Protection of cryptographic keys recodings against physical attacks - PowerPoint PPT Presentation

𝑆 Protection of cryptographic keys recodings against physical attacks Supervisor: Author: Simon RASTIKIAN Arnaud TISSERAND 𝑄 𝑧 2 = 𝑦 3 − 𝑦

𝑆 PLAN • Introduction • Elliptic Curve Cryptography • Side Channel Attacks • Application 𝑄 𝑧 2 = 𝑦 3 − 𝑦

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. 𝑄 • What about security? 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography • An elliptic curve over a field K is defined by the Weierstrass equation [1] E: 𝑧 2 + 𝑏 1 𝑦𝑧 + 𝑏 3 𝑧 = 𝑦 3 + 𝑏 2 𝑦 2 + 𝑏 4 𝑦 + 𝑏 6 Where 𝑏 1 , 𝑏 2 , 𝑏 3 , 𝑏 4 , 𝑏 6 ∈ 𝐿 𝑏𝑜𝑒 Δ ≠ 0 3 − 27𝑒 6 2 + 9𝑒 2 𝑒 4 𝑒 6 2 𝑒 8 − 8𝑒 4 Δ = −𝑒 2 2 + 4𝑏_2 𝑒 2 = 𝑏 1 𝑄 𝑒 4 = 2𝑏 4 + 𝑏 1 𝑏 3 2 + 4𝑏 6 𝑒 6 = 𝑏 3 2 − 𝑏 4 2 𝑏 6 + 4𝑏 2 𝑏 6 − 𝑏 1 𝑏 3 𝑏 4 + 𝑏 2 𝑏 3 2 𝑒 8 = 𝑏 1 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography • 𝐹 1 , 𝐹 2 are isomorphic over K if ∃𝑣, 𝑠, 𝑡, 𝑢 ∈ 𝐿 𝑥𝑗𝑢ℎ 𝑣 ≠ 0 such that [1]: Φ ∶ 𝐿 2 → 𝐿 2 𝑦, 𝑧 → (𝑣 2 𝑦 + 𝑠, 𝑣 3 𝑧 + 𝑣 2 𝑡𝑦 + 𝑢) Transforms equation 𝐹 1 into equation 𝐹 2 . 𝑄 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography Over prime fields F p p > 3 : y 2 = x 3 + ax + b where a, b ∈ K Over binary fields F 2 𝑛 : If 𝑏 1 ≠ 0 then 𝑧 2 + 𝑦𝑧 = 𝑦 3 + 𝑏𝑦 2 + 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐 ∈ 𝐿 Δ = −16 (4a 3 + 27b 2 ) Δ = 𝑐 If 𝑏 1 = 0 then 𝑧 2 + 𝑑𝑧 = 𝑦 3 + 𝑏𝑦 + 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐, 𝑑 ∈ 𝐿 Δ = 𝑑 4 Over optimal extension fields F 3 𝑛 : 2 ≠ −𝑏 2 then 𝑧 2 = 𝑦 3 + 𝑏𝑦 2 + 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐 ∈ 𝐿 If 𝑏 1 Δ = −𝑏 3 𝑐 𝑄 2 = −𝑏 2 then 𝑧 2 = 𝑦 3 + 𝑏𝑦 If 𝑏 1 + 𝑐 𝑥ℎ𝑓𝑠𝑓 𝑏, 𝑐 ∈ 𝐿 Δ = −𝑏 3 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography • Additive law +: 𝐹 𝐿 → 𝐹(𝐿) defined by the chord-and-tangent rule 𝑄 Point addition and point doubling on the curve 𝑧 2 = 𝑦 3 − 𝑦 + 1 defined over R [5]. 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography • Mathematically ∀𝑄, 𝑅 ∈ 𝐹(𝐿) : Identity : 𝑄 + ∞ = ∞ + 𝑄 = 𝑄 1. Negative : −𝑄 = − 𝑦, 𝑧 = 𝑦, −𝑧 and 𝑦, 𝑧 + 𝑦, −𝑧 = ∞ 2. Point addition : 𝑄 = 𝑦 1 , 𝑧 1 , 𝑅 = 𝑦 2 , 𝑧 2 𝑏𝑜𝑒 𝑄 ≠ ±𝑅 𝑢ℎ𝑓𝑜 3. 2 𝑧 2 −𝑧 1 𝑧 2 −𝑧 1 𝑄 + 𝑅 = 𝑦 3 , 𝑧 3 𝑥ℎ𝑓𝑠𝑓 𝑦 3 = and 𝑧 3 = 𝑦 1 − 𝑦 3 − 𝑧 1 𝑦 2 −𝑦 1 𝑦 2 −𝑦 1 4. Point doubling : if 𝑄 ≠ −𝑄 then 2 𝑄 = 𝑦 3 , 𝑧 3 where 𝑄 2 2 +𝑏 3𝑦 1 +𝑏 3𝑦 1 𝑦 3 = − 2𝑦 1 and 𝑧 3 = 𝑦 1 − 𝑦 3 − 𝑧 1 2𝑧 1 2𝑧 1 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography Let E be an elliptic curve defined over F p . Suppose P ∈ 𝐹( F p ) has a prime order 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. 𝑄 ECDLP problem : Given E, p, P, n (public) and Q=kP. Find k (secret key). 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography • No sub-exponential complexity algorithm for solving ECDLP. • Pollard’s rho attack and Shanks attack solve it in Ο(√𝑜) . 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 𝑄 NIST comparision of ECC, RSA and DH key for different security requierements. 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography • Projective coordinates: c,d postitive integers . An equivalence relation on the set 𝐿 3 \{(0,0,0)} noted as 2 , 𝑎 2 ) exists if 𝑌 1 = 𝜇 𝑑 𝑌 2 , 𝑍 1 = 𝜇 𝑒 𝑍 2 , 𝑎 1 = 𝜇𝑎 2 𝑔𝑝𝑠 𝜇 ∈ 𝐿 ∗ 𝑌 1 , 𝑍 1 , 𝑎 1 ~(𝑌 2 , 𝑍 The projective point is the representative class 𝑌: 𝑍: 𝑎 = {(𝜇 𝑑 𝑌, 𝜇 𝑒 𝑍, 𝜇𝑎)|𝜇 ∈ 𝐿 ∗ } 1-1 correspondance between the projective points such that 𝑎 ≠ 0 and the affine points. 𝑄 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Elliptic Curve Cryptography • Several projective coordinates : 1. Standard projective coordinates (c=1 and d=1): (X,Y,Z) with 𝑎 ≠ 0 𝑌 𝑍 corresponds to the affine point ( 𝑎 , 𝑎 ) and (0:1:0) to ∞ 2. Jacobian projective coordinates (c=2 and d=3): (X,Y,Z) with 𝑎 ≠ 0 𝑌 𝑍 corresponds to the affine point ( 𝑎 2 , 𝑎 3 ) and (1:1:0) to ∞ . 3. Chudnovsky coordiates: The Jacobian point is represented with redundancy (X:Y:Z:Z²:Z³) 𝑄 𝑧 2 = 𝑦 3 − 𝑦

