Algorithms and arithmetic for the implementation of cryptographic pairings
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1 Cairn seminar November 29th, 2013 Algorithms and arithmetic for the implementation of cryptographic pairings Nicolas Estibals CAIRN project-team, IRISA
2 What is an elliptic curve? O N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 1 / 21
3 What is an elliptic curve? O Q P N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 1 / 21
4 What is an elliptic curve? O L P,Q Q P N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 1 / 21
5 What is an elliptic curve? O Q R L P,Q P N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 1 / 21
6 What is an elliptic curve? O V R Q R L P,Q P N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 1 / 21
7 What is an elliptic curve? O V R Q R L P,Q P R = P + Q N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 1 / 21
8 Elliptic Curve Cryptography Discrete Logarithm Problem (DLP) Let G be a cyclic group, P a generator, given Q G, it is supposed to be hard to compute a such that Q = [a]p N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 2 / 21
9 Elliptic Curve Cryptography Discrete Logarithm Problem (DLP) Let G be a cyclic group, P a generator, given Q G, it is supposed to be hard to compute a such that Q = [a]p Use this hard problem to design cryptographic protocols Diffie Hellman key exchange: Alice generates a secret integer a Alice sends [a]p to Bob Bob generates a secret integer b Bob sends [b]p to Alice Alice computes [a][b]p Bob computes [b][a]p They both share the same secret: [ab]p N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 2 / 21
10 What is a pairing? Pairing e(.,.) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 3 / 21
11 What is a pairing? Pairing e(.,.) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 3 / 21
12 What is a pairing? Pairing e(.,.) Cryptographic interest: Mixing two secrets without having to know them e([a]p, [b]q) = e(p, Q) ab [ a]p [ b]q e e(p, Q) ab N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 3 / 21
13 What is a pairing? Pairing e(.,.) Cryptographic interest: Mixing two secrets without having to know them e([a]p, [b]q) = e(p, Q) ab [ a]p [ b]q e e(p, Q) ab Useful for advanced protocols short signature electronic voting electronic money... N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 3 / 21
14 Example: Short signature PKI P ap Alice a Bob N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
15 Example: Short signature PKI P ap Alice a Bob N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
16 Example: Short signature PKI P ap Alice a Bob Message digest D N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
17 Example: Short signature PKI P ap Alice a Bob Signature: Message digest ad D N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
18 Example: Short signature PKI P ap Alice a Bob Signature: Message digest ad D N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
19 Example: Short signature PKI P ap Alice a Bob Signature: Message digest ad D N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
20 Example: Short signature PKI P ap Alice a Bob Signature: Message digest ad D N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
21 Example: Short signature PKI P ap Alice a Bob Signature: Message digest ad D D ad N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
22 Example: Short signature PKI P ap Alice a Bob Signature: Message digest ad D D ap ad P N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
23 Example: Short signature PKI P ap Alice a Bob Signature: Message digest ad D D ad ê(d, apap) ê(ad, P P) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 4 / 21
24 Security considerations DLP should be hard on all the groups involved N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 5 / 21
25 Security considerations DLP should be hard on all the groups involved Security measurement number of operations to break a cryptosystem today s recommendation: 128-bit security operations N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 5 / 21
26 Why cryptography and hardware implementations? I Growth of numeric exchanges many applications? bank services? secure firmware updates? personal communications?... many targets? embedded electronics? smart cards? smartphones? computers, servers I Security implies non-trivial computations I Need for hardware implementations CPUs may be inadequate limited resources N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 6 / 21
27 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
28 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection Performance metric time (ms) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
29 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection Performance metric time (ms) area (slices) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
30 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection Performance metric time (ms) area (slices) Different designs for the same computation optimized for latency optimized for compactness Computation time Area N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
31 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection Performance metric time (ms) area (slices) Different designs for the same computation optimized for latency optimized for compactness Computation time Area N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
