TYPES Workshop, june 2006 p. 1/22. The Elliptic Curve Factorization method
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1 Ä ÙÖ ÒØ ÓÙ Ð ÙÖ ÒØ ÓÑ Ø ºÒ Ø TYPES Workshop, june 2006 p. 1/22 ÄÇÊÁ ÍÒ Ú Ö Ø À ÒÖ ÈÓ Ò Ö Æ ÒÝÁ. The Elliptic Curve Factorization method
2 Outline 1. Introduction 2. Factorization method principle 3. Elliptic Curves 4. Fast modular multiplication 5. ECM in practice: GMP-ECM 6. Conclusion TYPES Workshop, june 2006 p. 2/22
3 interesting arithmetic problems Why factor? TYPES Workshop, june 2006 p. 3/22
4 interesting arithmetic problems Why factor? factorize numbers of a given family (Fermat, Cunningham) TYPES Workshop, june 2006 p. 3/22
5 Why factor? interesting arithmetic problems factorize numbers of a given family (Fermat, Cunningham) link to cryptography TYPES Workshop, june 2006 p. 3/22
6 Why factor? interesting arithmetic problems factorize numbers of a given family (Fermat, Cunningham) link to cryptography used for primality proving TYPES Workshop, june 2006 p. 3/22
7 Why factor? interesting arithmetic problems factorize numbers of a given family (Fermat, Cunningham) link to cryptography used for primality proving for fun and records breaking TYPES Workshop, june 2006 p. 3/22
8 Factorization method principle Let n be the number to be factored, and p an unknown prime factor of n. TYPES Workshop, june 2006 p. 4/22
9 Factorization method principle Let n be the number to be factored, and p an unknown prime factor of n. 1. Pick an abelian group G p where operations are compatible with modular arithmetics. TYPES Workshop, june 2006 p. 4/22
10 Factorization method principle Let n be the number to be factored, and p an unknown prime factor of n. 1. Pick an abelian group G p where operations are compatible with modular arithmetics. 2. Perform all operations mod n in the structure G n. TYPES Workshop, june 2006 p. 4/22
11 Factorization method principle Let n be the number to be factored, and p an unknown prime factor of n. 1. Pick an abelian group G p where operations are compatible with modular arithmetics. 2. Perform all operations mod n in the structure G n. 3. Operations done mod n reduce naturally to G p. TYPES Workshop, june 2006 p. 4/22
12 Factorization method principle Let n be the number to be factored, and p an unknown prime factor of n. 1. Pick an abelian group G p where operations are compatible with modular arithmetics. 2. Perform all operations mod n in the structure G n. 3. Operations done mod n reduce naturally to G p. 4. Pick an element P 0 G n, a positive integer x and compute x P. TYPES Workshop, june 2006 p. 4/22
13 Factorization method principle Let n be the number to be factored, and p an unknown prime factor of n. 1. Pick an abelian group G p where operations are compatible with modular arithmetics. 2. Perform all operations mod n in the structure G n. 3. Operations done mod n reduce naturally to G p. 4. Pick an element P 0 G n, a positive integer x and compute x P. 5. If o Gp (P 0 ) x the computation of x P 0 will give a factor of n (probably). TYPES Workshop, june 2006 p. 4/22
14 Factorization method principle Example: TYPES Workshop, june 2006 p. 5/22
15 Factorization method principle Example: For the P-1 factorization method, we have G p = (Z/pZ). TYPES Workshop, june 2006 p. 5/22
16 Factorization method principle Example: For the P-1 factorization method, we have G p = (Z/pZ). For the P+1 factorization method, we have G p = GF(p 2 ) TYPES Workshop, june 2006 p. 5/22
