The Discrete Logarithm Problem on non-hyperelliptic Curves of Genus g 4
|
|
- Walter Moody
- 7 years ago
- Views:
Transcription
1 on Curves The Discrete Logarithm Problem on non-hyperelliptic Curves of Genus g 4 Sebastian Kochinke November 20, 2014 Sebastian Kochinke The Discrete Logarithm Problem on Curves 1 / 32
2 on Curves 1 The Discrete Logarithm Double Large Prime Variation 2 on Curves 3 Sebastian Kochinke The Discrete Logarithm Problem on Curves 2 / 32
3 on Curves The Discrete Logarithm Double Large Prime Variation The Discrete Logarithm Sebastian Kochinke The Discrete Logarithm Problem on Curves 3 / 32
4 on Curves The Discrete Logarithm Double Large Prime Variation Definition Let (G, ) be a finite group, and let a, b G with b < a >. The discrete logarithm of b with respect to a is the smallest non-negative integer e with a e = b. Sebastian Kochinke The Discrete Logarithm Problem on Curves 4 / 32
5 on Curves The Discrete Logarithm Double Large Prime Variation Definition Let (G, ) be a finite group, and let a, b G with b < a >. The discrete logarithm of b with respect to a is the smallest non-negative integer e with a e = b. The task to find e given G, a, b is called the Discrete Logarithm Problem. Sebastian Kochinke The Discrete Logarithm Problem on Curves 4 / 32
6 on Curves The Discrete Logarithm Double Large Prime Variation Definition Let (G, ) be a finite group, and let a, b G with b < a >. The discrete logarithm of b with respect to a is the smallest non-negative integer e with a e = b. The task to find e given G, a, b is called the Discrete Logarithm Problem. Examples for G are: F q Elliptic Curves Pic 0 (C) for some curve C of higher genus Sebastian Kochinke The Discrete Logarithm Problem on Curves 4 / 32
7 on Curves The Discrete Logarithm Double Large Prime Variation Definition Let (G, ) be a finite group, and let a, b G with b < a >. The discrete logarithm of b with respect to a is the smallest non-negative integer e with a e = b. The task to find e given G, a, b is called the Discrete Logarithm Problem. Examples for G are: F q Elliptic Curves Pic 0 (C) for some curve C of higher genus We will focus on G = Pic 0 (C). Sebastian Kochinke The Discrete Logarithm Problem on Curves 4 / 32
8 on Curves The Discrete Logarithm Double Large Prime Variation Sebastian Kochinke The Discrete Logarithm Problem on Curves 5 / 32
9 on Curves The Discrete Logarithm Double Large Prime Variation Write the group law in G additively and assume ord(a) is known. Sebastian Kochinke The Discrete Logarithm Problem on Curves 6 / 32
10 on Curves The Discrete Logarithm Double Large Prime Variation Write the group law in G additively and assume ord(a) is known. Then we have the following basic Algorithm: 1 Fix a so-called factor base F := {a 1,... a n } G. Sebastian Kochinke The Discrete Logarithm Problem on Curves 6 / 32
11 on Curves The Discrete Logarithm Double Large Prime Variation Write the group law in G additively and assume ord(a) is known. Then we have the following basic Algorithm: 1 Fix a so-called factor base F := {a 1,... a n } G. 2 Find (n + 1) relations j r i,j a j = α i a + β i b for r i,j, α i, β i (Z/ord(a)Z). Sebastian Kochinke The Discrete Logarithm Problem on Curves 6 / 32
12 on Curves The Discrete Logarithm Double Large Prime Variation Write the group law in G additively and assume ord(a) is known. Then we have the following basic Algorithm: 1 Fix a so-called factor base F := {a 1,... a n } G. 2 Find (n + 1) relations j r i,j a j = α i a + β i b for r i,j, α i, β i (Z/ord(a)Z). 3 Let R := (r i,j ), α := (α i ), β := (β i ). Sebastian Kochinke The Discrete Logarithm Problem on Curves 6 / 32
13 on Curves The Discrete Logarithm Double Large Prime Variation Write the group law in G additively and assume ord(a) is known. Then we have the following basic Algorithm: 1 Fix a so-called factor base F := {a 1,... a n } G. 2 Find (n + 1) relations j r i,j a j = α i a + β i b for r i,j, α i, β i (Z/ord(a)Z). 3 Let R := (r i,j ), α := (α i ), β := (β i ). 4 Find γ (Z/ord(a)Z) 1 (n+1) such that γr = 0. Sebastian Kochinke The Discrete Logarithm Problem on Curves 6 / 32
14 on Curves The Discrete Logarithm Double Large Prime Variation Write the group law in G additively and assume ord(a) is known. Then we have the following basic Algorithm: 1 Fix a so-called factor base F := {a 1,... a n } G. 2 Find (n + 1) relations j r i,j a j = α i a + β i b for r i,j, α i, β i (Z/ord(a)Z). 3 Let R := (r i,j ), α := (α i ), β := (β i ). 4 Find γ (Z/ord(a)Z) 1 (n+1) such that γr = 0. 5 If i γ iβ i (Z/ord(a)Z) then e := ( i γ iα i )( i γ iβ i ) 1 is the discrete logarithm of b with respect to a. Sebastian Kochinke The Discrete Logarithm Problem on Curves 6 / 32