32 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection Performance metric time (ms) area (slices) Different designs for the same computation optimized for latency optimized for compactness Computation time Area N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
33 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection Performance metric time (ms) area (slices) Different designs for the same computation optimized for latency optimized for compactness Computation time Area N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
34 Hardware implementation Our target: Field Programmable Gate Array (FPGA) integrated circuit matrix of simple configurable logic cells programmable interconnection Performance metric time (ms) area (slices) time area product Different designs for the same computation optimized for latency optimized for compactness optimized for throughput Computation time Area N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 7 / 21
35 Contributions Fast accelerator for pairings [CHES 2009, IEEE TC 2011] Joint work with Beuchat, Detrey, Okamoto and Rodríguez-Henríquez parallel architecture pipelined subquadratic multiplier Compact design for pairings reaching 128-bit security composite extension fields [Paring 2010] hyperelliptic curves [CT-RSA 2012] Joint work with Aranha, Beuchat and Detrey Formulae for sub-quadratic multiplication [WAIFI 2012] Joint work with Barbulescu, Detrey and Zimmermann exhaustive search improved formulae for F 3 5m N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 8 / 21
36 Contributions Fast accelerator for pairings [CHES 2009, IEEE TC 2011] Joint work with Beuchat, Detrey, Okamoto and Rodríguez-Henríquez parallel architecture pipelined subquadratic multiplier Compact design for pairings reaching 128-bit security composite extension fields [Paring 2010] hyperelliptic curves [CT-RSA 2012] Joint work with Aranha, Beuchat and Detrey Formulae for sub-quadratic multiplication [WAIFI 2012] Joint work with Barbulescu, Detrey and Zimmermann exhaustive search improved formulae for F 3 5m N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 8 / 21
37 Outline of the talk Compact design through composite extension fields Pipelined subquadratic multiplier Conclusion and Perspectives N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 9 / 21
38 An arithmetic coprocessor Register file Linear operations addition / subtraction Frobenius ( ) 3 Parallel serial multiplier D coeffs / cycle 509/D cycles / product For a supersingular elliptic curve over F Only need arithmetic operations in F implement a specialized processor Multiplication is critical separate linear operations and multiplications careful scheduling to keep multiplier busy Operation count (.) (.) 1 1 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 10 / 21
39 An arithmetic coprocessor Register file Linear operations addition / subtraction Frobenius ( ) 3 Parallel serial multiplier D coeffs / cycle 509/D cycles / product For a supersingular elliptic curve over F Only need arithmetic operations in F implement a specialized processor Multiplication is critical separate linear operations and multiplications careful scheduling to keep multiplier busy Operation count (.) (.) 1 1 Inverse is only needed once: Itoh Tsujii algorithm no need for hardware support N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 10 / 21
40 An arithmetic coprocessor Register file Linear operations addition / subtraction Frobenius ( ) 3 Parallel serial multiplier D coeffs / cycle 509/D cycles / product For a supersingular elliptic curve over F Only need arithmetic operations in F implement a specialized processor Multiplication is critical separate linear operations and multiplications careful scheduling to keep multiplier busy Operation count (.) (.) 1 1 Inverse is only needed once: Itoh Tsujii algorithm no need for hardware support Synthesis results for F 3 509: 9625 slices almost fully occupy a Virtex 6 LX 75 T (82%) computation time: 4 ms N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 10 / 21
41 Detailed architecture of the coprocessor (char. 3) c 0 c 5 c 6 A $0 $1 $2 $3 A 1 c 14 c c 17 c c c 25 c 26 c c 28 c 29 c 30 c 7 c 12 c 13 B $62 1 B c x 3 c 21 c 20 x (mod f ) x 2 x 3 x 13 x 14 (mod f ) (mod f ) (mod f ) (mod f ) DPRAM c 22 c x 3 c c 24 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 11 / 21
42 Detailed architecture of the coprocessor (char. 3) c 0 c 5 c 6 A $0 $1 $2 $3 A 1 c 14 c c 17 c c c 25 c 26 c c 28 c 29 c 30 c 7 c 12 c 13 B $62 1 B c x 3 c 21 c 20 x (mod f ) x 2 x 3 x 13 x 14 (mod f ) (mod f ) (mod f ) (mod f ) DPRAM c 22 c x 3 c c 24 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 11 / 21
43 Detailed architecture of the coprocessor (char. 3) 0 1 c 17 c c 0 c 5 c 6 A $0 $1 $2 $3 A c 14 c c 18 c c 26 c c 28 c 29 c 30 c 7 c 12 c 13 B $62 1 B c x 3 c 21 c 20 x (mod f ) x 2 x 3 x 13 x 14 (mod f ) (mod f ) (mod f ) (mod f ) DPRAM c x 3 c c c 24 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 11 / 21
44 Detailed architecture of the coprocessor (char. 3) 0 1 c 17 c c 0 c 5 c 6 A $0 $1 $2 $3 A c 14 c c 18 c c 26 c c 28 c 29 c 30 c 7 c 12 c 13 B $62 1 B c x 3 c 21 c 20 x (mod f ) x 2 x 3 x 13 x 14 (mod f ) (mod f ) (mod f ) (mod f ) DPRAM c x 3 c c c 24 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 11 / 21
