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𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Protection of cryptographic keys recodings against physical attacks

Author: Simon RASTIKIAN Supervisor: Arnaud TISSERAND

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

PLAN

• Introduction
• Elliptic Curve Cryptography
• Side Channel Attacks
• Application

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Introduction

• Public-key cryptography conceived by W. Diffie & M. Hellman.
• Then comes RSA.
• Then ECC by N. Koblitz & V. Miller basing their schemes on ECDLP.
• What about security?

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Elliptic Curve Cryptography

• An elliptic curve over a field K is defined by the Weierstrass equation [1]

E: 𝑧2 + 𝑏1𝑦𝑧 + 𝑏3𝑧 = 𝑦3 + 𝑏2𝑦2 + 𝑏4𝑦 + 𝑏6 Where 𝑏1, 𝑏2, 𝑏3, 𝑏4, 𝑏6 ∈ 𝐿 𝑏𝑜𝑒 Δ ≠ 0 Δ = −𝑒2

2𝑒8 − 8𝑒4 3 − 27𝑒6 2 + 9𝑒2𝑒4𝑒6

𝑒2 = 𝑏1

2 + 4𝑏_2

𝑒4 = 2𝑏4 + 𝑏1𝑏3 𝑒6 = 𝑏3

2 + 4𝑏6

𝑒8 = 𝑏1

2𝑏6 + 4𝑏2𝑏6 − 𝑏1𝑏3𝑏4 + 𝑏2𝑏3 2 − 𝑏4 2

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Elliptic Curve Cryptography

• 𝐹1, 𝐹2 are isomorphic over K if ∃𝑣, 𝑠, 𝑡, 𝑢 ∈ 𝐿 𝑥𝑗𝑢ℎ 𝑣 ≠ 0 such that

[1]: Φ ∶ 𝐿2 → 𝐿2 Transforms equation 𝐹1 into equation 𝐹2. 𝑦, 𝑧 → (𝑣2𝑦 + 𝑠, 𝑣3𝑧 + 𝑣2𝑡𝑦 + 𝑢)

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Elliptic Curve Cryptography

Over prime fields Fp p > 3 : y2 = x3 + ax + b where a, b ∈ K Δ = −16 (4a3 + 27b2) Over binary fields F2𝑛: If 𝑏1 ≠ 0 then 𝑧2 + 𝑦𝑧 = 𝑦3 + 𝑏𝑦2 + 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐 ∈ 𝐿 Δ = 𝑐 If 𝑏1 = 0 then 𝑧2 + 𝑑𝑧 = 𝑦3 + 𝑏𝑦 + 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐, 𝑑 ∈ 𝐿 Δ = 𝑑4 Over optimal extension fields F3𝑛: If 𝑏1

2 ≠ −𝑏2 then 𝑧2 = 𝑦3 + 𝑏𝑦2 + 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐 ∈ 𝐿

Δ = −𝑏3𝑐 If 𝑏1

2 = −𝑏2 then 𝑧2 = 𝑦3 + 𝑏𝑦

+ 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐 ∈ 𝐿 Δ = −𝑏3

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Elliptic Curve Cryptography

• Additive law +: 𝐹 𝐿 → 𝐹(𝐿) defined by the chord-and-tangent rule

Point addition and point doubling on the curve 𝑧2 = 𝑦3 − 𝑦 + 1 defined over R [5].

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Mathematically ∀𝑄, 𝑅 ∈ 𝐹(𝐿) :

1. Identity : 𝑄 + ∞ = ∞ + 𝑄 = 𝑄 2. Negative : −𝑄 = − 𝑦, 𝑧 = 𝑦, −𝑧 and 𝑦, 𝑧 + 𝑦, −𝑧 = ∞ 3. Point addition : 𝑄 = 𝑦1, 𝑧1 , 𝑅 = 𝑦2, 𝑧2 𝑏𝑜𝑒 𝑄 ≠ ±𝑅 𝑢ℎ𝑓𝑜 4. Point doubling : if 𝑄 ≠ −𝑄 then 2 𝑄 = 𝑦3, 𝑧3 where

Elliptic Curve Cryptography

𝑄 + 𝑅 = 𝑦3, 𝑧3 𝑥ℎ𝑓𝑠𝑓 𝑦3 =

𝑧2−𝑧1 𝑦2−𝑦1 2

and 𝑧3 =

𝑧2−𝑧1 𝑦2−𝑦1

𝑦1 − 𝑦3 − 𝑧1 𝑦3 =

3𝑦1+𝑏 2𝑧1 2

− 2𝑦1 and 𝑧3 =

3𝑦1

2+𝑏

2𝑧1

𝑦1 − 𝑦3 − 𝑧1

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Let E be an elliptic curve defined over Fp . Suppose P ∈ 𝐹(Fp) has a prime

• rder n.