15 on Curves The Discrete Logarithm Double Large Prime Variation Double Large Prime Variation Sebastian Kochinke The Discrete Logarithm Problem on Curves 7 / 32
16 on Curves The Discrete Logarithm Double Large Prime Variation We can improve upon the basic algorithm by using double large prime variation. Sebastian Kochinke The Discrete Logarithm Problem on Curves 8 / 32
17 on Curves The Discrete Logarithm Double Large Prime Variation We can improve upon the basic algorithm by using double large prime variation. The relation generation process then consists of two steps. Sebastian Kochinke The Discrete Logarithm Problem on Curves 8 / 32
18 on Curves The Discrete Logarithm Double Large Prime Variation We can improve upon the basic algorithm by using double large prime variation. The relation generation process then consists of two steps. Step 1 Allow relations involving at most two elements g i,1, g i,2 G F to construct a so-called tree of large prime relations T. Sebastian Kochinke The Discrete Logarithm Problem on Curves 8 / 32
19 on Curves The Discrete Logarithm Double Large Prime Variation We can improve upon the basic algorithm by using double large prime variation. The relation generation process then consists of two steps. Step 1 Allow relations involving at most two elements g i,1, g i,2 G F to construct a so-called tree of large prime relations T. Step 2 Generate relations over a factor base enlarged by the g i,1, g i,2 and use T to substitute relations involving the g i,j. Sebastian Kochinke The Discrete Logarithm Problem on Curves 8 / 32
20 on Curves The Discrete Logarithm Double Large Prime Variation We can improve upon the basic algorithm by using double large prime variation. The relation generation process then consists of two steps. Step 1 Allow relations involving at most two elements g i,1, g i,2 G F to construct a so-called tree of large prime relations T. Step 2 Generate relations over a factor base enlarged by the g i,1, g i,2 and use T to substitute relations involving the g i,j. We will focus on the generation of T since this step is crucial for the over all running time. Sebastian Kochinke The Discrete Logarithm Problem on Curves 8 / 32
21 on Curves Sebastian Kochinke The Discrete Logarithm Problem on Curves 9 / 32
22 on Curves In this talk curves will always be non-hyperelliptic, smooth and geometrically irreducible. Sebastian Kochinke The Discrete Logarithm Problem on Curves 10 / 32
23 on Curves In this talk curves will always be non-hyperelliptic, smooth and geometrically irreducible. Curves can be represented by their plane models, that is a 1-dimensional closed subscheme C pm P 2 k birational to C. Sebastian Kochinke The Discrete Logarithm Problem on Curves 10 / 32
24 on Curves In this talk curves will always be non-hyperelliptic, smooth and geometrically irreducible. Curves can be represented by their plane models, that is a 1-dimensional closed subscheme C pm P 2 k birational to C. Figure: A Curve and its Plane Model Sebastian Kochinke The Discrete Logarithm Problem on Curves 10 / 32
25 on Curves Let C be a curve of genus g 3 over F q represented by a plane model C pm of degree d. Sebastian Kochinke The Discrete Logarithm Problem on Curves 11 / 32
26 on Curves Let C be a curve of genus g 3 over F q represented by a plane model C pm of degree d. The idea: Figure: A Net on a Curve Sebastian Kochinke The Discrete Logarithm Problem on Curves 11 / 32
27 on Curves This leads to the following algorithm for construction the tree of large prime relations T. Sebastian Kochinke The Discrete Logarithm Problem on Curves 12 / 32
28 on Curves This leads to the following algorithm for construction the tree of large prime relations T. Constructing T via a Net 1 Construct a factor base F C(F q ) on C of size q 1 1 d 2. Sebastian Kochinke The Discrete Logarithm Problem on Curves 12 / 32
29 on Curves This leads to the following algorithm for construction the tree of large prime relations T. Constructing T via a Net 1 Construct a factor base F C(F q ) on C of size q 1 1 d 2. 2 Choose a line l in P 2 through two points in F, viewed as points on C pm. Sebastian Kochinke The Discrete Logarithm Problem on Curves 12 / 32
30 on Curves This leads to the following algorithm for construction the tree of large prime relations T. Constructing T via a Net 1 Construct a factor base F C(F q ) on C of size q 1 1 d 2. 2 Choose a line l in P 2 through two points in F, viewed as points on C pm. 3 Calculate those points on C pm at which l meets C pm. Sebastian Kochinke The Discrete Logarithm Problem on Curves 12 / 32
