasz (LLL) algorithm and simultaneous Diophantine approximations method for =1,2,…,
A Unified Method for Private Exponent Attacks on RSA using Lattices
International audienceLet (n = pq, e = n^β) be an RSA public key with private exponent d = n^δ , where p and q are large primes of the same bit size. At Eurocrypt 96, Coppersmith presented a polynomial-time algorithm for finding small roots of univariate modular equations based on lattice reduction and then succussed to factorize the RSA modulus. Since then, a series of attacks on the key equation ed − kφ(n) = 1 of RSA have been presented. In this paper, we show that many of such attacks can be unified in a single attack using a new notion called Coppersmith's interval. We determine a Coppersmith's interval for a given RSA public key (n, e). The interval is valid for any variant of RSA, such as Multi-Prime RSA, that uses the key equation. Then we show that RSA is insecure if δ < β + 1/3 α − 1/3 √ (12αβ + 4α^2) provided that we have approximation p0 ≥ √ n of p with |p − p0| ≤ 1/2 n^α , α ≤ 1/2. The attack is an extension of Coppersmith's result
Improved Results on Factoring General RSA Moduli with Known Bits
We revisit the factoring with known bits problem on general RSA moduli in the forms of for , where two primes and are of the same bit-size. The relevant moduli are inclusive of , for , and for , which are used in the standard RSA scheme and other RSA-type variants. Previous works acquired the results mainly by solving univariate modular equations.
In contrast, we investigate how to efficiently factor with given leakage of the primes by the integer method using the lattice-based technique in this paper. More precisely, factoring general RSA moduli with known most significant bits (MSBs) of the primes can be reduced to solving bivariate integer equations, which was first proposed by Coppersmith to factor with known high bits. Our results provide a unifying solution to the factoring with known bits problem on general RSA moduli. Furthermore, we reveal that there exists an improved factoring attack via the integer method for particular RSA moduli like and
An analysis of key generation efficiency of RSA cryptosystem in distributed environments
Thesis (Master)--Izmir Institute of Technology, Computer Engineering, Izmir, 2005Includes bibliographical references (leaves: 68)Text in English Abstract: Turkish and Englishix, 74 leavesAs the size of the communication through networks and especially through Internet grew, there became a huge need for securing these connections. The symmetric and asymmetric cryptosystems formed a good complementary approach for providing this security. While the asymmetric cryptosystems were a perfect solution for the distribution of the keys used by the communicating parties, they were very slow for the actual encryption and decryption of the data flowing between them. Therefore, the symmetric cryptosystems perfectly filled this space and were used for the encryption and decryption process once the session keys had been exchanged securely. Parallelism is a hot research topic area in many different fields and being used to deal with problems whose solutions take a considerable amount of time. Cryptography is no exception and, computer scientists have discovered that parallelism could certainly be used for making the algorithms for asymmetric cryptosystems go faster and the experimental results have shown a good promise so far. This thesis is based on the parallelization of a famous public-key algorithm, namely RSA
On the Security of Some Variants of RSA
The RSA cryptosystem, named after its inventors, Rivest, Shamir and Adleman, is the most widely known and widely used public-key cryptosystem in the world today. Compared to other public-key cryptosystems, such as
elliptic curve cryptography, RSA requires longer keylengths and is computationally more expensive. In order to address these shortcomings, many variants of RSA have been proposed over the years. While the security
of RSA has been well studied since it was proposed in 1977, many of these variants have not. In this thesis, we investigate the security of five of these variants of RSA. In particular, we provide detailed analyses of the best known algebraic attacks (including some new attacks) on instances of
RSA with certain special private exponents, multiple instances of RSA sharing a common small private exponent, Multi-prime RSA, Common Prime RSA and Dual RSA
Generalised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and
feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve
cryptography. Their form is such that modular reduction is extremely efficient,
thus making them an attractive choice for modular multiplication
implementation. However, the issue of residue multiplication efficiency seems
to have been overlooked. Asymptotically, using a cyclic rather than a linear