<P>={∞,P, 2P, …, (n-1) P} is a cyclic group. ECDLP: Key pair generation: Given E, p, P, n (public parameter). Choose random integer k in [1,n-1] (secret key). Compute Q=kP. ECDLP problem : Given E, p, P, n (public) and Q=kP. Find k (secret key).

Elliptic Curve Cryptography

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• No sub-exponential complexity algorithm for solving ECDLP.
• Pollard’s rho attack and Shanks attack solve it in Ο(√𝑜).

Elliptic Curve Cryptography

Symmetric key size (bits) RSA and DH key size (bits) ECC key size (bits) 80 (SKIPJACK ) 1024 160 112 (Triple-DES) 2048 224 128 (AES-Small) 3072 256 192 (AES-Medium) 7680 384 256 (AES-Large) 15360 521 NIST comparision of ECC, RSA and DH key for different security requierements.

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Projective coordinates: c,d postitive integers .

An equivalence relation on the set 𝐿3\{(0,0,0)} noted as 𝑌1, 𝑍

1, 𝑎1 ~(𝑌2, 𝑍 2, 𝑎2) exists if 𝑌1 = 𝜇𝑑𝑌2, 𝑍 1 = 𝜇𝑒𝑍 2, 𝑎1 = 𝜇𝑎2 𝑔𝑝𝑠 𝜇 ∈ 𝐿∗

The projective point is the representative class 𝑌: 𝑍: 𝑎 = {(𝜇𝑑𝑌, 𝜇𝑒𝑍, 𝜇𝑎)|𝜇 ∈ 𝐿∗} 1-1 correspondance between the projective points such that 𝑎 ≠ 0 and the affine points.

Elliptic Curve Cryptography

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Several projective coordinates :
• 1. Standard projective coordinates (c=1 and d=1): (X,Y,Z) with 𝑎 ≠ 0

corresponds to the affine point (

𝑌 𝑎 , 𝑍 𝑎) and (0:1:0) to ∞

• 2. Jacobian projective coordinates (c=2 and d=3): (X,Y,Z) with 𝑎 ≠ 0

corresponds to the affine point(

𝑌 𝑎2 , 𝑍 𝑎3) and (1:1:0) to ∞.

• 3. Chudnovsky coordiates: The Jacobian point is represented with redundancy

(X:Y:Z:Z²:Z³)

Elliptic Curve Cryptography

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Side Channel Attacks

• Making assumption about the knowledge that an attacker has about the

security.

• It is best to make stronger assumption than Kerckhoff’s principle.
• Electronic circuits are enherently leaky.

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Power analysis attack is the observation and the study of the power

consuption of the cryptegraphic device.

• Two types of power analysis attacks are well-known:
• 1. Simple power attack (SPA): Visual examination of graphs of the current

used by a device overtime. Small number of power traces is needed.

• 2. Differential power attack (DPA): Does not require detailed knowledge

about the device. It is a statistical analysis of the power consumption measurements from a cyptosystem. Large number of power traces is needed.

Side Channel Attacks

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• How to compute Q=kP?
• Classical algorithm : Double-and-Add

Side Channel Attacks

Input : 𝑙 = 𝑙𝑜−1𝑙𝑜−2 … 𝑙0 , 𝑄 ∈ 𝐹(F𝑞) Output: Q=kP 𝑅 ← ∞ For i form n-1 to 0 do Q ← 2 𝑅 (DBL) If ki = 1 then 𝑅 ← 𝑅 + 𝑄 (ADD)

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Side Channel Attacks

Power consumption measure of Double-and-Add algorithm from left to right coded on FPGA [5].

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

In Jacobian coordinates ADD = 12 M + 4 S DBL = 4M + 4S

Side Channel Attacks

Power consumption measure of Double-and-Add algorithm from left to right coded on FPGA [5].

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

In Jacobian coordinates ADD = 12 M + 4 S DBL = 4M + 4S

Side Channel Attacks

Power consumption measure of Double-and-Add algorithm from left to right coded on FPGA [5].