31 on Curves This leads to the following algorithm for construction the tree of large prime relations T. Constructing T via a Net 1 Construct a factor base F C(F q ) on C of size q 1 1 d 2. 2 Choose a line l in P 2 through two points in F, viewed as points on C pm. 3 Calculate those points on C pm at which l meets C pm. 4 If all these points are rational and nonsingular on C pm and all but at most two of them factor over F, store the corresponding effective divisor on C. Sebastian Kochinke The Discrete Logarithm Problem on Curves 12 / 32
32 on Curves This leads to the following algorithm for construction the tree of large prime relations T. Constructing T via a Net 1 Construct a factor base F C(F q ) on C of size q 1 1 d 2. 2 Choose a line l in P 2 through two points in F, viewed as points on C pm. 3 Calculate those points on C pm at which l meets C pm. 4 If all these points are rational and nonsingular on C pm and all but at most two of them factor over F, store the corresponding effective divisor on C. 5 Repeat step 2 to 4 until q divisors as above are found. Sebastian Kochinke The Discrete Logarithm Problem on Curves 12 / 32
33 on Curves This leads to the following algorithm for construction the tree of large prime relations T. Constructing T via a Net 1 Construct a factor base F C(F q ) on C of size q 1 1 d 2. 2 Choose a line l in P 2 through two points in F, viewed as points on C pm. 3 Calculate those points on C pm at which l meets C pm. 4 If all these points are rational and nonsingular on C pm and all but at most two of them factor over F, store the corresponding effective divisor on C. 5 Repeat step 2 to 4 until q divisors as above are found. 6 Construct T from these relations. Sebastian Kochinke The Discrete Logarithm Problem on Curves 12 / 32
34 on Curves Sebastian Kochinke The Discrete Logarithm Problem on Curves 13 / 32
35 on Curves Based on the relation generation method described above, Claus Diem was able to prove the following. Sebastian Kochinke The Discrete Logarithm Problem on Curves 14 / 32
36 on Curves Based on the relation generation method described above, Claus Diem was able to prove the following. Theorem (Diem) The discrete logarithm problem in the degree 0 class group of curves C as above given by plane models of degree d such that there is an effective divisor given by a line l in P 2 that splits completely into distinct points can be solved in an expected time of Õ (q 2 2 d 2 ). Sebastian Kochinke The Discrete Logarithm Problem on Curves 14 / 32
37 on Curves Based on the relation generation method described above, Claus Diem was able to prove the following. Theorem (Diem) The discrete logarithm problem in the degree 0 class group of curves C as above given by plane models of degree d such that there is an effective divisor given by a line l in P 2 that splits completely into distinct points can be solved in an expected time of Õ (q 2 2 d 2 ). The line l ensures there is a separable cover C P 1 to which we can apply the Chebotaryov density theorem. Sebastian Kochinke The Discrete Logarithm Problem on Curves 14 / 32
38 on Curves Question: Are there enough (or any) divisors given by lines that split completely over the base field? Sebastian Kochinke The Discrete Logarithm Problem on Curves 15 / 32
39 on Curves Question: Are there enough (or any) divisors given by lines that split completely over the base field? Theorem (Diem) We consider curves as above given by plane models of degree d, where for d 4 we restrict ourselves to reflexive plane models. Then the number of divisors on C that are given by lines in P 2 and split completely into distinct points is in 1 ( ) d! q2 + O q 3 2. Sebastian Kochinke The Discrete Logarithm Problem on Curves 15 / 32
40 on Curves Question: Are there enough (or any) divisors given by lines that split completely over the base field? Theorem (Diem) We consider curves as above given by plane models of degree d, where for d 4 we restrict ourselves to reflexive plane models. Then the number of divisors on C that are given by lines in P 2 and split completely into distinct points is in 1 ( ) d! q2 + O q 3 2. Briefly, reflexivity means that the classical duality theory holds. Sebastian Kochinke The Discrete Logarithm Problem on Curves 15 / 32
41 on Curves Question: Are there enough (or any) divisors given by lines that split completely over the base field? Theorem (Diem) We consider curves as above given by plane models of degree d, where for d 4 we restrict ourselves to reflexive plane models. Then the number of divisors on C that are given by lines in P 2 and split completely into distinct points is in 1 ( ) d! q2 + O q 3 2. Briefly, reflexivity means that the classical duality theory holds. The reflexivity ensures there are rational, nonsingular points p on the plane model s.t. only ordinary tangents pass through p. This enables us to apply the Chebotaryov density theorem again. Sebastian Kochinke The Discrete Logarithm Problem on Curves 15 / 32