17 Factorization method principle Example: For the P-1 factorization method, we have G p = (Z/pZ). For the P+1 factorization method, we have G p = GF(p 2 ) (more correctly GF(p 2 ) /GF(p). TYPES Workshop, june 2006 p. 5/22
18 Factorization method principle Example: For the P-1 factorization method, we have G p = (Z/pZ). For the P+1 factorization method, we have G p = GF(p 2 ) (more correctly GF(p 2 ) /GF(p). For ECM, we chose an elliptic curve E a,b. TYPES Workshop, june 2006 p. 5/22
19 A simple P-1 example Take n = , P 0 = 42 and x = 42. TYPES Workshop, june 2006 p. 6/22
20 A simple P-1 example Take n = , P 0 = 42 and x = 42. We notive that gcd(p0 x 1, n) = 1009 so we found a non trivial factor p = 1009 of n. TYPES Workshop, june 2006 p. 6/22
21 A simple P-1 example Take n = , P 0 = 42 and x = 42. We notive that gcd(p x 0 1, n) = 1009 so we found a non trivial factor p = 1009 of n. The order of 42 in (Z/pZ) is 21 and TYPES Workshop, june 2006 p. 6/22
22 A simple P-1 example Take n = , P 0 = 42 and x = 42. We notive that gcd(p x 0 1, n) = 1009 so we found a non trivial factor p = 1009 of n. The order of 42 in (Z/pZ) is 21 and Moreover p 1 = TYPES Workshop, june 2006 p. 6/22
23 A simple P-1 example Take n = , P 0 = 42 and x = 42. We notive that gcd(p x 0 1, n) = 1009 so we found a non trivial factor p = 1009 of n. The order of 42 in (Z/pZ) is 21 and Moreover p 1 = n = p q where q is prime and q 1 = TYPES Workshop, june 2006 p. 6/22
24 A simple P-1 example Take n = , P 0 = 42 and x = 42. We notive that gcd(p x 0 1, n) = 1009 so we found a non trivial factor p = 1009 of n. The order of 42 in (Z/pZ) is 21 and Moreover p 1 = n = p q where q is prime and q 1 = and the order of 42 in (Z/qZ) is > 42. TYPES Workshop, june 2006 p. 6/22
25 A simple P-1 example Take n = , P 0 = 42 and x = 42. We notive that gcd(p x 0 1, n) = 1009 so we found a non trivial factor p = 1009 of n. The order of 42 in (Z/pZ) is 21 and Moreover p 1 = n = p q where q is prime and q 1 = and the order of 42 in (Z/qZ) is > 42. So we were able to factorize n. TYPES Workshop, june 2006 p. 6/22
26 Elliptic Curve An elliptic curve over a field K of characteristic other than 2 or 3 is the set of point (X, Y ) such that Y 2 = X 3 + AX + B where A, B K and 4A B 3 0, and a point at infinity 0 E. TYPES Workshop, june 2006 p. 7/22
27 Elliptic Curve An elliptic curve over a field K of characteristic other than 2 or 3 is the set of point (X, Y ) such that Y 2 = X 3 + AX + B where A, B K and 4A B 3 0, and a point at infinity 0 E. This is called the Weierstrass representation of the curve. TYPES Workshop, june 2006 p. 7/22
28 Elliptic Curve An elliptic curve over a field K of characteristic other than 2 or 3 is the set of point (X, Y ) such that Y 2 = X 3 + AX + B where A, B K and 4A B 3 0, and a point at infinity 0 E. This is called the Weierstrass representation of the curve. Switching to Montgomery and homogeneous form the equation of the curve becomes by 2 z = x 3 + ax 2 z + xz 2 which is more suited for computations. TYPES Workshop, june 2006 p. 7/22
29 Elliptic Curve The geometric interpretation of the group operation is that three aligned points sum to zero. (infinity) : 0 E = (0 : 1 : 0) TYPES Workshop, june 2006 p. 8/22
30 Elliptic Curve The geometric interpretation of the group operation is that three aligned points sum to zero. Q P (infinity) : 0 E = (0 : 1 : 0) TYPES Workshop, june 2006 p. 8/22
