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Cryptographic coprocessor

Cryptographic coprocessor. Tomáš Davidovič 16. 6. 2008. 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

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Cryptographic coprocessor

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  1. Cryptographiccoprocessor Tomáš Davidovič 16. 6. 2008

  2. Introduction • Asymmetric cryptography • RSA – Integer factorization • ECC – Elliptic Curve Cryptography • Pointson 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 Cryptographic coprocessor

  3. EC – point addition – real numbers Cryptographic coprocessor

  4. 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 Cryptographic coprocessor

  5. 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) Cryptographic coprocessor

  6. Block diagram Previous work Cryptographic coprocessor

  7. Polynomial squaring Cryptographic coprocessor

  8. Multiplication – bit-serial Cryptographic coprocessor

  9. 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 Cryptographic coprocessor

  10. Micro-controller Cryptographic coprocessor

  11. 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 Cryptographic coprocessor

  12. Results – area Cryptographic coprocessor

  13. Results – speed (cycles) Cryptographic coprocessor

  14. 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 Cryptographic coprocessor

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