convolution, residue multiplication modulo a Mersenne number is twice as fast
as integer multiplication; this property does not hold for prime GMNs, unless
they are of Mersenne's form. In this work we exploit an alternative
generalisation of Mersenne numbers for which an analogue of the above property
--- and hence the same efficiency ratio --- holds, even at bitlengths for which
schoolbook multiplication is optimal, while also maintaining very efficient
reduction. Moreover, our proposed primes are abundant at any bitlength, whereas
GMNs are extremely rare. Our multiplication and reduction algorithms can also
be easily parallelised, making our arithmetic particularly suitable for
hardware implementation. Furthermore, the field representation we propose also
naturally protects against side-channel attacks, including timing attacks,
simple power analysis and differential power analysis, which is essential in
many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio
Pendekatan konstruktif dalam inovasi pengajaran dan pembelajaran Bahasa Melayu di Kolej Vokasional
Pendekatan konstruktif adalah pendekatan pengajaran dan pembelajaran yang
berpusatkan pelajar manakala inovasi pengajaran pula dikaitkan dengan kaedah
pengajaran yang terbaru demi mengukuhkan pemahaman pelajar. Pembelajaran
berasaskan pendekatan konstruktif merupakan elemen yang penting dan perlu
difahami oleh guru-guru bagi memantapkan proses pengajaran dan pembelajaran
sesuai dengan peredaran masa dan menjayakan proses tranformasi pendidikan
negara. Objektif kajian ini dijalankan untuk mengenal pasti pemahaman guru-guru
bahasa Melayu berkaitan inovasi, mengenal pasti perbezaan yang wujud antara guru
lelaki dan guru perempuan dalam mengamalkan inovasi, pengkaji juga melihat
adakah wujud perbezaan antara guru baru dan guru yang sudah berpengalaman
dalam aspek mengaplikasikan inovasi serta mengenal pasti kekangan-kekangan yang
dialami oleh para guru untuk mengaplikasikan inovasi di sekolah. Seramai 63 orang
guru bahasa Melayu dari lapan buah kolej vokasional telah dipilih sebagai responden
dalam kajian ini. Data dianalisis menggunakan perisian Winsteps 3.69.1.11 dengan
pendekatan Model Pengukuran Rasch. Hasil analisis menunjukkan bahawa guru�guru bahasa Melayu memahami kepentingan inovasi dalam pengajaran dan
pembelajaran. Hasil kajian juga menunjukkan guru-guru perempuan lebih banyak
menerapkan unsur inovasi dalam pengajaran berbanding guru lelaki. Walaupun
begitu, aspek pengalaman tidak menunjukkan perbezaan dari segi pengamalan
inovasi sama ada guru baru ataupun guru yang sudah berpengalaman. Pengkaji juga
mengenal pasti beberapa kekangan yang dialami oleh guru-guru untuk mengamalkan
inovasi ini. Sebagai langkah untuk menangani masalah berkenaan, beberapa
cadangan telah dikemukakan oleh pengkaji bagi memastikan guru-guru dapat
merealisasikan proses pengajaran berkesan dengan penerapan inovasi mengikut
model pendekatan konstruktif. Pengkaji berharap, kajian ini dapat dijadikan sebagai
satu panduan kepada pelaksana kurikulum bagi memastikan budaya inovasi sentiasa
menjadi amalan dalam kalangan guru demi mengangkat profesionalisme guru di
Malaysia
The Interpolating Random Spline Cryptosystem and the Chaotic-Map Public-Key Cryptosystem
The feasibility of implementing the interpolating cubic spline function as encryption and decryption transformations is presented. The encryption method can be viewed as computing a transposed polynomial. The main characteristic of the spline cryptosystem is that the domain and range of encryption are defined over real numbers, instead of the traditional integer numbers. Moreover, the spline cryptosystem can be implemented in terms of inexpensive multiplications and additions.
Using spline functions, a series of discontiguous spline segments can execute the modular arithmetic of the RSA system. The similarity of the RSA and spline functions within the integer domain is demonstrated. Furthermore, we observe that such a reformulation of RSA cryptosystem can be characterized as polynomials with random offsets between ciphertext values and plaintext values. This contrasts with the spline cryptosystems, so that a random spline system has been developed. The random spline cryptosystem is an advanced structure of spline cryptosystem. Its mathematical indeterminacy on computing keys with interpolants no more than 4 and numerical sensitivity to the random offset t( increases its utility.
This article also presents a chaotic public-key cryptosystem employing a one-dimensional difference equation as well as a quadratic difference equation. This system makes use of the El Gamal’s scheme to accomplish the encryption process. We note that breaking this system requires the identical work factor that is needed in solving discrete logarithm with the same size of moduli
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