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| | If you were to picture the idea body proportions plus form you've probably got an image of wide shoulders and a narrow waist (or at least you SHOULD have this image inside a head). But what we may not be concentrating in on is the exact size of your shoulder compared to your waist.<br><br>Balance- Like flexibility, balance is additionally an significant element in wise fitness. An total healthy body relies heavily on being perfectly balanced, and the risk or injury and broken bones from falls increases drastically with age. To asses the fitness level inside this region, try standing on one foot with a arms at your waist hip ratio sides for a period of one minute. If you feel as should you might fall, stand close to a wall, table or chair. Work on improving fitness degrees inside balance, try practicing exercises that focus on and promote superior balance like yoga or Pilates.<br><br>To this end we will need to do several exercises. It is significant to do some aerobic exercises inside purchase to improve your metabolism. You must equally consider starting a weight training system because this can help waist hip ratio calculator you to build more muscle and lose belly fat a lot faster. In most cases you will want to have several exercises for about 30 to 40 minute each day.<br><br>At the additional side of the scale, this Real Woman movement is questioning the femininity of millions. While its doubtful that Victoria Beckham plus Kate Moss can receive much sympathy in this debate, what of the women inside developing countries? Many are starving and might therefore has the low body fat needed to fit a size zero; this suggests which they cannot potentially be Real Women appropriate? Needless to say not, Real Women have hips and breasts; they dont starve right down to ridiculously unattainable sizes.<br><br>You can additionally walk up hills. However I can't snap my fingers plus put a hill outside your house. So the initially 2 exercises are more calorie plus fat burners that annihilate belly fat by stealing it for power.<br><br>Even more exact than the scale and BMI is your [http://safedietplansforwomen.com/waist-to-hip-ratio waist hip ratio]. This may provide we a general idea of where you stand at the moment and how far we have to go, without offering we an actual number. Just measure the hips and your waist plus then divide the hip measurement by the waist. Ideally, females should aim for around 0.80 plus guys for about 0.95.<br><br>This logic is clearly self-contradictory, as countless folks absolutely know plus as a growing number are coming to realise; not each woman above a size zero has curvy hips or breasts. Case inside point, Beth Ditto (singer inside the band Gossip), Beth is clearly a greater woman and due to her brief stature is presumably clinically obese; when you were to assume that all women above a zero are Real then Beth would fit into this category, be watched as a positive character model and be praised for her figure worldwide. But it you were to use the more recent description, Beth isnt Real at all as her waist-hip ratio appears to be quite significant.<br><br>Along with these exercises, you need to go for usual cardiovascular exercises plus a balanced diet. Another important thing to remember is to do these exercises regularly. Regularity is the key to success. Happy working out! |
| In [[mathematics]], a '''discrete logarithm''' is an integer ''k'' solving the equation {{nowrap|1=''b''<sup>''k''</sup> = ''g''}}, where ''b'' and ''g'' are elements of a [[Group (mathematics)|group]]. Discrete logarithms are thus the group-theoretic analogue of ordinary [[logarithm]]s, which solve the same equation for [[real number]]s ''b'' and ''g'', where ''b'' is the base of the logarithm and ''g'' is the value whose logarithm is being taken.
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| Computing discrete logarithms is believed to be difficult. No efficient general method for computing discrete logarithms on conventional computers is known, and several important algorithms in [[public-key cryptography]] base their security on the assumption that the discrete logarithm problem has no efficient solution.
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| == Example ==
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| Discrete logarithms are perhaps simplest to understand in the group [[Multiplicative group of integers modulo n|('''Z'''<sub>''p''</sub>)<sup>×</sup>]]. This is the group of multiplication [[modular arithmetic|modulo]] the [[prime number|prime]] ''p''. Its elements are [[congruence class]]es modulo ''p'', and the group product of two elements may be obtained by ordinary integer multiplication of the elements followed by reduction modulo ''p''.
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| The ''k''th [[exponentiation|power]] of one of the numbers in this group may be computed by finding its ''k''th power as an integer and then finding the remainder after division by ''p''. This process is called [[modular exponentiation]]. For example, consider ('''Z'''<sub>17</sub>)<sup>×</sup>. To compute 3<sup>4</sup> in this group, compute 3<sup>4</sup> = 81, and then divide 81 by 17, obtaining a remainder of 13. Thus 3<sup>4</sup> = 13 in the group ('''Z'''<sub>17</sub>)<sup>×</sup>.