31 Elliptic Curve The geometric interpretation of the group operation is that three aligned points sum to zero. Q P (infinity) : 0 E = (0 : 1 : 0) TYPES Workshop, june 2006 p. 8/22
32 Elliptic Curve The geometric interpretation of the group operation is that three aligned points sum to zero. Q P P + Q (infinity) : 0 E = (0 : 1 : 0) TYPES Workshop, june 2006 p. 8/22
33 Arithmetic on the curve A point on the curve is a triplet (x : y : z). TYPES Workshop, june 2006 p. 9/22
34 Arithmetic on the curve A point on the curve is a triplet (x : y : z). We can forget y and identify P with P to get simpler formulas. TYPES Workshop, june 2006 p. 9/22
35 Arithmetic on the curve A point on the curve is a triplet (x : y : z). We can forget y and identify P with P to get simpler formulas. To compute P + Q: x P+Q = 4z P Q (x P x Q z P z Q ) 2 z P+Q = 4x P Q (x P z Q z P x Q ) 2 so we need to have computed P Q already. Formulas for doubling omitted. TYPES Workshop, june 2006 p. 9/22
36 Comparison of ECM with other methods Group Z/pZ GF(p 2 ) Generic cyclotomic E a,b mod p Order p 1 = Π 1 (p) p + 1 = Π 2 (p) Π d (p) o (p + 1) < 2 p TYPES Workshop, june 2006 p. 10/22
37 Complexity of ECM Let L α,c (p) = e c(log p)α (log log p) 1 α. Then the expected time complexity of ECM to find a factor p of n is O(L1 2, 2 (p)m(log n)) where M(log n) is the complexity of multiplication mod n. TYPES Workshop, june 2006 p. 11/22
38 Complexity of ECM Let L α,c (p) = e c(log p)α (log log p) 1 α. Then the expected time complexity of ECM to find a factor p of n is O(L1 2, 2 (p)m(log n)) where M(log n) is the complexity of multiplication mod n. for NFS O(L1,c(n)) (where c < 2). 3 TYPES Workshop, june 2006 p. 11/22
39 Complexity of ECM Let L α,c (p) = e c(log p)α (log log p) 1 α. Then the expected time complexity of ECM to find a factor p of n is O(L1 2, 2 (p)m(log n)) where M(log n) is the complexity of multiplication mod n. for NFS O(L1,c(n)) (where c < 2). 3 ECM won t break RSA any time soon. TYPES Workshop, june 2006 p. 11/22
40 Complexity of ECM Let L α,c (p) = e c(log p)α (log log p) 1 α. Then the expected time complexity of ECM to find a factor p of n is O(L1 2, 2 (p)m(log n)) where M(log n) is the complexity of multiplication mod n. for NFS O(L1,c(n)) (where c < 2). 3 ECM won t break RSA any time soon. huge numbers with expected relatively small factors are out of reach for NSF: ECM can be used there. TYPES Workshop, june 2006 p. 11/22
41 How to chose x The hope is to pick P 0 and x such that x o Gp (P 0 ). Use two bounds B1, B2 TYPES Workshop, june 2006 p. 12/22
42 How to chose x The hope is to pick P 0 and x such that x o Gp (P 0 ). Use two bounds B1, B2 cover all primes up to B 1, TYPES Workshop, june 2006 p. 12/22
43 How to chose x The hope is to pick P 0 and x such that x o Gp (P 0 ). Use two bounds B1, B2 cover all primes up to B 1, permit an additional prime factor in [B 1, B 2 ]. TYPES Workshop, june 2006 p. 12/22
44 How to chose x The hope is to pick P 0 and x such that x o Gp (P 0 ). Use two bounds B1, B2 cover all primes up to B 1, permit an additional prime factor in [B 1, B 2 ]. This explains naturally the two stages method used in GMP-ECM. TYPES Workshop, june 2006 p. 12/22
45 High level algorithm description INPUT: a number n, integer bounds B 1 B 2. OUTPUT: a factor p of n or FAIL. 1: Choose a random elliptic curve E a,b mod n and a point P 0 = (x 0 : y 0 : z 0 ) on it. 2: Compute Q = Q π B 1 π log B 1/ log π P 0. 3: for π prime, B 1 < π B 2 do 4: (x π : y π : z π ) πq 5: g gcd(n, z π ) 6: if g 1 then 7: return g 8: end if 9: end for 10: return FAIL TYPES Workshop, june 2006 p. 13/22