42 on Curves We want to get rid of the reflexivity assumption. Sebastian Kochinke The Discrete Logarithm Problem on Curves 16 / 32
43 on Curves We want to get rid of the reflexivity assumption. The following theorem is currently under construction. Theorem (K.) We consider curves of genus at least 3 over fields F q of characteristic c 2. Then the discrete logarithm problem in the degree 0 class group of such curves C can be solved in an expected time of Õ (q 2 2 g 1 ). Sebastian Kochinke The Discrete Logarithm Problem on Curves 16 / 32
44 on Curves We want to get rid of the reflexivity assumption. The following theorem is currently under construction. Theorem (K.) We consider curves of genus at least 3 over fields F q of characteristic c 2. Then the discrete logarithm problem in the degree 0 class group of such curves C can be solved in an expected time of Õ (q 2 2 g 1 ). The statement above relies on the following observation: Sebastian Kochinke The Discrete Logarithm Problem on Curves 16 / 32
45 on Curves We want to get rid of the reflexivity assumption. The following theorem is currently under construction. Theorem (K.) We consider curves of genus at least 3 over fields F q of characteristic c 2. Then the discrete logarithm problem in the degree 0 class group of such curves C can be solved in an expected time of Õ (q 2 2 g 1 ). The statement above relies on the following observation: If each plane model of degree (g + 1) was non-reflexive, then for all but a fixed number of rational points p C each hyperplane containing the tangent to p on the canonical model meets this model at p with multiplicity at least c. Sebastian Kochinke The Discrete Logarithm Problem on Curves 16 / 32
46 on Curves Sebastian Kochinke The Discrete Logarithm Problem on Curves 17 / 32
47 on Curves We have seen that the running times above depend on the degree of the plane model. Sebastian Kochinke The Discrete Logarithm Problem on Curves 18 / 32
48 on Curves We have seen that the running times above depend on the degree of the plane model. Question: How can we generate plane models of small degree? Sebastian Kochinke The Discrete Logarithm Problem on Curves 18 / 32
49 on Curves We have seen that the running times above depend on the degree of the plane model. Question: How can we generate plane models of small degree? Let C be a curve of genus g 4 over k = k. We represent C by its canonical model C P g 1. Sebastian Kochinke The Discrete Logarithm Problem on Curves 18 / 32
50 on Curves We have seen that the running times above depend on the degree of the plane model. Question: How can we generate plane models of small degree? Let C be a curve of genus g 4 over k = k. We represent C by its canonical model C P g 1. Then for an effective divisors p p g 3 C g 3 we expect the successive projection from p 1,..., p g 3 to define a birational embedding of C into P 2 and therefore a plane model C pm of C of degree (g + 1). Sebastian Kochinke The Discrete Logarithm Problem on Curves 18 / 32
51 on Curves We have seen that the running times above depend on the degree of the plane model. Question: How can we generate plane models of small degree? Let C be a curve of genus g 4 over k = k. We represent C by its canonical model C P g 1. Then for an effective divisors p p g 3 C g 3 we expect the successive projection from p 1,..., p g 3 to define a birational embedding of C into P 2 and therefore a plane model C pm of C of degree (g + 1). In fact, this is the case for most divisors on any curve over finite fields. Sebastian Kochinke The Discrete Logarithm Problem on Curves 18 / 32
52 on Curves Theorem (K.) Fix a characteristic c > 0 and a genus g 4. For a curve C of genus g over F c n let P C be the probability that a divisor D on C chosen uniformly at random from C g 3 (F c n) does not lead to a birational embedding via successive projection. Then P C converges to 0 independent of the specific curve chosen. Sebastian Kochinke The Discrete Logarithm Problem on Curves 19 / 32
53 on Curves Theorem (K.) Fix a characteristic c > 0 and a genus g 4. For a curve C of genus g over F c n let P C be the probability that a divisor D on C chosen uniformly at random from C g 3 (F c n) does not lead to a birational embedding via successive projection. Then P C converges to 0 independent of the specific curve chosen. Above theorem is proven in 2 steps. Sebastian Kochinke The Discrete Logarithm Problem on Curves 19 / 32