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| The discrete logarithm is just the inverse operation. For example, consider the equation 3<sup>''k''</sup> ≡ 13 (mod 17) for ''k''. From the example above, one solution is ''k'' = 4, but it is not the only solution. Since 3<sup>16</sup> ≡ 1 (mod 17) — as follows from [[Fermat's little theorem]] — it also follows that if ''n'' is an integer then 3<sup>4+16''n''</sup> ≡ 3<sup>4</sup> × (3<sup>16</sup>)<sup>''n''</sup> ≡ 13 × 1<sup>''n''</sup> ≡ 13 (mod 17). Hence the equation has infinitely many solutions of the form 4 + 16''n''. Moreover, since 16 is the smallest positive integer ''m'' satisfying 3<sup>''m''</sup> ≡ 1 (mod 17), i.e. 16 is the [[Multiplicative order|order]] of 3 in ('''Z'''<sub>17</sub>)<sup>×</sup>, these are the only solutions. Equivalently, the set of all possible solutions can be expressed by the constraint that ''k'' ≡ 4 (mod 16).
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| == Definition ==
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| In general, let ''G'' be any group, with its group operation denoted by multiplication. Let ''b'' and ''g'' be any elements of ''G''. Then any integer ''k'' that solves ''b''<sup>''k''</sup> = ''g'' is termed a '''discrete logarithm''' (or simply '''logarithm''', in this context) of ''g'' to the base ''b''. We write ''k'' = log<sub>''b''</sub> ''g''. Depending on ''b'' and ''g'', it is possible that no discrete logarithm exists, or that more than one discrete logarithm exists. Let ''H'' be the [[subgroup]] of ''G'' [[generating set of a group|generated]] by ''b''. Then ''H'' is a [[cyclic group]], and log<sub>''b''</sub> ''g'' exists for all ''g'' in ''H''. If ''H'' is infinite, then log<sub>''b''</sub> ''g'' is also unique, and the discrete logarithm amounts to a [[group isomorphism]]
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| :<math>\log_b \colon H \rightarrow \mathbf{Z}.</math>
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| On the other hand, if ''H'' is finite of size ''n'', then log<sub>''b''</sub> ''g'' is unique only up to congruence modulo ''n'', and the discrete logarithm amounts to a group isomorphism
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| :<math>\log_b\colon H \rightarrow \mathbf{Z}_n,</math>
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| where '''Z'''<sub>''n''</sub> denotes the [[ring (algebra)|ring]] of integers modulo ''n''. The familiar base change formula for ordinary logarithms remains valid: If ''c'' is another generator of ''H'', then
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| :<math>\log_c (g) = \log_c (b) \cdot \log_b (g).</math>
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| == Algorithms ==
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| {{See also|Discrete logarithm records}}
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| {{unsolved|computer science|Can the discrete logarithm be computed in polynomial time on a classical computer?}}
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| No efficient classical algorithm for computing general discrete logarithms log<sub>''b''</sub> ''g'' is known. The naive algorithm is to raise ''b'' to higher and higher powers ''k'' until the desired ''g'' is found; this is sometimes called ''trial multiplication''. This algorithm requires [[running time]] linear in the size of the group ''G'' and thus exponential in the number of digits in the size of the group. There exists an efficient quantum algorithm due to [[Peter Shor]].<ref>{{cite journal |arxiv=quant-ph/9508027 |title=Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer |first=Peter |last=Shor |journal=SIAM Journal on Computing |volume=26 |issue=5 |year=1997 |pages=1484–1509 }}</ref>
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| More sophisticated algorithms exist, usually inspired by similar algorithms for integer factorization. These algorithms run faster than the naive algorithm, some of them linear in the ''square root'' of the size of the group, and thus exponential in half the number of digits in the size of the group. However none of them runs in [[polynomial time]] (in the number of digits in the size of the group).
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| * [[Baby-step giant-step]]
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| * [[Pollard's rho algorithm for logarithms]]
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| * [[Pollard's kangaroo algorithm]] (aka Pollard's lambda algorithm)
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| * [[Pohlig–Hellman algorithm]]
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| * [[Index calculus algorithm]]
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| * [[Number field sieve]]
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| * [[Function field sieve]]
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| == Comparison with integer factorization ==
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| While computing discrete logarithms and [[integer factorization|factoring integers]] are distinct problems, they share some properties:
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| *both problems are difficult (no efficient [[algorithm]]s are known for non-[[quantum computer]]s),
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| *for both problems efficient algorithms on quantum computers are known,
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| *algorithms from one problem are often adapted to the other, and
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| *the difficulty of both problems has been used to construct various [[cryptography|cryptographic]] systems.