45 Field of composite extension degree F Software F Hardware F 3 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 12 / 21
46 Field of composite extension degree F F Provides some arithmetic advantages smaller datapath Software F Hardware F F 3 97 F 3 F 3 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 12 / 21
47 Field of composite extension degree F F Provides some arithmetic advantages smaller datapath efficient multiplication algorithm Software F Hardware F Subquadratic multiplication F 3 97 Quadratic multiplication F 3 F 3 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 12 / 21
48 Field of composite extension degree F F Provides some arithmetic advantages smaller datapath efficient multiplication algorithm Allows supplementary attacks on the curve with limited effect on security Software F Hardware F Subquadratic multiplication F 3 97 Quadratic multiplication F 3 F 3 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 12 / 21
49 Field of composite extension degree F F Provides some arithmetic advantages smaller datapath efficient multiplication algorithm Allows supplementary attacks on the curve with limited effect on security Results 1848 slices of the same Virtex 6 LX (15%) 5.2 times smaller compute a pairing in 1.6 ms 2.5 times faster Software F Hardware F F 3 97 Subquadratic multiplication Quadratic multiplication F 3 F 3 N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 12 / 21
50 Benchmarks F Computation time [ms] Pairing implementations at 128 bits of security on Virtex Area [ 10 3 slices] N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 13 / 21
51 Benchmarks F Computation time [ms] [Est10] Pairing implementations at 128 bits of security on Virtex Area [ 10 3 slices] N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 13 / 21
52 Benchmarks [Est10] F Computation time [ms] [Est10] Pairing implementations at 128 bits of security on Virtex Area [ 10 3 slices] N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 13 / 21
53 Benchmarks [Est10] F Computation time [ms] [Est10] [FVV12] [Che+11] [GVR13] Pairing implementations at 128 bits of security on Virtex 6 [Yao+13] Area [ 10 3 slices] N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 13 / 21
54 Benchmarks [Est10] F Computation time [ms] 3 2 [Est10] [FVV12] Pairing implementations at 128 bits of security on Virtex 6 1 [Che+11] [GVR13] [Yao+13] [AHN13] [GRD11] Area [ 10 3 slices] [AHN13] N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 13 / 21
55 Benchmarks [Est10] F Computation time [ms] 3 2 [Ara+12] [Est10] [FVV12] Pairing implementations at 128 bits of security on Virtex 6 1 [Che+11] [GVR13] [Yao+13] [AHN13] [GRD11] Area [ 10 3 slices] [AHN13] N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 13 / 21
56 Outline of the talk Compact design through composite extension fields Pipelined subquadratic multiplier Conclusion and Perspectives N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 14 / 21
57 Finite field representation and Karatsuba s formula Small characteristic fields Polynomial basis representation N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 15 / 21
58 Finite field representation and Karatsuba s formula Small characteristic fields Polynomial basis representation Multiplication is critical make use of fast arithmetic Karatsuba algorithm for polynomials A B A B N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 15 / 21
59 Finite field representation and Karatsuba s formula Small characteristic fields Polynomial basis representation Multiplication is critical make use of fast arithmetic Karatsuba algorithm for polynomials A H A A L B H B B L A B A H B H X 2n + (A H B L + A L B H )X n + A L B L N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 15 / 21
60 Finite field representation and Karatsuba s formula Small characteristic fields Polynomial basis representation Multiplication is critical make use of fast arithmetic Karatsuba algorithm for polynomials A H A A L B H B B L A B A H B H X 2n + (A H B L + A L B H )X n + A L B L ab + a b = (a + a )(b + b ) ab a b A H B H X 2n + ((A H + A L )(B H + B L ) A H B H A L B L )X n + A L B L N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 15 / 21
61 Finite field representation and Karatsuba s formula Small characteristic fields Polynomial basis representation Multiplication is critical make use of fast arithmetic Karatsuba algorithm for polynomials A H A A L B H B B L A B A H B H X 2n + (A H B L + A L B H )X n + A L B L ab + a b = (a + a )(b + b ) ab a b A H B H X 2n + ((A H + A L )(B H + B L ) A H B H A L B L )X n + A L B L N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 15 / 21
62 Finite field representation and Karatsuba s formula Small characteristic fields Polynomial basis representation Multiplication is critical make use of fast arithmetic Karatsuba algorithm for polynomials A H A A L B H B B L A B A H B H X 2n + (A H B L + A L B H )X n + A L B L ab + a b = (a + a )(b + b ) ab a b A H B H X 2n + ((A H + A L )(B H + B L ) A H B H A L B L )X n + A L B L N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 15 / 21