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• NAF algorithms coded in C language.
• w-NAF algorithm for point multiplication ressembles to Double-and-Add but

with different secret key representation.

• Subtracting a point is easy beacause -(X,Y,Z) = (X,-Y,Z).
• A width-w NAF of k is the expression 𝑙 = 𝑗=0

𝑚−1 𝑙𝑗2𝑗 𝑥ℎ𝑓𝑠𝑓 𝑙𝑗 < 2𝑥−1 and

𝑙𝑗 are either odd or zero except 𝑙𝑚−1 ≠ 0. At most one of any consecutive digits is nonzero.

• Unique representation given k and w noted 𝑂𝐵𝐺

𝑥(𝑙).

• 𝑀𝑓𝑜𝑕𝑢ℎ 𝑂𝐵𝐺

𝑥 𝑙

= Length k + 1

Side Channel Attacks

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Width-w NAF algorithm:

Side Channel Attacks

Input: k positive integer, w Output: 𝑂𝐵𝐺

𝑥 𝑙

𝑗 ← 0 While 𝑙 ≥ 0 do : if k is odd then 𝑙𝑗 ← 𝑙 𝑛𝑝𝑒𝑡 2𝑥, 𝑙 ← 𝑙 − 𝑙𝑗 else 𝑙𝑗 ← 0 𝑙 ←

𝑙 2 , 𝑗 ← i + 1

Return 𝑂𝐵𝐺

𝑥 𝑙 = (𝑙𝑗−1… 𝑙0)

mods is a function that keeps 𝑙𝑗 ∈ −2𝑥−1 , 2𝑥−1 − 1

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Window NAF method for point multiplication algorithm:

Side Channel Attacks

Input: k positive integer, w, 𝑄 ∈ 𝐹 F𝑟 Output: 𝑙𝑄 Calculate 𝑂𝐵𝐺𝑥 𝑙 Compute and store all 𝑄

𝑗 = 𝑗𝑄 ∀𝑗 𝑝𝑒𝑒 𝑏𝑜𝑒 𝑗 < 2𝑥−1

𝑅 ← ∞ For i from l-1 downto 0 do : 𝑅 ← 2𝑅 if 𝑙𝑗 ≠ 0 then if 𝑙𝑗 > 0 then 𝑅 ← 𝑅 + 𝑄𝑙𝑗 else 𝑅 ← 𝑅 − 𝑄−𝑙𝑗

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Window NAF method for point multiplication algorithm:

Side Channel Attacks

Input: k positive integer, w, 𝑄 ∈ 𝐹 F𝑟 Output: 𝑙𝑄 Calculate 𝑂𝐵𝐺𝑥 𝑙 Compute and store all 𝑄

𝑗 = 𝑗𝑄 ∀𝑗 𝑝𝑒𝑒 𝑏𝑜𝑒 𝑗 < 2𝑥−1

𝑅 ← ∞ For i from l-1 downto 0 do : 𝑅 ← 2𝑅 if 𝑙𝑗 ≠ 0 then if 𝑙𝑗 > 0 then 𝑅 ← 𝑅 + 𝑄𝑙𝑗 else 𝑅 ← 𝑅 − 𝑄−𝑙𝑗

• Faster computation of kP.
• Is it safe against SPA?

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Cryptographic device STM32L053R8 Nucleo [3]
• Make the led twinkle.

Side Channel Attacks

• Ultra-Low power consumption platform.
• Processor ARM 32-bit Cortex-M0+.
• 64 Kbytes Flash.
• 8Kbytes RAM.
• 32MHz CPU.
• 1 user led and 2 buttons.
• Mbed Enabled.
• Etc …

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Measurement Setup: Digital Sampling Oscilloscope. [2]
• Records the power consumption.
• 3 characteristics :
• Input Bandwidth: difference between the highest and the lowest frequency component.

1GHz

• Sampling rate: Number of points recorded per second.
• Resolution: conversion between time-discrete signal into time and value-discrete. 8bits

means each value is mapped to one of 256 possible value.

Side Channel Attacks

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Application

• 𝜈𝑂𝑏𝐷𝑚 is an ongoing library made for ECC.
• The core coded by P. Schwabe & M. Hutter.
• The standalone Curve25519 implementation for ARM Cortex-M0 coded by B.

Haase & A. H. Sánchez.

• Few problems at compilation that were surpassed.