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| == Cryptography ==
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| There exist groups for which computing discrete logarithms is apparently difficult. In some cases (e.g. large prime order subgroups of groups ('''Z'''<sub>''p''</sub>)<sup>×</sup>) there is not only no efficient algorithm known for the worst case, but the [[average-case complexity]] can be shown to be about as hard as the worst case using [[random self-reducibility]].
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| At the same time, the inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using [[exponentiation by squaring]], for example). This asymmetry is analogous to the one between integer factorization and integer [[multiplication]]. Both asymmetries have been exploited in the construction of cryptographic systems.
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| Popular choices for the group ''G'' in discrete logarithm [[cryptography]] are the cyclic groups ('''Z'''<sub>''p''</sub>)<sup>×</sup> (e.g. [[ElGamal encryption]], [[Diffie–Hellman key exchange]], and the [[Digital Signature Algorithm]]) and cyclic subgroups of [[elliptic curve]]s over [[finite field]]s (''see'' [[elliptic curve cryptography]]).
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| ==References==
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| {{Reflist}}
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| * [[Richard Crandall]]; [[Carl Pomerance]]. Chapter 5, ''Prime Numbers: A computational perspective'', 2nd ed., Springer.
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| * {{Citation | last1=Stinson | first1=Douglas Robert | title=Cryptography: Theory and Practice | publisher=[[CRC Press]] | location=London | edition=3rd | isbn=978-1-58488-508-5 | year=2006}}
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| {{Number theoretic algorithms}}
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| {{DEFAULTSORT:Discrete Logarithm}}
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| [[Category:Modular arithmetic]]
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| [[Category:Group theory]]
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| [[Category:Cryptography]]
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| [[Category:Logarithms]]
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| [[Category:Finite fields]]
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| [[Category:Binary operations]]
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| [[Category:Computational hardness assumptions]]
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| [[Category:Unsolved problems in computer science]]
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If you were to picture the idea body proportions plus form you've probably got an image of wide shoulders and a narrow waist (or at least you SHOULD have this image inside a head). But what we may not be concentrating in on is the exact size of your shoulder compared to your waist.
Balance- Like flexibility, balance is additionally an significant element in wise fitness. An total healthy body relies heavily on being perfectly balanced, and the risk or injury and broken bones from falls increases drastically with age. To asses the fitness level inside this region, try standing on one foot with a arms at your waist hip ratio sides for a period of one minute. If you feel as should you might fall, stand close to a wall, table or chair. Work on improving fitness degrees inside balance, try practicing exercises that focus on and promote superior balance like yoga or Pilates.
To this end we will need to do several exercises. It is significant to do some aerobic exercises inside purchase to improve your metabolism. You must equally consider starting a weight training system because this can help waist hip ratio calculator you to build more muscle and lose belly fat a lot faster. In most cases you will want to have several exercises for about 30 to 40 minute each day.
At the additional side of the scale, this Real Woman movement is questioning the femininity of millions. While its doubtful that Victoria Beckham plus Kate Moss can receive much sympathy in this debate, what of the women inside developing countries? Many are starving and might therefore has the low body fat needed to fit a size zero; this suggests which they cannot potentially be Real Women appropriate? Needless to say not, Real Women have hips and breasts; they dont starve right down to ridiculously unattainable sizes.
You can additionally walk up hills. However I can't snap my fingers plus put a hill outside your house. So the initially 2 exercises are more calorie plus fat burners that annihilate belly fat by stealing it for power.
Even more exact than the scale and BMI is your waist hip ratio. This may provide we a general idea of where you stand at the moment and how far we have to go, without offering we an actual number. Just measure the hips and your waist plus then divide the hip measurement by the waist. Ideally, females should aim for around 0.80 plus guys for about 0.95.
This logic is clearly self-contradictory, as countless folks absolutely know plus as a growing number are coming to realise; not each woman above a size zero has curvy hips or breasts. Case inside point, Beth Ditto (singer inside the band Gossip), Beth is clearly a greater woman and due to her brief stature is presumably clinically obese; when you were to assume that all women above a zero are Real then Beth would fit into this category, be watched as a positive character model and be praised for her figure worldwide. But it you were to use the more recent description, Beth isnt Real at all as her waist-hip ratio appears to be quite significant.
Along with these exercises, you need to go for usual cardiovascular exercises plus a balanced diet. Another important thing to remember is to do these exercises regularly. Regularity is the key to success. Happy working out!