54 on Curves Theorem (K.) Fix a characteristic c > 0 and a genus g 4. For a curve C of genus g over F c n let P C be the probability that a divisor D on C chosen uniformly at random from C g 3 (F c n) does not lead to a birational embedding via successive projection. Then P C converges to 0 independent of the specific curve chosen. Above theorem is proven in 2 steps. Step 1 We show that for each C as above there is a divisor D C g 3 ( k) that leads to a plane model via successive projection. Sebastian Kochinke The Discrete Logarithm Problem on Curves 19 / 32
55 on Curves Theorem (K.) Fix a characteristic c > 0 and a genus g 4. For a curve C of genus g over F c n let P C be the probability that a divisor D on C chosen uniformly at random from C g 3 (F c n) does not lead to a birational embedding via successive projection. Then P C converges to 0 independent of the specific curve chosen. Above theorem is proven in 2 steps. Step 1 We show that for each C as above there is a divisor D C g 3 ( k) that leads to a plane model via successive projection. Step 2 We then prove that for each C as above the subset M C g 3 of divisors that lead to a birational embedding is open (in the Zariski topology) of bounded degree. Sebastian Kochinke The Discrete Logarithm Problem on Curves 19 / 32
56 on Curves Sebastian Kochinke The Discrete Logarithm Problem on Curves 20 / 32
57 on Curves The idea: Figure: A Net on a Curve Sebastian Kochinke The Discrete Logarithm Problem on Curves 21 / 32
58 on Curves The idea: Figure: A Pencil on a Curve Sebastian Kochinke The Discrete Logarithm Problem on Curves 22 / 32
59 on Curves The idea: Figure: A Pencil on a Curve Sebastian Kochinke The Discrete Logarithm Problem on Curves 23 / 32
60 on Curves The idea: Figure: A Pencil on a Curve Sebastian Kochinke The Discrete Logarithm Problem on Curves 24 / 32
61 on Curves The idea: Figure: A Pencil on a Curve Sebastian Kochinke The Discrete Logarithm Problem on Curves 25 / 32
62 on Curves The idea: Figure: A Pencil on a Curve Sebastian Kochinke The Discrete Logarithm Problem on Curves 26 / 32
63 on Curves Let C be a curve of genus g 5 over F q. Sebastian Kochinke The Discrete Logarithm Problem on Curves 27 / 32
64 on Curves Let C be a curve of genus g 5 over F q. Constructing T via Pencils 1 Construct a factor base F C(F q ) on C of size q 1 1 g 2 Sebastian Kochinke The Discrete Logarithm Problem on Curves 27 / 32
65 on Curves Let C be a curve of genus g 5 over F q. Constructing T via Pencils 1 Construct a factor base F C(F q ) on C of size q 1 1 g 2 2 Construct plane models C pm of C of degree (g + 1) until C pm possesses a rational singularity p. Sebastian Kochinke The Discrete Logarithm Problem on Curves 27 / 32
66 on Curves Let C be a curve of genus g 5 over F q. Constructing T via Pencils 1 Construct a factor base F C(F q ) on C of size q 1 1 g 2 2 Construct plane models C pm of C of degree (g + 1) until C pm possesses a rational singularity p. 3 Consider the pencil g 1 g 1 on C given by the pullback of lines in P 2 through p. Sebastian Kochinke The Discrete Logarithm Problem on Curves 27 / 32
67 on Curves Let C be a curve of genus g 5 over F q. Constructing T via Pencils 1 Construct a factor base F C(F q ) on C of size q 1 1 g 2 2 Construct plane models C pm of C of degree (g + 1) until C pm possesses a rational singularity p. 3 Consider the pencil g 1 g 1 on C given by the pullback of lines in P 2 through p. 4 Fix some D g 1 g 1 that splits completely over F. Store relations of the form [D D] = 0 for those D g 1 g 1 that split into elements in F and up to 2 additional rational points. Sebastian Kochinke The Discrete Logarithm Problem on Curves 27 / 32
68 on Curves Let C be a curve of genus g 5 over F q. Constructing T via Pencils 1 Construct a factor base F C(F q ) on C of size q 1 1 g 2 2 Construct plane models C pm of C of degree (g + 1) until C pm possesses a rational singularity p. 3 Consider the pencil g 1 g 1 on C given by the pullback of lines in P 2 through p. 4 Fix some D g 1 g 1 that splits completely over F. Store relations of the form [D D] = 0 for those D g 1 g 1 that split into elements in F and up to 2 additional rational points. 5 Repeat step 2 to 4 until q relations are found. Sebastian Kochinke The Discrete Logarithm Problem on Curves 27 / 32
69 on Curves Let C be a curve of genus g 5 over F q. Constructing T via Pencils 1 Construct a factor base F C(F q ) on C of size q 1 1 g 2 2 Construct plane models C pm of C of degree (g + 1) until C pm possesses a rational singularity p. 3 Consider the pencil g 1 g 1 on C given by the pullback of lines in P 2 through p. 4 Fix some D g 1 g 1 that splits completely over F. Store relations of the form [D D] = 0 for those D g 1 g 1 that split into elements in F and up to 2 additional rational points. 5 Repeat step 2 to 4 until q relations are found. 6 Construct T from these relations. Sebastian Kochinke The Discrete Logarithm Problem on Curves 27 / 32
