An Inverter Architecture for ECC-GF(2 m ) Based on the Stein’s Algorithm

An Inverter Architecture for ECC-GF(2m)Based on theStein’s Algorithm

Objectives • To present the development of a modular inverter for elliptic curves, implemented by programmable circuit; • To show the viability to implement this inverter by combinatorial circuit; • To talk about the difficulties found during the development of the inverter; • To explain the solutions found to allow the development of the inverter.

Justifications • The interest about digital circuits; • The today importance of projects that work with cryptography implemented by hardware; • The need to create a device in order to increase the performance of programs that work with asymmetric cryptography; • Challenges: speed X area; combinatorial circuits X affine coordinates.

Summary • Introduction; • ECC-GF(2m); • A combinatorial Circuit for Modular Inversion; • A combinatorial Circuit for Point Doubling and Point Addition; • A Cryptosystem Implemented by Programmable Circuits; • Results; • Conclusions.

Introduction • Information; • QS (symmetric cryptography); • QP and k (asymmetric cryptography); • Q = kP; • 2P and +P: doubling and addition of P; • +, *, x2, mod, x-1: finite field arithmetic. Introduction

A Key-Exchange Example • cli – the client side of the communication; • serv – the server side of the communication; • P – an element previously chosen by cli and serv; • k – a private key; • QP – a public key; • QS – a secret key; • Q and P are elements of the same type. Introduction

Key-Exchange • The cli generates at random an integer number kcli; • The serv generates at random an integer number kserv; • The cli calculates: QPcli = kcli P; • The serv calculates: QPserv = kserv P; • The cli sends QPcli to the serv; • The serv sends QPserv to the cli; • The cli calculates: QS = kcliQPserv; • The serv calculates: QS = kserv QPcli. Introduction

Elliptic Curves • They are represented by the equation: y2 + xy = x3 + ax2 + b for which: • x and y represent the point coordinates; • a and b define an elliptic curve over GF(2m). ECC-GF()

The Main Operation of the Elliptic Curve Algorithms Q = kP for which: • k is an integer number; • P is an elliptic curve point of coordinates x and y; • Q is an elliptic curve point of coordinates x and y; • P(Px,Py) and Q(Qx,Qy) are points represented by affine coordinates and polynomial basis. ECC-GF()

Double-and-Add Algorithm Q = 10P 1010 => ((2P)2+P)2 => 10P Q = 17P 10001 => (((2P)2)2)2+P => 17P ECC-GF()

Point Doubling S = PX + ((P’Y) / (P’X)) mod p QX = (S2 + S + a) mod p QY = (S(PX + QX) + PY + QX) mod p ECC-GF()

Point Addition S = ((PY + P’Y) / (PX + P’X)) mod p QX = (S2 + S + PX + P’X + a) mod p QY = (S(PX + QX) + PY + QX) mod p ECC-GF()

Comparison BetweenDoublings .and Additions . • S = Px + (P’y / P’x) mod p. • S = ((Py + P’y) / (Px + P’x)) mod p. • Qx = (S2 + S + a) mod p. • Qx = (S2 + S + Px + P’x + a) mod p. • Qy = (S(Px + Qx) + Py + Qx) mod p. • Qy = (S(Px + Qx) + Py + Qx) mod p. ECC-GF()

Unification of the Equations S = F + ((G + P’Y) / (H + P’X)) mod p QX = (S2 + S + PX + P’X + a) mod p QY = (S(PX + QX) + PY + QX) mod p ECC-GF()

Point Doubling: F = PX; G = 0; H = 0. Point Addition: F = 0; G = PY; H = PX. Attribution of Values ECC-GF()

Finite Field Arithmetic • Sum; • Multiplication; • Module; • Squaring; • Modular Inversion. ECC-GF()

Unification of the Equations S = F + ((G + P’Y) / (H + P’X)) mod p QX = (S2 + S + PX + P’X + a) mod p QY = (S(PX + QX) + PY + QX) mod p ECC-GF()

