The Discrete Logarithm Problem on non-hyperelliptic Curves of Genus g 4

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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

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