70 on Curves Heuristic Result (Diem, K.) We consider curves of genus at least 5 over fields F q. Then the discrete logarithm problem in the degree 0 class group of nearly all such curves C can be solved in an expected time of Õ (q 2 2 g 2 ). Sebastian Kochinke The Discrete Logarithm Problem on Curves 28 / 32
71 on Curves Heuristic Result (Diem, K.) We consider curves of genus at least 5 over fields F q. Then the discrete logarithm problem in the degree 0 class group of nearly all such curves C can be solved in an expected time of Õ (q 2 2 g 2 ). Things we know from Brill-Noether theory: Sebastian Kochinke The Discrete Logarithm Problem on Curves 28 / 32
72 on Curves Heuristic Result (Diem, K.) We consider curves of genus at least 5 over fields F q. Then the discrete logarithm problem in the degree 0 class group of nearly all such curves C can be solved in an expected time of Õ (q 2 2 g 2 ). Things we know from Brill-Noether theory: For g = 3 there are no g 1 2 s. Sebastian Kochinke The Discrete Logarithm Problem on Curves 28 / 32
73 on Curves Heuristic Result (Diem, K.) We consider curves of genus at least 5 over fields F q. Then the discrete logarithm problem in the degree 0 class group of nearly all such curves C can be solved in an expected time of Õ (q 2 2 g 2 ). Things we know from Brill-Noether theory: For g = 3 there are no g 1 2 s. For g = 4 there are at most two g 1 3 s. Sebastian Kochinke The Discrete Logarithm Problem on Curves 28 / 32
74 on Curves Heuristic Result (Diem, K.) We consider curves of genus at least 5 over fields F q. Then the discrete logarithm problem in the degree 0 class group of nearly all such curves C can be solved in an expected time of Õ (q 2 2 g 2 ). Things we know from Brill-Noether theory: For g = 3 there are no g 1 2 s. For g = 4 there are at most two g 1 3 s. For g 5 the dimension of the space of complete g 1 g 1 s is at least g 4. Sebastian Kochinke The Discrete Logarithm Problem on Curves 28 / 32
75 on Curves Heuristic Result (Diem, K.) We consider curves of genus at least 5 over fields F q. Then the discrete logarithm problem in the degree 0 class group of nearly all such curves C can be solved in an expected time of Õ (q 2 2 g 2 ). Things we know from Brill-Noether theory: For g = 3 there are no g 1 2 s. For g = 4 there are at most two g 1 3 s. For g 5 the dimension of the space of complete g 1 g 1 s is at least g 4. Over an algebraically closed field all g 1 g 1 s can be constructed the way described above. Sebastian Kochinke The Discrete Logarithm Problem on Curves 28 / 32
76 on Curves For a rigorous proof, amongst others, we need to answer the following questions: Sebastian Kochinke The Discrete Logarithm Problem on Curves 29 / 32
77 on Curves For a rigorous proof, amongst others, we need to answer the following questions: Is there a lower bound on the probability that a plane model yields a rational singularity? Sebastian Kochinke The Discrete Logarithm Problem on Curves 29 / 32
78 on Curves For a rigorous proof, amongst others, we need to answer the following questions: Is there a lower bound on the probability that a plane model yields a rational singularity? Is there a lower bound on the probability that a pencil through a rational singularity yields a completely split divisor? Sebastian Kochinke The Discrete Logarithm Problem on Curves 29 / 32
79 on Curves For a rigorous proof, amongst others, we need to answer the following questions: Is there a lower bound on the probability that a plane model yields a rational singularity? Is there a lower bound on the probability that a pencil through a rational singularity yields a completely split divisor? Is T sufficiently nice? Sebastian Kochinke The Discrete Logarithm Problem on Curves 29 / 32
80 on Curves Sebastian Kochinke The Discrete Logarithm Problem on Curves 30 / 32
81 on Curves Some practical data (rounded) can be found in the following tables. T rel and T la stand for the time (in hours) needed to create relations and perform the linear algebra step. F 3 11 = F genus size of F T rel T la Rel. Gen. via a Net Rel. Gen. via Pencils Sebastian Kochinke The Discrete Logarithm Problem on Curves 31 / 32
82 on Curves Some practical data (rounded) can be found in the following tables. T rel and T la stand for the time (in hours) needed to create relations and perform the linear algebra step. F 3 11 = F genus size of F T rel T la Rel. Gen. via a Net Rel. Gen. via Pencils F 7 7 = F genus size of F T rel T la Rel. Gen. via Lines Rel. Gen. via Pencils Sebastian Kochinke The Discrete Logarithm Problem on Curves 31 / 32