46 Algorithmic challenges An implementation of ECM is faced with the following algorithmic challenges: TYPES Workshop, june 2006 p. 14/22
47 Algorithmic challenges An implementation of ECM is faced with the following algorithmic challenges: fast modular arithmetic TYPES Workshop, june 2006 p. 14/22
48 Algorithmic challenges An implementation of ECM is faced with the following algorithmic challenges: fast modular arithmetic efficient curve operations TYPES Workshop, june 2006 p. 14/22
49 Algorithmic challenges An implementation of ECM is faced with the following algorithmic challenges: fast modular arithmetic efficient curve operations fast polynomial evaluation (stage 2) TYPES Workshop, june 2006 p. 14/22
50 Algorithmic challenges An implementation of ECM is faced with the following algorithmic challenges: fast modular arithmetic efficient curve operations fast polynomial evaluation (stage 2) In order to be fast, we use algorithmic improvements tailored to the size of the numbers used (with thresholds) as well as assembly code. TYPES Workshop, june 2006 p. 14/22
51 Modular arithmetic Operations on the curve translate into residue arithmetic operations (addition, multiplication and division mod n). TYPES Workshop, june 2006 p. 15/22
52 Modular arithmetic Operations on the curve translate into residue arithmetic operations (addition, multiplication and division mod n). divisions can be performed by way of multiplications TYPES Workshop, june 2006 p. 15/22
53 Modular arithmetic Operations on the curve translate into residue arithmetic operations (addition, multiplication and division mod n). divisions can be performed by way of multiplications the most interesting operation to optimize is therefore multiplication. TYPES Workshop, june 2006 p. 15/22
54 Modular arithmetic An example of algorithmic improvement for the operation: TYPES Workshop, june 2006 p. 16/22
55 Modular arithmetic An example of algorithmic improvement for the operation: c a b mod n TYPES Workshop, june 2006 p. 16/22
56 Modular arithmetic An example of algorithmic improvement for the operation: c a b mod n The naive approach uses: a full log n log n 2 log n multiplication, TYPES Workshop, june 2006 p. 16/22
57 Modular arithmetic An example of algorithmic improvement for the operation: c a b mod n The naive approach uses: a full log n log n 2 log n multiplication, a modular reduction 2 log n by log n (as costly as a division). TYPES Workshop, june 2006 p. 16/22
58 Modular arithmetic Instead we can use Montgomery representation. Let β be the integer base (usually β = 2 32 ) and l smallest integer such that β l > n TYPES Workshop, june 2006 p. 17/22
59 Modular arithmetic Instead we can use Montgomery representation. Let β be the integer base (usually β = 2 32 ) and l smallest integer such that β l > n The Montgomery representation gives: a a = β l a mod n TYPES Workshop, june 2006 p. 17/22
60 Modular arithmetic Instead we can use Montgomery representation. Let β be the integer base (usually β = 2 32 ) and l smallest integer such that β l > n The Montgomery representation gives: a b a = β l a mod n b = β l b mod n TYPES Workshop, june 2006 p. 17/22