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. 𝑄 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks • 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 𝑄 needed. 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks • How to compute Q=kP? • Classical algorithm : Double-and-Add Input : 𝑙 = 𝑙 𝑜−1 𝑙 𝑜−2 … 𝑙 0 , 𝑄 ∈ 𝐹(F 𝑞 ) Output: Q=kP 𝑅 ← ∞ For i form n-1 to 0 do 𝑄 Q ← 2 𝑅 (DBL) If k i = 1 then 𝑅 ← 𝑅 + 𝑄 (ADD) 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks 𝑄 Power consumption measure of Double-and-Add algorithm from left to right coded on FPGA [5]. 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks 𝑄 Power consumption measure of Double-and-Add algorithm from left to right coded on FPGA [5]. In Jacobian coordinates ADD = 12 M + 4 S DBL = 4M + 4S 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks 𝑄 Power consumption measure of Double-and-Add algorithm from left to right coded on FPGA [5]. In Jacobian coordinates ADD = 12 M + 4 S DBL = 4M + 4S 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks • 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). 𝑚−1 𝑙 𝑗 2 𝑗 𝑥ℎ𝑓𝑠𝑓 𝑙 𝑗 < 2 𝑥−1 and • A width-w NAF of k is the expression 𝑙 = 𝑗=0 𝑙 𝑗 are either odd or zero except 𝑙 𝑚−1 ≠ 0 . At most one of any consecutive digits is nonzero. 𝑄 • Unique representation given k and w noted 𝑂𝐵𝐺 𝑥 (𝑙) . • 𝑀𝑓𝑜𝑕𝑢ℎ 𝑂𝐵𝐺 𝑥 𝑙 = Length k + 1 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks Width-w NAF algorithm: Input: k positive integer, w Output: 𝑂𝐵𝐺 𝑥 𝑙 𝑗 ← 0 While 𝑙 ≥ 0 do : if k is odd then 𝑙 𝑗 ← 𝑙 𝑛𝑝𝑒𝑡 2 𝑥 , 𝑙 ← 𝑙 − 𝑙 𝑗 else 𝑙 𝑗 ← 0 𝑙 𝑙 ← 2 , 𝑗 ← i + 1 𝑄 Return 𝑂𝐵𝐺 𝑥 𝑙 = (𝑙 𝑗−1 … 𝑙 0 ) mods is a function that keeps 𝑙 𝑗 ∈ −2 𝑥−1 , 2 𝑥−1 − 1 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks Window NAF method for point multiplication algorithm: Input: k positive integer, w, 𝑄 ∈ 𝐹 F 𝑟 Output: 𝑙𝑄 Calculate 𝑂𝐵𝐺 𝑥 𝑙 𝑗 = 𝑗𝑄 ∀𝑗 𝑝𝑒𝑒 𝑏𝑜𝑒 𝑗 < 2 𝑥−1 Compute and store all 𝑄 𝑅 ← ∞ For i from l-1 downto 0 do : 𝑅 ← 2𝑅 𝑄 if 𝑙 𝑗 ≠ 0 then if 𝑙 𝑗 > 0 then 𝑅 ← 𝑅 + 𝑄 𝑙 𝑗 else 𝑅 ← 𝑅 − 𝑄 −𝑙 𝑗 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks Window NAF method for point multiplication algorithm: Input: k positive integer, w, 𝑄 ∈ 𝐹 F 𝑟 Output: 𝑙𝑄 Calculate 𝑂𝐵𝐺 𝑥 𝑙 𝑗 = 𝑗𝑄 ∀𝑗 𝑝𝑒𝑒 𝑏𝑜𝑒 𝑗 < 2 𝑥−1 Compute and store all 𝑄 𝑅 ← ∞ For i from l-1 downto 0 do : • Faster computation of kP. 𝑅 ← 2𝑅 • Is it safe against SPA? 𝑄 if 𝑙 𝑗 ≠ 0 then if 𝑙 𝑗 > 0 then 𝑅 ← 𝑅 + 𝑄 𝑙 𝑗 else 𝑅 ← 𝑅 − 𝑄 −𝑙 𝑗 𝑧 2 = 𝑦 3 − 𝑦

𝑆 Side Channel Attacks • Cryptographic device STM32L053R8 Nucleo [3] • 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 … 𝑄 • Make the led twinkle. 𝑧 2 = 𝑦 3 − 𝑦