63 Finite field representation and Karatsuba s formula Small characteristic fields Polynomial basis representation Multiplication is critical make use of fast arithmetic Karatsuba algorithm for polynomials A H A A L B H B B L A B A H B H X 2n + (A H B L + A L B H )X n + A L B L ab + a b = (a + a )(b + b ) ab a b A B A H B H X 2n + ((A H + A L )(B H + B L ) A H B H A L B L )X n + A L B L N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 15 / 21
64 Odd even split for Karatsuba multiplication A B A B N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 16 / 21
65 Odd even split for Karatsuba multiplication A B A O A E B O B E A B (A O B O X 2 + A E B E ) + X (A O B E + A E B O ) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 16 / 21
66 Odd even split for Karatsuba multiplication A B A O A E B O B E A B (A O B O X 2 + A E B E ) + X (A O B E + A E B O ) ab + a b = (a + a )(b + b ) ab a b (A O B O X 2 + A E B E ) + X ((A O + A E )(B O + B E ) A O B O A E B E ) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 16 / 21
67 Odd even split for Karatsuba multiplication A B A O A E B O B E A B (A O B O X 2 + A E B E ) + X (A O B E + A E B O ) ab + a b = (a + a )(b + b ) ab a b (A O B O X 2 + A E B E ) + X ((A O + A E )(B O + B E ) A O B O A E B E ) N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 16 / 21
68 Odd even split for Karatsuba multiplication A B A O A E B O B E A B (A O B O X 2 + A E B E ) + X.(A.. O B E + A E B O ) ab + a b = (a + a )(b + b ) ab a b... (A O B O X 2 + A E B E ) + X ((A O + A E )(B O + B E ) A O B O A E B E )... A B N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 16 / 21
69 Multiplier architecture Karatsuba-like algorithm: split the operands compute the subproducts recompose the result Fully parallel evaluation of the subproducts A B Splitter Recomposer A B N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 17 / 21
70 Multiplier architecture Karatsuba-like algorithm: split the operands compute the subproducts recompose the result Fully parallel evaluation of the subproducts A B Recursive scheme eventually use different multiplication algorithms end with the quadratic paper-and-pencil algorithm Splitter Recomposer Splitter Splitter Recomposer Recomposer A B N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 17 / 21
71 Multiplier architecture Karatsuba-like algorithm: split the operands compute the subproducts recompose the result Fully parallel evaluation of the subproducts A B Recursive scheme eventually use different multiplication algorithms end with the quadratic paper-and-pencil algorithm Splitter Recomposer Splitter Splitter Recomposer Pipelined with the help of optional registers cut the critical path increase the frequency Recomposer A B N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 17 / 21
72 Multiplier architecture Karatsuba-like algorithm: split the operands compute the subproducts recompose the result Fully parallel evaluation of the subproducts A B Recursive scheme eventually use different multiplication algorithms end with the quadratic paper-and-pencil algorithm Splitter Recomposer Splitter Splitter Recomposer Pipelined with the help of optional registers cut the critical path increase the frequency Generated VHDL code work for all small-characteristic fields compare different recursions configurable pipeline depth Recomposer N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 17 / 21 A B
73 Outline of the talk Compact design through composite extension fields Pipelined subquadratic multiplier Conclusion and Perspectives N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 18 / 21
74 Conclusion Hardware implementations of pairing N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 19 / 21
75 Conclusion Hardware implementations of pairing General method for cryptographic implementations study mathematical structures fix parameters thanks to cryptanalysis algorithmic optimizations choose the right arithmetic representation implement different hardware accelerators N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 19 / 21
76 Perspectives Lower-level architecture FPGA is a good prototyping platform but with limited uses in real-life devices develop skills in ASIC designs power consumption awareness N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 20 / 21
77 Perspectives Lower-level architecture FPGA is a good prototyping platform but with limited uses in real-life devices develop skills in ASIC designs power consumption awareness Integrate side-channel counter-measures side-channel attacks are very effective threats embedded systems need to be protected N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 20 / 21
78 Perspectives Lower-level architecture FPGA is a good prototyping platform but with limited uses in real-life devices develop skills in ASIC designs power consumption awareness Integrate side-channel counter-measures side-channel attacks are very effective threats embedded systems need to be protected Use this method on different cryptographic primitives scalar multiplication on hyperelliptic curves lattice-based cryptography N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 20 / 21
79 Thank you for your attention! Questions? N. Estibals Algorithms and arithmetic for the implementation of cryptographic pairings 21 / 21
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