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| | The treadmill provides we absolutely awesome weight reduction workouts. Currently most fitness treadmills equipped with a functioning computer offer fat loss or weight loss programs.<br><br>Leafy veggies are excellent inside fiber, low inside calories, packed full of compounds, and around half their digestible calories come from protein. Eating at least 10 servings of greens a day may replace a great deal of the high calorie foods in your diet, however leave you simply because full. If you require to lose weight fast, eat the greens. If you have to lose weight faster, eat MORE greens. I very recommend drinking a green smoothie for breakfast every morning.<br><br>First, think about the difficult truth. How could you lose weight? The just method to do this is by burning more calories than we eat. Many health experts state that how to approach this radical thought is by simply shaving off 500 calories from your diet every day. That sounds doable, appropriate?<br><br>Eat plenty of raw veggies. Start a meal with lots of salad having carrots, peppers, celery, broccoli cucumber and cabbage. Eating raw vegetables may give we a feeling of fullness, causing we to eat less at standard food.<br><br>After a healthy lifestyle is crucial for preserving a healthy weight. Our body requirements rest to rejuvenate and so does the mind. Make sure you get enough rest, follow the old adage 'early to bed plus early to rise makes a man healthy, rich plus [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight] wise'. If your schedule permits you, take a force nap of 10 - 15 minutes in the afternoon, this might keep a vitality degrees excellent. A word of caution here is to not exceed a force naps, otherwise you will feel sleepy. Supplement your habit of drinking soda with fruit juices or better yet with water. Avoid processed plus sugary snacks like candies, cookies, rather choose snacks that are high inside fiber and important compounds.<br><br>These meals arrive as single servings so you eat just the appropriate amount to remain inside the calorie count. All you need to do is warm the meal in the microwave plus enjoy. Along with removing temptation, diet plans also remove the difficulty associated with meal planning and buying when you go on a diet. Forget all complicated stuff. Just order the meals that appeal to we plus they may arrive at your door. If you don't wish to pick plus choose your meals you can even let the business choose the food for we so you have a balanced diet.<br><br>All in every, this really is a quite fast weight loss so you need to be willing to accept the possibility which 2 weeks from now you can not be 10 lbs lighter. However, should you choose the right fat loss plan plus stick with it, we will see swiftly plus continuous results. |
| 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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The treadmill provides we absolutely awesome weight reduction workouts. Currently most fitness treadmills equipped with a functioning computer offer fat loss or weight loss programs.
Leafy veggies are excellent inside fiber, low inside calories, packed full of compounds, and around half their digestible calories come from protein. Eating at least 10 servings of greens a day may replace a great deal of the high calorie foods in your diet, however leave you simply because full. If you require to lose weight fast, eat the greens. If you have to lose weight faster, eat MORE greens. I very recommend drinking a green smoothie for breakfast every morning.
First, think about the difficult truth. How could you lose weight? The just method to do this is by burning more calories than we eat. Many health experts state that how to approach this radical thought is by simply shaving off 500 calories from your diet every day. That sounds doable, appropriate?
Eat plenty of raw veggies. Start a meal with lots of salad having carrots, peppers, celery, broccoli cucumber and cabbage. Eating raw vegetables may give we a feeling of fullness, causing we to eat less at standard food.
After a healthy lifestyle is crucial for preserving a healthy weight. Our body requirements rest to rejuvenate and so does the mind. Make sure you get enough rest, follow the old adage 'early to bed plus early to rise makes a man healthy, rich plus lose weight wise'. If your schedule permits you, take a force nap of 10 - 15 minutes in the afternoon, this might keep a vitality degrees excellent. A word of caution here is to not exceed a force naps, otherwise you will feel sleepy. Supplement your habit of drinking soda with fruit juices or better yet with water. Avoid processed plus sugary snacks like candies, cookies, rather choose snacks that are high inside fiber and important compounds.
These meals arrive as single servings so you eat just the appropriate amount to remain inside the calorie count. All you need to do is warm the meal in the microwave plus enjoy. Along with removing temptation, diet plans also remove the difficulty associated with meal planning and buying when you go on a diet. Forget all complicated stuff. Just order the meals that appeal to we plus they may arrive at your door. If you don't wish to pick plus choose your meals you can even let the business choose the food for we so you have a balanced diet.
All in every, this really is a quite fast weight loss so you need to be willing to accept the possibility which 2 weeks from now you can not be 10 lbs lighter. However, should you choose the right fat loss plan plus stick with it, we will see swiftly plus continuous results.