83 on Curves Some practical data (rounded) can be found in the following tables. T rel and T la stand for the time (in hours) needed to create relations and perform the linear algebra step. F 3 11 = F genus size of F T rel T la Rel. Gen. via a Net Rel. Gen. via Pencils F 7 7 = F genus size of F T rel T la Rel. Gen. via Lines Rel. Gen. via Pencils F 3 13 = F genus size of F T rel T la Rel. Gen. via Lines Rel. Gen. via Pencils Sebastian Kochinke The Discrete Logarithm Problem on Curves 31 / 32
84 on Curves The experiments indicate the following: Sebastian Kochinke The Discrete Logarithm Problem on Curves 32 / 32
85 on Curves The experiments indicate the following: For fixed genus g 5, except for a constant factor the 2 nd Algorithm performs for curves of genus g 5 as fast as the 1 st one for curves of genus (g 1). Sebastian Kochinke The Discrete Logarithm Problem on Curves 32 / 32
86 on Curves The experiments indicate the following: For fixed genus g 5, except for a constant factor the 2 nd Algorithm performs for curves of genus g 5 as fast as the 1 st one for curves of genus (g 1). In particular, as of right now the best algorithm for genus 4 and 5 have (asymptotically) the same running time. Sebastian Kochinke The Discrete Logarithm Problem on Curves 32 / 32
87 on Curves The experiments indicate the following: For fixed genus g 5, except for a constant factor the 2 nd Algorithm performs for curves of genus g 5 as fast as the 1 st one for curves of genus (g 1). In particular, as of right now the best algorithm for genus 4 and 5 have (asymptotically) the same running time. The difference comes from the way relations are generated. The 1 st algorithm only uses one plane model whereas the 2 nd algorithm varies the plane model and maps the factor base back and forth between these models. Sebastian Kochinke The Discrete Logarithm Problem on Curves 32 / 32
SINTESI DELLA TESI. Enriques-Kodaira classification of Complex Algebraic Surfaces
Università degli Studi Roma Tre Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica SINTESI DELLA TESI Enriques-Kodaira classification of Complex Algebraic Surfaces
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationOSTROWSKI FOR NUMBER FIELDS
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for
More informationOn The Existence Of Flips
On The Existence Of Flips Hacon and McKernan s paper, arxiv alg-geom/0507597 Brian Lehmann, February 2007 1 Introduction References: Hacon and McKernan s paper, as above. Kollár and Mori, Birational Geometry
More information1 Chapter I Solutions
1 Chapter I Solutions 1.1 Section 1 (TODO) 1 2 Chapter II Solutions 2.1 Section 1 1.16b. Given an exact sequence of sheaves 0 F F F 0 over a topological space X with F flasque show that for every open
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationGEOMETRIC PROPERTIES OF PROJECTIVE MANIFOLDS OF SMALL DEGREE
GEOMETRIC PROPERTIES OF PROJECTIVE MANIFOLDS OF SMALL DEGREE SIJONG KWAK AND JINHYUNG PARK Abstract. We study geometric structures of smooth projective varieties of small degree in birational geometric
More informationElliptic Curve Cryptography
Elliptic Curve Cryptography Elaine Brow, December 2010 Math 189A: Algebraic Geometry 1. Introduction to Public Key Cryptography To understand the motivation for elliptic curve cryptography, we must first
More informationON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE
Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 145 153 S 0019-2082 ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE J. HUISMAN Abstract. Let C be a
More informationCONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents
More informationOn the complexity of some computational problems in the Turing model
On the complexity of some computational problems in the Turing model Claus Diem November 18, 2013 Abstract Algorithms for concrete problems are usually described and analyzed in some random access machine
More informationShort Programs for functions on Curves
Short Programs for functions on Curves Victor S. Miller Exploratory Computer Science IBM, Thomas J. Watson Research Center Yorktown Heights, NY 10598 May 6, 1986 Abstract The problem of deducing a function
More informationThe Taxman Game. Robert K. Moniot September 5, 2003
The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationRESEARCH STATEMENT AMANDA KNECHT
RESEARCH STATEMENT AMANDA KNECHT 1. Introduction A variety X over a field K is the vanishing set of a finite number of polynomials whose coefficients are elements of K: X := {(x 1,..., x n ) K n : f i
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationLecture 4 Cohomological operations on theories of rational type.