61 Modular arithmetic Instead we can use Montgomery representation. Let β be the integer base (usually β = 2 32 ) and l smallest integer such that β l > n The Montgomery representation gives: a a = β l a mod n b b = β l b mod n a b β l a b = (a b β l ) mod n = REDC(a b ) TYPES Workshop, june 2006 p. 17/22
62 Algorithm REDC INPUT: numbers T, β l, n, n such that 0 T < β l n and nn = 1 mod β l. OUTPUT: Tβ l mod n. 1: m Tn mod β l Lower half multiplication 2: t (T + mn)/β l Upper half multiplication 3: if t n then 4: return t n 5: else 6: return t 7: end if TYPES Workshop, june 2006 p. 18/22
63 Algorithm REDC INPUT: numbers T, β l, n, n such that 0 T < β l n and nn = 1 mod β l. OUTPUT: Tβ l mod n. 1: m Tn mod β l Lower half multiplication 2: t (T + mn)/β l Upper half multiplication 3: if t n then 4: return t n 5: else 6: return t 7: end if mn = Tn n = T mod β l TYPES Workshop, june 2006 p. 18/22
64 Algorithm REDC INPUT: numbers T, β l, n, n such that 0 T < β l n and nn = 1 mod β l. OUTPUT: Tβ l mod n. 1: m Tn mod β l Lower half multiplication 2: t (T + mn)/β l Upper half multiplication 3: if t n then 4: return t n 5: else 6: return t 7: end if mn = Tn n = T mod β l tβ l = T mod n so t = Tβ l mod n TYPES Workshop, june 2006 p. 18/22
65 Algorithm REDC INPUT: numbers T, β l, n, n such that 0 T < β l n and nn = 1 mod β l. OUTPUT: Tβ l mod n. 1: m Tn mod β l Lower half multiplication 2: t (T + mn)/β l Upper half multiplication 3: if t n then 4: return t n 5: else 6: return t 7: end if mn = Tn n = T mod β l tβ l = T mod n so t = Tβ l mod n 0 T + mn < β l n + β l n so 0 t < 2n. TYPES Workshop, june 2006 p. 18/22
66 An example ECM factorization with GMP-ECM Let N = of 187 digits. We chose B 1 = , the curve parameterized by σ = (Suyama). TYPES Workshop, june 2006 p. 19/22
67 Åȹ Å º½ ÔÓÛ Ö Ý ÅÈ º¾ Å ÁÒÔÙØÒÙÑ Ö ººº ½ Ø µ ѹڹڹ Ñ ¼¼ ½ ÓÑÔÓ Ø Í Ò ½ ¾ ½ ¾ ÔÓÐÝÒÓÑ Ð ÓÒ µ Ñ ¼¼ ½ ÜÔ Ø ÒÙÑ ÖÓ ÙÖÚ ØÓ Ò ØÓÖÓ Ò Ø Í Ò ÅÇ ÅÍÄÆ ¾ ¼ ¼ ¼ ¼ ¾ ½ ¾ ¾¾ ¾ ¼½ ½ ¾ ½º ¼ ¾º ¼ º½ ½¼ ¾¼ ËØ Ô¾ØÓÓ ½¾¾¼ Ñ ØÓÖ ÓÙÒ Ò Ø Ô¾ ½ ¼ ½¼¼½¾ ¾¼ ¾ ½ ËØ Ô½ØÓÓ Ñ ÓÑÔÓ Ø Ó ØÓÖ ½ººº½ ½ Ø ÓÙÒ ÔÖÓ Ð ÔÖ Ñ ØÓÖÓ Ø ½ ¼ ½¼¼½¾ ¾¼ ¾ ½ An example ECM factorization with GMP-ECM TYPES Workshop, june 2006 p. 20/22
68 Conclusion ECM has seen numerous improvements, both mathematical and in the implementations since its discovery by H. W. Lenstra, Jr in TYPES Workshop, june 2006 p. 21/22
69 Conclusion ECM has seen numerous improvements, both mathematical and in the implementations since its discovery by H. W. Lenstra, Jr in easy to adapt to distributed computing (each computer runs a curve) TYPES Workshop, june 2006 p. 21/22
70 Conclusion ECM has seen numerous improvements, both mathematical and in the implementations since its discovery by H. W. Lenstra, Jr in easy to adapt to distributed computing (each computer runs a curve) packaged in Debian TYPES Workshop, june 2006 p. 21/22
71 Conclusion ECM has seen numerous improvements, both mathematical and in the implementations since its discovery by H. W. Lenstra, Jr in easy to adapt to distributed computing (each computer runs a curve) packaged in Debian why not try it today? TYPES Workshop, june 2006 p. 21/22
72 Some words from our sponsors RNC7 Conference More Digits competition LORIA, Nancy July sequel to the "Many Digits" competition ØØÔ»»ÖÒ ºÐÓÖ º Ö»ÓÑÔ Ø Ø ÓÒº ØÑÐ improve your numerical package measure your accuracy (don t be the little guy in the club anymore) it is not too late to join! only doctor approved formulas! satisfaction guaranteed! TYPES Workshop, june 2006 p. 22/22
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