Finding small roots of univariate modular equations revisited

1997 ◽  
pp. 131-142 ◽  
Author(s):  
Nicholas Howgrave-Graham
Keyword(s):  
Modular Equations ◽  
Small Roots

scholarly journals A Unified Method for Private Exponent Attacks on RSA Using Lattices

2020 ◽  
Vol 31 (02) ◽  
pp. 207-231
Author(s):  
Hatem M. Bahig ◽  
Dieaa I. Nassr ◽  
Ashraf Bhery ◽  
Abderrahmane Nitaj
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Lattice Reduction ◽  
Public Key ◽  
Approximation Formula ◽  
Modular Equations ◽  
Key Equation ◽  
Small Roots ◽  
Unified Method

Let [Formula: see text] be an RSA public key with private exponent [Formula: see text] where [Formula: see text] and [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] provided that we have approximation [Formula: see text] of [Formula: see text] with [Formula: see text] [Formula: see text] The attack is an extension of Coppersmith’s result.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

Water transport in European Beech roots (<5mm) under drought in the south of Lower Saxony

2021 ◽  
Author(s):  
Eva Messinger ◽  
Heinz Coners ◽  
Dietrich Hertel ◽  
Christoph Leuschner
Keyword(s):  
Soil Moisture ◽  
Water Transport ◽  
Water Uptake ◽  
Sap Flow ◽  
Climate Models ◽  
European Beech ◽  
Root Surface Area ◽  
Lower Saxony ◽  
Tree Roots ◽  
Small Roots

&lt;p&gt;Climate models predict hotter and dryer summers in Germany, with longer periods of extreme droughts like in summer 2018. How does this affect the water uptake and transport in tree roots growing in the top- and subsoil?&lt;/p&gt;&lt;p&gt;In summer 2018 and 2019 we measured the water transport in fine roots (&lt;5mm) of European Beech on tertiary sand and triassic sandstone up to 2 m depth. We adapted the well-established HRM technique to enable measurements of very small sap flow rates in small roots. Thus, we measured the water transport as a temperature ratio of a stretching heat pulse.&lt;/p&gt;&lt;p&gt;Relating sap flow to root surface area, root depth, anatomy, soil moisture, and VPD allows for interesting insights in tree water uptake rates: Where are the limits of drought intensity and duration, for water uptake and recovery of small roots? Are there differences in the function of top- and subsoil roots? Are roots specialized for water transport or nutrient uptake? The investigated data gives a first hint on how the water transport in Beech roots differs with changes in the soil moisture and VPD under changing climate.&lt;/p&gt;

Belowground morphology and population dynamics of two forest understory herbs of contrasting growth form

Botany ◽  
2021 ◽  
Author(s):  
Joseph A. Antos ◽  
Donald B. Zobel ◽  
Dylan Fischer
Keyword(s):  
Pacific Northwest ◽  
Large Range ◽  
Forest Understory ◽  
Old Growth Forest ◽  
Forest Herbs ◽  
Understory Herbs ◽  
Small Roots ◽  
Chimaphila Umbellata ◽  
Range Of Variation ◽  
Xerophyllum Tenax

Forest understory herbs exhibit a large range of variation in morphology and life history. Here we expand the reported range of variation by describing the belowground structures of two very different species, Xerophyllum tenax and Chimaphila umbellata. We excavated individuals in forests of the Cascade Mountains, Pacific Northwest, USA. Xerophyllum tenax has short rhizomes, but an extensive root system that is exceptionally large among forest understory species. The roots reach 4 m in length and may occupy an area 50 times that of the aboveground canopy. In contrast, Chimaphila umbellata has very small roots, but an extensive rhizome system. The largest plant we excavated had 57 m of connected rhizomes and still had a seedling source. Both species have long-lived individuals but differ in response to disturbance. Based on monitoring of 151 permanent 1 m2 plots in an old-growth forest, X. tenax increased only minimally in density over 40 years following tephra deposition from the 1980 eruption of Mount St. Helens, whereas density of C. umbellata increased substantially. The very different morphology of these two species highlights the large range of variation found among forest herbs, which needs to be considered when examining the forest understory.

Sign in / Sign up

Export Citation Format

Share Document