Lecture 4 Cohomological operations on theories of rational type. 4.1 Main Theorem The Main Result which permits to describe operations from a theory of rational type elsewhere is the following: Theorem
More informationA New Generic Digital Signature Algorithm
Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationOn Smooth Surfaces of Degree 10 in the Projective Fourspace
On Smooth Surfaces of Degree 10 in the Projective Fourspace Kristian Ranestad Contents 0. Introduction 2 1. A rational surface with π = 8 16 2. A rational surface with π = 9 24 3. A K3-surface with π =
More informationIs n a Prime Number? Manindra Agrawal. March 27, 2006, Delft. IIT Kanpur
Is n a Prime Number? Manindra Agrawal IIT Kanpur March 27, 2006, Delft Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 1 / 47 Overview 1 The Problem 2 Two Simple, and Slow, Methods
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationHow To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationPlanar Curve Intersection
Chapter 7 Planar Curve Intersection Curve intersection involves finding the points at which two planar curves intersect. If the two curves are parametric, the solution also identifies the parameter values
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationArithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJ-PRG. R. Barbulescu Sieves 0 / 28
Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJ-PRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer
More informationFaster deterministic integer factorisation
David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers
More informationMaster of Arts in Mathematics
Master of Arts in Mathematics Administrative Unit The program is administered by the Office of Graduate Studies and Research through the Faculty of Mathematics and Mathematics Education, Department of
More informationFiber sums of genus 2 Lefschetz fibrations
Proceedings of 9 th Gökova Geometry-Topology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity
More informationThe Multiple Conical Surfaces
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 1, 71-87. The Multiple Conical Surfaces Josef Schicho Research Institute for Symbolic Computation Johannes
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationNumber of Divisors. Terms. Factors, prime factorization, exponents, Materials. Transparencies Activity Sheets Calculators
of Divisors Purpose: Participants will investigate the relationship between the prime-factored form of a number and its total number of factors. Overview: In small groups, participants will generate the
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationMA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
More informationSection IV.1: Recursive Algorithms and Recursion Trees
Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More information3.6 The Real Zeros of a Polynomial Function
SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,
More informationMATHEMATICS (MATH) 3. Provides experiences that enable graduates to find employment in sciencerelated
194 / Department of Natural Sciences and Mathematics MATHEMATICS (MATH) The Mathematics Program: 1. Provides challenging experiences in Mathematics, Physics, and Physical Science, which prepare graduates
More informationGröbner Bases and their Applications
Gröbner Bases and their Applications Kaitlyn Moran July 30, 2008 1 Introduction We know from the Hilbert Basis Theorem that any ideal in a polynomial ring over a field is finitely generated [3]. However,
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the
More informationBenford s Law: Tables of Logarithms, Tax Cheats, and The Leading Digit Phenomenon
Benford s Law: Tables of Logarithms, Tax Cheats, and The Leading Digit Phenomenon Michelle Manes (mmanes@math.hawaii.edu) 9 September, 2008 History (1881) Simon Newcomb publishes Note on the frequency
More informationMICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationAlgebraic and Transcendental Numbers
Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)
More informationPrime numbers and prime polynomials. Paul Pollack Dartmouth College
Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationKevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm
MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More informationMATHEMATICS BONUS FILES for faculty and students http://www2.onu.edu/~mcaragiu1/bonus_files.html
MATHEMATICS BONUS FILES for faculty and students http://www2onuedu/~mcaragiu1/bonus_fileshtml RECEIVED: November 1 2007 PUBLISHED: November 7 2007 Solving integrals by differentiation with respect to a
More informationECE 842 Report Implementation of Elliptic Curve Cryptography
ECE 842 Report Implementation of Elliptic Curve Cryptography Wei-Yang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic
More information2.4 Real Zeros of Polynomial Functions
SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003
MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld
More informationStudy of algorithms for factoring integers and computing discrete logarithms
Study of algorithms for factoring integers and computing discrete logarithms First Indo-French Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department
More informationLecture 10: Distinct Degree Factoring
CS681 Computational Number Theory Lecture 10: Distinct Degree Factoring Instructor: Piyush P Kurur Scribe: Ramprasad Saptharishi Overview Last class we left of with a glimpse into distant degree factorization.
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationCommunication on the Grassmann Manifold: A Geometric Approach to the Noncoherent Multiple-Antenna Channel
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 2, FEBRUARY 2002 359 Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent Multiple-Antenna Channel Lizhong Zheng, Student
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationThe van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
More informationTHE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
More informationApplication. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:
More informationI. Introduction. MPRI Cours 2-12-2. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F.
F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 3/22 F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 4/22 MPRI Cours 2-12-2 I. Introduction Input: an integer N; logox F. Morain logocnrs
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationThe Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationCongruent Number Problem
University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases
More informationMCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials
Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from
More informationz 0 and y even had the form
Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 1-14, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO
More informationMany algorithms, particularly divide and conquer algorithms, have time complexities which are naturally
Recurrence Relations Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally modeled by recurrence relations. A recurrence relation is an equation which
More informationDimension Theory for Ordinary Differential Equations
Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskii-norms 15 1 Singular values
More information