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Width-w NAF coded for keys of 255 bits and w between 2 and 8.
• Basic operation used.
• Some optimisation with the size of 𝑂𝐵𝐺

2 𝑙 .

Application

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Started coding window NAF method for point multiplication.
• Noticed that Curve25519 coded in 𝜈𝑂𝑏𝐷𝑚 has no Y projective coordinate.
• No subtraction because -(X,Y,Z)=(X,-Y,Z).
• Curve25519 is a Montgomery curve. [4]
• Curve25519 is defined over the quadratic extension of the prime field

F 2255−19

Application

𝑁

𝐵,𝐶: 𝐶𝑧2 = 𝑦3 + 𝐵𝑦2 + 𝑦 𝑥ℎ𝑓𝑠𝑓 𝐵, 𝐶 ∈ F𝑟 𝑏𝑜𝑒 𝐶 𝐵2 − 4 ≠ 0

𝐷𝑣𝑠𝑤𝑓25519: 𝑧2 = 𝑦3 + 486662𝑦2 + 𝑦

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Montgomery coordinates (X:Z).
• Try coding the radix-8 algorithm.
• Non-possibility because of the use of differential addition.
• Differential addition 𝑄 + 𝑅 requires the presence of 𝑄, 𝑅 𝑏𝑜𝑒 𝑄 − 𝑅.
• On a Montgomery curve:
• 𝑦𝑄+𝑅𝑦 𝑄−𝑅

𝑦𝑄 − 𝑦𝑅

2 = 𝑦𝑄𝑦𝑅 − 1 𝑗𝑔 𝑄 ≠ 𝑅

• 4𝑦 2 𝑄𝑦𝑄 𝑦𝑄

2 + 𝐵𝑦𝑄 + 1 = 𝑦𝑄 2 − 1 2 𝑗𝑔 𝑄 = 𝑅

Application

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• How to use the differential addition?
• Secure Montgomery ladder algorithm:

Application

Input : 𝑙 = 𝑙𝑚−1𝑙𝑚−2 … 𝑙0 𝑏𝑜𝑒 𝑙𝑚−1 = 1, 𝑄 ∈ 𝐹(F𝑞) Output: kP 𝑅 = 𝑅0, 𝑅1 ← 𝑄, 2 𝑄 For i from l-2 to 0 do If ki = 1 then Q ← (𝑅0+𝑅1, 2 𝑅1) else Q ← ( 2 𝑅0, 𝑅0+𝑅1) Return 𝑅0

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

• Combining algorithms : working idea but w-1 additional point storing and w-1

additional point addition in each loop.

• Example: width-2 NAF & Montgomery ladder algorithm:

Application

Input : 𝑙 = 𝑙𝑚−1𝑙𝑚−2 … 𝑙0 𝑏𝑜𝑒 𝑙𝑚−1 = 1, 𝑄 ∈ 𝐹(F𝑞) Output: kP Compute 𝑂𝐵𝐺2 𝑙 𝑅 = 𝑅0, 𝑅1, 𝑅2 ← 𝑄, 2 𝑄, ∞ For i form length(𝑂𝐵𝐺2 𝑙 )-2 to 0 do If ki = 1 then Q ← (𝑅0+𝑅1, 2 𝑅1, 𝑅2 + 𝑅1) elsif ki = 0 then Q ← ( 2 𝑅0, 𝑅0+𝑅1, 𝑅2 + 𝑅0) else Q ← (𝑅0 + 𝑅2, 𝑅1+𝑅2, [2]𝑅2) Return 𝑅0

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

Thank you for your attentio ion

𝑧2 = 𝑦3 − 𝑦

𝑄 𝑆

References

• [1] D. Hankerson, A. Menezes and S. Vanstone. Guide to Elliptic Curve Cryptography. Springer,

2004

• [2] S. Mangard, E. Oswald and T. Popp. Power Analysis Attacks: Revealing the Secrets of

Smart Cards. Springer, 2007.

• [3] STM32L053R8. URL: https://www.st.com/en/microcontrollers/stm32l053r8.html.
• [4] C. Costello and B. Smith. Montgomery curves and their arithmetic: The case of large

characteristics fields. 2017. URL: https://arxiv.org/pdf/1703.01863.pdf.

• [5] T. Chabrier. Arithmetic recodings for ECC cryptoprocessors with protection against side-

channel attacks. 2013. URL: https://www.theses.fr/174580924