Cryptographic coprocessor

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Presentation transcript:

Cryptographic coprocessor Tomáš Davidovič 16. 6. 2008

Cryptographic coprocessor Introduction Asymmetric cryptography RSA – Integer factorization ECC – Elliptic Curve Cryptography Points on an Elliptic Curve Basic operation – scalar point multiplication Q = k*P – compute via add-and-double ECDLP – Elliptic Curve Discrete Logarithm Problem, determine k from Q and P 16. 6. 2008 Cryptographic coprocessor

EC – point addition – real numbers 16. 6. 2008 Cryptographic coprocessor

Cryptographic coprocessor EC – discrete Cannot use real numbers Coordinates from GF(2m) Two coordinate systems Affine coordinates (x, y) – mul & div Projective coordinates (x, y, z) – mul only Two bases in GF(2m) Polynomial – am-1xm-1+am-2xm-2+…+a1x+a0 Normal – am-1x2^(m-1)+am-2x2^(m-2)+…+a1x2+a0x 16. 6. 2008 Cryptographic coprocessor

EC – required operations Addition, subtraction Bitwise XOR in both bases Squaring Simple (but different) comb. logic in both Multiplication Bit-serial (m cycles) Digit-serial multiplier in both (m/D cycles) Division Polynomial b. via Extended Euclid’s Algorithm Normal b. via Little Fermat Theorem (costly) 16. 6. 2008 Cryptographic coprocessor

Cryptographic coprocessor Block diagram Previous work 16. 6. 2008 Cryptographic coprocessor

Cryptographic coprocessor Polynomial squaring 16. 6. 2008 Cryptographic coprocessor

Multiplication – bit-serial 16. 6. 2008 Cryptographic coprocessor

Multiplication – digit-serial Bit-serial – C = A*B Multiplies by 1 bit at a time Digit-serial – C = A*B; Digit – multiply by D bits at a time C = 0; i = 0; (D = 2) C = C + A*B[i] + (A<<1)*B[i+1] Shift A left by 2; i = i + 2; Repeat until i = m 16. 6. 2008 Cryptographic coprocessor

Cryptographic coprocessor Micro-controller 16. 6. 2008 Cryptographic coprocessor

Cryptographic coprocessor Verification Some functions base specific Wrappers Algorithms universal Test algorithms Use algorithms to verify design Quality – code coverage Statement coverage – each line used Branch coverage – each if taken both ways 16. 6. 2008 Cryptographic coprocessor

Cryptographic coprocessor Results – area 16. 6. 2008 Cryptographic coprocessor

Results – speed (cycles) 16. 6. 2008 Cryptographic coprocessor

Cryptographic coprocessor Wrap Up Coprocessor Both bases implemented Both coordinate systems evaluated Verification 100% Branch and statement coverage Everything passes Comparison Normal D=6 and poly D=1 equal in size Normal faster than poly when equal size 16. 6. 2008 Cryptographic coprocessor