Optimization of the Equations S = F + ((G + P’Y) * (H + P’X)-1) mod p QX = (S2 + S + PX + P’X + a) mod p QY = (S(PX + QX) + PY + QX) mod p ECC-GF()

Modular Division Methods • Gaussian Elimination; • Fermat’s Theorem; • MDC (Euclides or Stein). A Combinatorial Circuit for Modular Inversion

The Stein’s Algorithm (A, B, U, V)  (P’x, p, 1, 0) while A != 0 and B != 1 if A0 = 1 if deg(A) >= deg(B) (A, B)  (A + B, U + V) else (A, B, U, V)  (A + B, A, U + V, U) endif endif (A, U)  (A / 2, (U / 2) mod p) endwhile A Combinatorial Circuit for Modular Inversion

The Stein’s Algorithm Optimized (A, B, U, V, DCC, Flag, slice)  (P’x, p, 1, 0, 2, 1, 2m-1) while slice > 0 if A0 = 1 if Flag = 1 and DCC0 = 0 (A, B, U, V, Flag)  (A + B, A, U + V, U, 0) else (A, B)  (A + B, U + V) endif endif (A, U)  (A / 2, (U / 2) mod p) if Flag = 0 and DCC0 = 0 DCC  DCC / 2 else (DCC, Flag)  ((DCC * 2), 1) endif slice  slice -1 endwhile A Combinatorial Circuit for Modular Inversion

DCCin0 FLAGin FLAGout Ain0 FLAGin DCCin0 AUX FLAG and AUX handling A Combinatorial Circuit for Modular Inversion

DCCL[m..0] DCCout[m..0] DCCR[m..0] FLAGout DCCin[m-1..0] DCCin[m..1] DCCL[m..1] DCCR[m-1..0] GND GND DCCL0 DCCRm DCC handling A Combinatorial Circuit for Modular Inversion

Ain[m+1..1] Aout[m..0] Bin[m+1..1] Ain0 GND GND Ainm+1 Binm+1 A handling A Combinatorial Circuit for Modular Inversion

Uin[m..0] U[m..0] Vin[m..0] Ain0 GND Um+1 U[m+1..1] Uout[m..0] p[m+1..1] GND pm+1 U handling A Combinational Circuit for Modular Inversion

Ain[m..0] Bout[m..0] Bin[m..0] AUX B handling A Combinational Circuit for Modular Inversion

Uin[m..0] Vout[m..0] Vin[m..0] AUX V handling A Combinatorial Circuit for Modular Inversion

A combinatorial Circuit forPoint Doubling and Point Addition Modular Inversion Other Operations Qx P’x P’x Qy P’y A Combinatorial Circuit for Point Doubling and Point Addition

Unification of the Equations S = F + ((G + P’Y) * (H + P’X)-1) mod p QX = (S2 + S + PX + P’X + a) mod p QY = (S(PX + QX) + PY + QX) mod p A Combinatorial Circuit for Point Doubling and Point Addition

k k Temp. Reg. Algorithm of ECC-GF(2m) RNG CAPTION data flow control signal P’y The Cryptosystem CPU PC-board Modular Inversion Other Operations EP2S180F1020C4 EP2S90F1508C3 Registers Key-Exchange Example P P P’x Qx S P’x Qy PC Bus A Cryptosystem Implemented by Programmable Circuits

General Results Results

Modular Inversion Time for Different Implementations Results

Scalar Multiplication Time for Different Implementations Results

Results of Our Key-Exchange Example Results

Conclusions • The Objectives was successfully achieved; • High performance to the detriment of a small area; • High-speed and high-density combinatorial circuits; • Our inverter and cryptosystem were made valid; • Our cryptosystem allows high frequency of key exchange. Conclusions

The End

An Inverter Architecture for ECC-GF(2 m ) Based on the Stein’s Algorithm